Bures and Sjöqvist Metrics over Thermal State Manifolds for Spin Qubits and Superconducting Flux Qubits

Carlo Cafaro¹ and Paul M. Alsing² ¹SUNY Polytechnic Institute, 12203 Albany, New York, USA ²Air Force Research Laboratory, Information Directorate, 13441 Rome, New York, USA

Abstract

The interplay among differential geometry, statistical physics, and quantum information science has been increasingly gaining theoretical interest in recent years.

In this paper, we present an explicit analysis of the Bures and Sjöqvist metrics over the manifolds of thermal states for specific spin qubit and the superconducting flux qubit Hamiltonian models. While the two metrics equally reduce to the Fubini-Study metric in the asymptotic limiting case of the inverse temperature approaching infinity for both Hamiltonian models, we observe that the two metrics are generally different when departing from the zero-temperature limit. In particular, we discuss this discrepancy in the case of the superconducting flux Hamiltonian model. We conclude the two metrics differ in the presence of a nonclassical behavior specified by the noncommutativity of neighboring mixed quantum states. Such a noncommutativity, in turn, is quantified by the two metrics in different manners. Finally, we briefly discuss possible observable consequences of this discrepancy between the two metrics when using them to predict critical and/or complex behavior of physical systems of interest in quantum information science.

pacs:

Quantum Computation (03.67.Lx), Quantum Information (03.67.Ac), Quantum Mechanics (03.65.-w), Riemannian Geometry (02.40.Ky), Statistical Mechanics (05.20.-y).

I Introduction

Geometry plays a special role in the description and, to a certain extent, in the understanding of various physical phenomena pettini07 ; karol06 . The concepts of length, area, and volume are ubiquitous in physics and their meaning can prove quite helpful in explaining physical phenomena from a more intuitive perspective cafaroprd22 ; cafaropre22 . The notions of “longer” and “shorter” are extensively used in virtually all disciplines cafarophysicaa22 . Indeed, geometric formulations of classical and quantum evolutions along with geometric descriptions of classical and quantum mechanical aspects of thermal phenomena are becoming increasingly important in science. Concepts, such as thermodynamic length, area law, and statistical volumes are omnipresent in geometric thermodynamics, general relativity, and statistical physics, respectively. The concept of entropy finds application in essentially any realm of science, from classical thermodynamics to quantum information science. The notions of “hotter” and “cooler” are widely used in many fields. Entropy can be used to provide measures of distinguishability of classical probability distributions, as well as pure and mixed quantum states. It can also be used to propose measures of complexity for classical motion, quantum evolution, and entropic motion on curved statistical manifolds underlying the entropic dynamics of physical systems for which only partial knowledge of relevant information can be obtained cafaroPhD ; cafaroCSF ; felice18 . Furthermore, entropy can also be used to express the degree of entanglement in a quantum state specifying a composite quantum system. For instance, concepts such as Shannon entropy, von Neumann entropy, and Umegaki relative entropy are ubiquitous in classical information science, quantum information theory, and information geometric formulations of mixed quantum state evolutions amari , respectively. In this paper, inspired by the increasing theoretical interest in the interplay among differential geometry, statistical physics, and quantum information science zanardiprl07 ; zanardi07 ; pessoa21 ; silva21 ; silva21B ; mera22 , we present an explicit analysis of the Bures bures69 ; uhlman76 ; hubner92 and Sjöqvist erik20 metrics over the manifolds of thermal states for the spin qubit and the superconducting flux qubit Hamiltonian models. From a chronological standpoint, the first physical application of the Sjöqvist interferometric metric occurs in the original paper by Sjöqvist himself in Ref. erik20 . Here, the author considered his newly proposed interferometric metric to quantify changes in behavior of a magnetic system in a thermal state under modifications of temperature and magnetic field intensity. Keeping the temperature constant while changing the externally applied magnetic field, the Sjöqvist interferometric metric was shown to be physically linked to the magnetic susceptibility. This quantity, in turn, quantifies how much a material will become magnetized when immersed in a magnetic field. A second application of the Sjöqvist interferometric metric happens in Refs. silva21 ; silva21B . Here, this metric is used to characterize the finite-temperature phase transitions in the framework of band insulators. In particular, the authors considered the massive Dirac Hamiltonian model for a band insulator in two spatial dimensions. The corresponding Sjöqvist interferometric metric was calculated and expressed in terms of two physical parameters, the temperature, and the hopping parameter. Furthermore, the Sjöqvist interferometric metric was physically regarded as an interferometric susceptibility. Interestingly, it was observed in Refs. silva21 ; silva21B a dramatic difference between the Sjöqvist interferometric metric and the Bures metric when studying topological phase transitions in these types of systems. Specifically, while the topological phase transition is captured for all temperatures in the case of the Sjöqvist interferometric metric, the topological phase transition is captured only at zero temperature in the case of the Bures metric. Clearly, the authors leave as an unsolved question the experimental observation of the singular behavior of the Sjöqvist interferometric metric in actual laboratory experiments in Refs. silva21 ; silva21B . A third interesting work is the one presented in Ref. mera22 . Here the authors focus on the zero-temperature aspects of certain quantum systems in their (pure) ground quantum state. They consider two systems. The first system is described by the $XY$ anisotropic spin- $1/2$ chain with $N$ -sites on a circle in the presence of an external magnetic field. The second model is the Haldane model, a two-dimensional condensed-matter lattice model haldane88 . In the first case, the two parameters that determine both the Hamiltonian and the parameter manifold are the anisotropy degree and the magnetic field intensity. In the second case, instead, the two key parameters become the on-site energy and the phase in the model being considered. Expressing the so-called quantum metric in terms of these tunable parameters, they study the thermodynamical limit of this metric along critical submanifolds of the whole parameter manifold. They observe a singular (regular) behavior of the metric along normal (tangent) directions to the critical submanifolds. Therefore, they conclude that tangent directions to critical manifolds are special. Finally, the authors also point out that it would be interesting to understand how their findings generalize to the finite-temperature case where states become mixed. Interestingly, without reporting any explicit analysis, the authors state they expect the Bures and the Sjöqvist metrics to assume different functional forms.

In this paper, inspired by the previously mentioned relevance of comprehending the physical significance of choosing one metric over another one in such geometric characterizations of physical aspects of quantum systems, we report a complete and straightforward analysis of the link between the Sjöqvist interferometric metric and the Bures metric for two special classes of nondegenerate mixed quantum states. Specifically , focusing on manifolds of thermal states for the spin qubit and the superconducting flux qubit Hamiltonian models, we observe that while the two metrics both reduce to the Fubini-Study metric provost80 ; wootters81 ; braunstein94 in the zero-temperature asymptotic limiting case of the inverse temperature $\beta\overset{\text{def}}{=}\left(k_{B}T\right)^{-1}$ (with $k_{B}$ being the Boltzmann constant) approaching infinity for both Hamiltonian models, the two metrics are generally different. Furthermore, we observe this different behavior in the case of the superconducting flux Hamiltonian model. More generally, we note that the two metrics seem to differ when a nonclassical behavior is present since in this case, the metrics quantify noncommutativity of neighboring mixed quantum states in different manners. Finally, we briefly discuss the possible observable consequences of this discrepancy between the two metrics when using them to predict critical and/or complex behavior of physical systems of interest. We acknowledge that despite the fact that most of the preliminary background results presented in this paper are partially known in the literature, they appear in a scattered fashion throughout several papers written by researchers working in distinct fields of physics who may not be necessary aware of each other’s findings. For this reason, we present here an explicit and unified comparative formulation of the Bures and Sjöqvist metrics for simple quantum systems in mixed states. In particular, as mentioned earlier, we illustrate our results on the examples of thermal state manifolds for spin qubits and superconducting flux qubits. These applications are original and, to the best of our knowledge, do not appear anywhere in the literature.

The layout of the rest of this paper is as follows. In Section II, we present with explicit derivations the expressions of the Bures metric in Hübner’s hubner92 and Zanardi’s zanardi07 forms. In Section III, we present an explicit derivation of the Sjöqvist interferometric metric erik20 between two neighboring nondegenerate density matrices. In Section IV, we present two Hamiltonian models. The first Hamiltonian model describes a spin- $1/2$ particle in a uniform and time-independent external magnetic field oriented along the $z$ -axis. The second Hamiltonian model, instead, specifies a superconducting flux qubit. Bringing these two systems in thermal equilibrium with a reservoir at finite and non-zero temperature $T$ , we construct the two corresponding parametric families of thermal states. In Section V, we present an explicit calculation of both the Sjöqvist and the Bures metrics for each one of the two distinct families of parametric thermal states. From our comparative analysis, we find that the two metric coincide for the first Hamiltonian model (electron in a constant magnetic field along the $z$ -direction), while they differ for the second Hamiltonian model (superconducting flux qubit). In Section VI, we discuss the effects that arise from the comparative analysis carried out in Section V concerning the Bures and Sjöqvist metrics for spin qubits and superconducting flux qubits Hamiltonian models introduced in Section IV. Finally, we conclude with our final remarks along with a summary of our main findings in Section VII.

II The Bures metric

In this section, we present two explicit calculations. In the first calculation, we carry out a detailed derivation of the Bures metric by following the original work presented by Hübner in Ref. hubner92 . In the second calculation, we recast the expression of the Bures metric obtained by Hübner in a way that is more suitable in the framework of geometric analyses on thermal states manifolds. Here, we follow the original work presented by Zanardi and collaborators in Ref. zanardi07 .

II.1 The explicit derivation of Hübner’s general expression

We begin by carrying out an explicit derivation of the Bures metric inspired by Hübner hubner92 . Recall that the squared Bures distance between two density matrices infinitesimally far apart is given by,

\left[d_{\mathrm{Bures}}\left(\rho\text{, }\rho+d\rho\right)\right]^{2}=2-2\mathrm{tr}\left[\rho^{1/2}\left(\rho+d\rho\right)\rho^{1/2}\right]^{1/2}\text{.}

(1)

To find a useful expression for $\left[d_{\mathrm{Bures}}\left(\rho\text{, }\rho+d\rho\right)\right]^{2}$ , we follow the line of reasoning used by Hübner in Ref. hubner92 . Consider an Hermitian matrix $A\left(t\right)$ with $t\in\mathbb{R}$ defined as

A\left(t\right)\overset{\text{def}}{=}\left[\rho^{1/2}\left(\rho+td\rho\right)\rho^{1/2}\right]^{1/2}\text{,}

(2)

with $A\left(t\right)A\left(t\right)=\rho^{1/2}\left(\rho+td\rho\right)\rho^{1/2}$ . Note that $A\left(0\right)=\rho$ and, for later use, we assume $\rho$ to be invertible. At this point we observe that knowledge of the metric tensor $g_{ij}\left(\rho\right)$ at $\rho$ requires knowing $\left[d_{\mathrm{Bures}}\left(\rho\text{, }\rho+td\rho\right)\right]^{2}$ up to the second order in $t$ . Hübner’s ansatz (i.e., educated guess) is

\left[d_{\mathrm{Bures}}\left(\rho\text{, }\rho+td\rho\right)\right]^{2}=t^{2}g_{ij}\left(\rho\right)d\rho^{i}d\rho^{j}\text{,}

(3)

with $\left\{\rho^{i}\right\}$ denoting a given set of coordinates on the manifold of density matrices. From Eq. (3), we note that

g_{ij}\left(\rho\right)d\rho^{i}d\rho^{j}=\frac{1}{2}\left(\frac{d^{2}}{dt^{2}}\left[d_{\mathrm{Bures}}\left(\rho\text{, }\rho+td\rho\right)\right]^{2}\right)_{t=0}\text{.}

(4)

Observe that using Eq. (2), the RHS of Eq. (4) becomes

$\displaystyle\frac{1}{2}\left(\frac{d^{2}}{dt^{2}}\left[d_{\mathrm{Bures}}\left(\rho\text{, }\rho+td\rho\right)\right]^{2}\right)_{t=0}$	$\displaystyle=\frac{1}{2}\frac{d^{2}}{dt^{2}}\left\{2-2\mathrm{tr}\left[\rho^{1/2}\left(\rho+td\rho\right)\rho^{1/2}\right]^{1/2}\right\}_{t=0}$
	$\displaystyle=\frac{1}{2}\frac{d^{2}}{dt^{2}}\left\{2-2\mathrm{tr}\left[A\left(t\right)\right]\right\}_{t=0}$
	$\displaystyle=-\mathrm{tr}\left[\ddot{A}\left(t\right)\right]_{t=0}\text{,}$	(5)

that is,

\frac{1}{2}\left(\frac{d^{2}}{dt^{2}}\left[d_{\mathrm{Bures}}\left(\rho\text{, }\rho+td\rho\right)\right]^{2}\right)_{t=0}=-\mathrm{tr}\left[\ddot{A}\left(t\right)\right]_{t=0}\text{.}

(6)

From Eqs. (4) and (6), we get

g_{ij}\left(\rho\right)d\rho^{i}d\rho^{j}=-\mathrm{tr}\left[\ddot{A}\left(t\right)\right]_{t=0}\text{.}

(7)

Differentiating two times the relation $A\left(t\right)A\left(t\right)=\rho^{1/2}\left(\rho+td\rho\right)\rho^{1/2}$ , setting $t=0$ and, finally, assuming $\rho$ diagonalized in the form

\rho=\sum_{i}\lambda_{i}\left|i\right\rangle\left\langle i\right|\text{,}

(8)

we have

\left\{\frac{d^{2}}{dt^{2}}\left[A\left(t\right)A\left(t\right)\right]\right\}_{t=0}=\left\{\frac{d^{2}}{dt^{2}}\left[\rho^{1/2}\left(\rho+td\rho\right)\rho^{1/2}\right]\right\}_{t=0}\text{.}

(9)

More explicitly, we notice that

\frac{d}{dt}\left[A\left(t\right)A\left(t\right)\right]=\dot{A}\left(t\right)A\left(t\right)+A\left(t\right)\dot{A}\left(t\right)

(10)

and,

\frac{d}{dt}\left[\rho^{1/2}\left(\rho+td\rho\right)\rho^{1/2}\right]=\rho^{1/2}d\rho\rho^{1/2}\text{.}

(11)

Setting $t=0$ , from Eqs. (10) and (11) we obtain

\dot{A}\left(0\right)A\left(0\right)+A\left(0\right)\dot{A}\left(0\right)=\rho^{1/2}d\rho\rho^{1/2}\text{.}

(12)

After the second differentiation of $A\left(t\right)A\left(t\right)$ and $\rho^{1/2}\left(\rho+td\rho\right)\rho^{1/2}$ , we get

\ddot{A}\left(0\right)A\left(0\right)+2\text{ }\dot{A}\left(0\right)\dot{A}\left(0\right)+A\left(0\right)\ddot{A}\left(0\right)=0\text{.}

(13)

Multiplying both sides of Eq. (13) by $A^{-1}\left(0\right)$ from the right and using the cyclicity of the trace operation, we get

\mathrm{tr}\left[\ddot{A}\left(0\right)\right]=-\mathrm{tr}\left[A^{-1}\left(0\right)\dot{A}\left(0\right)^{2}\right]\text{.}

(14)

From Eqs. (7) and (14), the Bures metric $ds_{\mathrm{Bures}}^{2}$ becomes

$\displaystyle ds_{\mathrm{Bures}}^{2}\left(\rho\text{, }\rho+d\rho\right)$	$\displaystyle=g_{ij}\left(\rho\right)d\rho^{i}d\rho^{j}$
	$\displaystyle=-\mathrm{tr}\left[\ddot{A}\left(t\right)\right]_{t=0}$
	$\displaystyle=\mathrm{tr}\left[A^{-1}\left(0\right)\dot{A}\left(0\right)^{2}\right]$
	$\displaystyle=\sum_{i}\left\langle i\left\|A^{-1}\left(0\right)\dot{A}\left(0\right)^{2}\right\|i\right\rangle\text{,}$	(15)

that is,

ds_{\mathrm{Bures}}^{2}\left(\rho\text{, }\rho+d\rho\right)=\frac{1}{2}\sum_{i,k,l}\left[\left\langle i\left|A^{-1}\left(0\right)\right|k\right\rangle\left\langle k\left|\dot{A}\left(0\right)\right|l\right\rangle\left\langle l\left|\dot{A}\left(0\right)\right|i\right\rangle+\left\langle i\left|A^{-1}\left(0\right)\right|l\right\rangle\left\langle l\left|\dot{A}\left(0\right)\right|k\right\rangle\left\langle k\left|\dot{A}\left(0\right)\right|i\right\rangle\right]\text{.}

(16)

Observe that $A\left(0\right)\left|k\right\rangle=\rho\left|k\right\rangle=\lambda_{k}\left|k\right\rangle$ and, therefore, $A^{-1}\left(0\right)\left|k\right\rangle=\rho^{-1}\left|k\right\rangle=\lambda_{k}^{-1}\left|k\right\rangle$ with $\lambda_{k}\neq 0$ for any $k$ . We need to find an expression for $\left\langle i\left|\dot{A}\left(0\right)\right|j\right\rangle$ . From Eq. (12), we get

\left\langle i\left|\dot{A}\left(0\right)A\left(0\right)+A\left(0\right)\dot{A}\left(0\right)\right|j\right\rangle=\left\langle i\left|\rho^{1/2}d\rho\rho^{1/2}\right|j\right\rangle\text{.}

(17)

We note that

$\displaystyle\left\langle i\left\|\dot{A}\left(0\right)A\left(0\right)+A\left(0\right)\dot{A}\left(0\right)\right\|j\right\rangle$	$\displaystyle=\left\langle i\left\|\dot{A}\left(0\right)\rho+\rho\dot{A}\left(0\right)\right\|j\right\rangle$
	$\displaystyle={\displaystyle\sum\limits_{k}}\lambda_{k}\left\langle i\left\|\dot{A}\left(0\right)\right\|k\right\rangle\left\langle k\left\|j\right.\right\rangle+{\displaystyle\sum\limits_{k}}\lambda_{k}\left\langle i\left\|k\right.\right\rangle\left\langle k\left\|\dot{A}\left(0\right)\right\|j\right\rangle$
	$\displaystyle=\lambda_{j}\left\langle i\left\|\dot{A}\left(0\right)\right\|j\right\rangle+\lambda_{i}\left\langle i\left\|\dot{A}\left(0\right)\right\|j\right\rangle$
	$\displaystyle=\left(\lambda_{i}+\lambda_{j}\right)\left\langle i\left\|\dot{A}\left(0\right)\right\|j\right\rangle\text{,}$	(18)

that is,

\left\langle i\left|\dot{A}\left(0\right)A\left(0\right)+A\left(0\right)\dot{A}\left(0\right)\right|j\right\rangle=\left(\lambda_{i}+\lambda_{j}\right)\left\langle i\left|\dot{A}\left(0\right)\right|j\right\rangle\text{.}

(19)

Moreover, we observe that

$\displaystyle\left\langle i\left\|\rho^{1/2}d\rho\rho^{1/2}\right\|j\right\rangle$	$\displaystyle=\left\langle i\left\|\left(\sum_{k}\sqrt{\lambda_{k}}\left\|k\right\rangle\left\langle k\right\|\right)d\rho\left(\sum_{m}\sqrt{\lambda_{m}}\left\|m\right\rangle\left\langle m\right\|\right)\right\|\right\rangle$
	$\displaystyle=\sum_{k,m}\sqrt{\lambda_{k}}\sqrt{\lambda_{m}}\left\langle i\left\|k\right.\right\rangle\left\langle k\left\|d\rho\right\|m\right\rangle\left\langle m\left\|j\right.\right\rangle$
	$\displaystyle=\sum_{k,m}\sqrt{\lambda_{k}}\sqrt{\lambda_{m}}\delta_{ik}\delta_{mj}\left\langle k\left\|d\rho\right\|m\right\rangle$
	$\displaystyle=\sqrt{\lambda_{i}}\sqrt{\lambda_{j}}\left\langle i\left\|d\rho\right\|j\right\rangle\text{,}$	(20)

that is,

\left\langle i\left|\rho^{1/2}d\rho\rho^{1/2}\right|j\right\rangle=\sqrt{\lambda_{i}}\sqrt{\lambda_{j}}\left\langle i\left|d\rho\right|j\right\rangle\text{.}

(21)

Using Eqs. (17), (19), and (21), we have

\left\langle i\left|\dot{A}\left(0\right)\right|j\right\rangle=\frac{\sqrt{\lambda_{i}}\sqrt{\lambda_{j}}}{\lambda_{i}+\lambda_{j}}\left\langle i\left|d\rho\right|j\right\rangle\text{.}

(22)

Finally, using Eq. (22), $ds_{\mathrm{Bures}}^{2}$ in Eq. (16) becomes

ds_{\mathrm{Bures}}^{2}\left(\rho\text{, }\rho+d\rho\right)=\frac{1}{2}\sum_{k,l}\frac{\lambda_{l}+\lambda_{k}}{\left(\lambda_{l}+\lambda_{k}\right)^{2}}\left|\left\langle k\left|d\rho\right|l\right\rangle\right|^{2}\text{,}

(23)

that is, relabelling the dummy indices (i.e., $k\rightarrow i$ and $l\rightarrow j$ ),

ds_{\mathrm{Bures}}^{2}\left(\rho\text{, }\rho+d\rho\right)=\frac{1}{2}\sum_{i,j}\frac{\left|\left\langle i\left|d\rho\right|j\right\rangle\right|^{2}}{\lambda_{i}+\lambda_{j}}\text{.}

(24)

The derivation of Eq. (24) ends our revisitation of Hübner’s original analysis presented in Ref. hubner92 . Note that in obtaining Eq. (24), there is no need to introduce any Hamiltonian H that might be responsible for the changes from $\rho$ to $\rho+d\rho$ . For this reason, the expression of the Bures metric in Eq. (24) is said to be general.

II.2 The explicit derivation of Zanardi’s general expression

In Ref. zanardi07 , Zanardi and collaborators provided an alternative expression of the Bures metric in Eq. (24). Interestingly, in this alternative expression, the Bures metric in Eq. (24) can be decomposed in its classical and nonclassical parts. To begin, in view of future geometric investigations in statistical physics, let us use a different notation and rewrite the Bures metric in Eq. (24) as

ds_{\mathrm{Bures}}^{2}\left(\rho\text{, }\rho+d\rho\right)\overset{\text{def}}{=}\frac{1}{2}\sum_{n\text{, }m}\frac{\left|\left\langle m|d\rho|n\right\rangle\right|^{2}}{p_{m}+p_{n}}\text{,}

(25)

with $1\leq m$ , $n\leq N$ . Let us assume that the quantities $\rho$ and $d\rho$ in Eq. (25) are given by,

\rho\overset{\text{def}}{=}\sum_{n}p_{n}\left|n\right\rangle\left\langle n\right|\text{,}

(26)

and

d\rho\overset{\text{def}}{=}\sum_{n}\left[dp_{n}\left|n\right\rangle\left\langle n\right|+p_{n}\left|dn\right\rangle\left\langle n\right|+p_{n}\left|n\right\rangle\left\langle dn\right|\right]\text{,}

(27)

respectively, with $\left\langle n\left|m\right.\right\rangle=\delta_{n,m}$ . Let us use Eqs. (27) and (25) to find a more explicit expression for the Bures metric $ds_{\mathrm{Bures}}^{2}\left(\rho\text{, }\rho+d\rho\right)$ . Observe that the quantity $\left\langle i\left|d\rho\right|j\right\rangle$ can be recast as,

$\displaystyle\left\langle i\left\|d\rho\right\|j\right\rangle$	$\displaystyle=\left\langle i\left\|\left(\sum_{n}\left[dp_{n}\left\|n\right\rangle\left\langle n\right\|+p_{n}\left\|dn\right\rangle\left\langle n\right\|+p_{n}\left\|n\right\rangle\left\langle dn\right\|\right]\right)\right\|j\right\rangle$
	$\displaystyle=\sum_{n}dp_{n}\left\langle i\|n\right\rangle\left\langle n\|j\right\rangle+p_{n}\left\langle i\|dn\right\rangle\left\langle n\|j\right\rangle+p_{n}\left\langle i\|n\right\rangle\left\langle dn\|j\right\rangle$
	$\displaystyle=\sum_{n}dp_{n}\delta_{in}\delta_{nj}+p_{n}\left\langle i\|dn\right\rangle\delta_{nj}+p_{n}\delta_{in}\left\langle dn\|j\right\rangle$
	$\displaystyle=dp_{i}\delta_{ij}+p_{j}\left\langle i\|dj\right\rangle+p_{i}\left\langle di\|j\right\rangle\text{,}$	(28)

that is,

\left\langle i\left|d\rho\right|j\right\rangle=dp_{i}\delta_{ij}+p_{j}\left\langle i|dj\right\rangle+p_{i}\left\langle di|j\right\rangle\text{.}

(29)

Note that the orthonormality condition $\left\langle i|j\right\rangle=\delta_{ij}$ implies $\left\langle di|j\right\rangle+\left\langle i|dj\right\rangle=0$ , that is

\left\langle di|j\right\rangle=-\left\langle i|dj\right\rangle\text{.}

(30)

Using Eq. (30), $\left\langle i\left|d\rho\right|j\right\rangle$ in Eq. (29) becomes

\left\langle i\left|d\rho\right|j\right\rangle=dp_{i}\delta_{ij}+p_{j}\left\langle i|dj\right\rangle-p_{i}\left\langle i|dj\right\rangle=\delta_{ij}dp_{i}+\left(p_{j}-p_{i}\right)\left\langle i|dj\right\rangle\text{,}

that is,

\left\langle i\left|d\rho\right|j\right\rangle=\delta_{ij}dp_{i}+\left(p_{j}-p_{i}\right)\left\langle i|dj\right\rangle\text{.}

(31)

Making use of Eq. (31), we can now find an explicit expression of the quantity $\left|\left\langle m|d\rho|n\right\rangle\right|^{2}$ in Eq. (25). Indeed, from Eq. (31) we have

$\displaystyle\left\|\left\langle m\|d\rho\|n\right\rangle\right\|^{2}$	$\displaystyle=\left\langle m\|d\rho\|n\right\rangle\left\langle m\|d\rho\|n\right\rangle^{\ast}$
	$\displaystyle=\left\langle m\|d\rho\|n\right\rangle\left\langle n\|d\rho\|m\right\rangle$
	$\displaystyle=\left[\delta_{mn}dp_{m}+\left(p_{n}-p_{m}\right)\left\langle m\|dn\right\rangle\right]\left[\delta_{nm}dp_{n}+\left(p_{m}-p_{n}\right)\left\langle n\|dm\right\rangle\right]$
	$\displaystyle=\delta_{mn}dp_{m}\delta_{nm}dp_{n}+\delta_{mn}dp_{m}\left(p_{m}-p_{n}\right)\left\langle n\|dm\right\rangle+\left(p_{n}-p_{m}\right)\left\langle m\|dn\right\rangle\delta_{nm}dp_{n}+$
	$\displaystyle+\left(p_{n}-p_{m}\right)\left\langle m\|dn\right\rangle\left(p_{m}-p_{n}\right)\left\langle n\|dm\right\rangle$
	$\displaystyle=\delta_{nm}dp_{n}^{2}-\left(p_{n}-p_{m}\right)^{2}\left\langle m\|dn\right\rangle\left\langle n\|dm\right\rangle\text{,}$	(32)

that is,

\left|\left\langle m|d\rho|n\right\rangle\right|^{2}=\delta_{nm}dp_{n}^{2}-\left(p_{n}-p_{m}\right)^{2}\left\langle m|dn\right\rangle\left\langle n|dm\right\rangle\text{.}

(33)

Using Eq. (30), Eq. (33) reduces to

	$\displaystyle\left\|\left\langle m\|d\rho\|n\right\rangle\right\|^{2}$	$\displaystyle=\delta_{nm}dp_{n}^{2}+\left(p_{n}-p_{m}\right)^{2}\left\langle dm\|n\right\rangle\left\langle n\|dm\right\rangle$
		$\displaystyle=\delta_{nm}dp_{n}^{2}+\left(p_{n}-p_{m}\right)^{2}\left\|\left\langle n\|dm\right\rangle\right\|^{2}\text{,}$		(34)

that is,

\left|\left\langle m|d\rho|n\right\rangle\right|^{2}=\delta_{nm}dp_{n}^{2}+\left(p_{n}-p_{m}\right)^{2}\left|\left\langle n|dm\right\rangle\right|^{2}\text{.}

(35)

Finally, substituting Eq. (35) into Eq. (25), $ds_{\mathrm{Bures}}^{2}\left(\rho\text{, }\rho+d\rho\right)$ becomes

	$\displaystyle ds_{\mathrm{Bures}}^{2}\left(\rho\text{, }\rho+d\rho\right)$	$\displaystyle=\frac{1}{2}\sum_{n\text{, }m}\frac{\delta_{nm}dp_{n}^{2}+\left(p_{n}-p_{m}\right)^{2}\left\|\left\langle n\|dm\right\rangle\right\|^{2}}{p_{m}+p_{n}}$
		$\displaystyle=\frac{1}{4}\sum_{n}\frac{dp_{n}^{2}}{p_{n}}+\frac{1}{2}\sum_{n\neq m}\left\|\left\langle n\|dm\right\rangle\right\|^{2}\frac{\left(p_{n}-p_{m}\right)^{2}}{p_{m}+p_{n}}\text{,}$		(36)

that is,

ds_{\mathrm{Bures}}^{2}\left(\rho\text{, }\rho+d\rho\right)=\frac{1}{4}\sum_{n}\frac{dp_{n}^{2}}{p_{n}}+\frac{1}{2}\sum_{n\neq m}\left|\left\langle n|dm\right\rangle\right|^{2}\frac{\left(p_{n}-p_{m}\right)^{2}}{p_{m}+p_{n}}\text{.}

(37)

Eq. (37) is the explicit expression of $ds_{\mathrm{Bures}}^{2}\left(\rho\text{, }\rho+d\rho\right)$ we were searching for. As a side remark, note that if both $\left|n\right\rangle$ and $\left|m\right\rangle\in\ker\left(\rho\right)$ , we have that $\left\langle n|d\rho|m\right\rangle=0$ . Indeed, from $\rho\left|m\right\rangle=\left|0\right\rangle$ , we have $d\rho\left|m\right\rangle+\rho\left|dm\right\rangle=\left|0\right\rangle$ . Therefore, we have $\left\langle n|d\rho|m\right\rangle+\left\langle n|\rho|dm\right\rangle=\left\langle n|0\right\rangle$ , that is, $\left\langle n|d\rho|m\right\rangle=0$ since $\left\langle n|0\right\rangle=0$ and $\left\langle n|\rho|dm\right\rangle=0$ . The expression of the Bures metric in Eq. (37) can be regarded as given by two contribution, a classical and a nonclassical term. The first term in $ds_{\mathrm{Bures}}^{2}\left(\rho\text{, }\rho+d\rho\right)$ in Eq. (37) is the classical one and is represented by the classical Fisher-Rao information metric between the two probability distributions $\left\{p_{n}\right\}_{1\leq n\leq N}$ and $\left\{p_{n}+dp_{n}\right\}_{1\leq n\leq N}$ . The second term is the nonclassical one and emerges from the noncommutativity of the density matrices $\rho$ and $\rho+d\rho$ (i.e., $\left[\rho\text{, }\rho+d\rho\right]=\left[\rho\text{, }d\rho\right]\neq 0$ , in general). When $\left[\rho\text{, }\rho+d\rho\right]=0$ , the problem becomes classical and the Bures metric reduces to the classical Fisher-Rao metric.

II.3 The explicit derivation of Zanardi’s expression for thermal states

In what follows, we specialize on the functional form of $ds_{\mathrm{Bures}}^{2}\left(\rho\text{, }\rho+d\rho\right)$ in Eq. (37) for thermal states. Specifically, let us focus on mixed quantum states $\rho\left(\beta\text{, }\lambda\right)$ of the form,

\rho\left(\beta\text{, }\lambda\right)\overset{\text{def}}{=}\frac{e^{-\beta\mathrm{H}\left(\lambda\right)}}{\mathcal{Z}}=\frac{e^{-\beta\mathrm{H}\left(\lambda\right)}}{\mathrm{tr}\left(e^{-\beta\mathrm{H}\left(\lambda\right)}\right)}\text{,}

(38)

with $\mathcal{Z}\overset{\text{def}}{=}\mathrm{tr}\left(e^{-\beta\mathrm{H}\left(\lambda\right)}\right)$ denoting the partition function of the system. The Hamiltonian $\mathrm{H}$ in Eq. (38) depends on a set of parameters $\left\{\lambda\right\}$ and is such that $\mathrm{H}\left|n\right\rangle=E_{n}\left|n\right\rangle$ or, equivalently

\mathrm{H}=\sum_{n}E_{n}\left|n\right\rangle\left\langle n\right|\text{,}

(39)

with $1\leq n\leq N$ . Using the spectral decomposition of $\mathrm{H}$ in Eq. (39), $\rho\left(\beta\text{, }\lambda\right)$ in Eq. (38) can be recast as

\rho\left(\beta\text{, }\lambda\right)=\sum_{n}p_{n}\left|n\right\rangle\left\langle n\right|=\sum_{n}\frac{e^{-\beta E_{n}}}{\mathcal{Z}}\left|n\right\rangle\left\langle n\right|\text{,}

(40)

with $p_{n}\overset{\text{def}}{=}e^{-\beta E_{n}}/\mathcal{Z}$ . Note that the $\lambda$ -dependence of $\rho$ in Eq. (40) appears, in general, in both $E_{n}=E_{n}\left(\lambda\right)$ and $\left|n\right\rangle=\left|n\left(\lambda\right)\right\rangle$ . Before presenting the general case where both $\beta$ and the set of $\left\{\lambda\right\}$ can change, we focus on the sub-case where $\beta$ is kept constant while $\left\{\lambda\right\}$ is allowed to change.

II.3.1 Case: $\beta$ -constant and $\lambda$ -nonconstant

In what follows, assuming that $\beta$ is fixed, we wish to find the expression of $ds_{\mathrm{Bures}}^{2}\left(\rho\text{, }\rho+d\rho\right)$ in Eq. (37) when $\rho$ is given as in Eq. (40). Clearly, we note that the two key quantities that we need to find are $dp_{i}^{2}$ and $\left\langle i|dj\right\rangle$ . Let us start with the latter. From $\mathrm{H}\left|j\right\rangle=E_{j}\left|j\right\rangle$ , we have $d\mathrm{H}\left|j\right\rangle+\mathrm{H}\left|dj\right\rangle=dE_{j}\left|j\right\rangle+E_{j}\left|dj\right\rangle$ . Assuming $i\neq j$ , we have

\left\langle i\right|d\mathrm{H}\left|j\right\rangle+\left\langle i\right|\mathrm{H}\left|dj\right\rangle=\left\langle i\right|dE_{j}\left|j\right\rangle+\left\langle i\right|E_{j}\left|dj\right\rangle=dE_{j}\delta_{ij}+E_{j}\left\langle i|dj\right\rangle=E_{j}\left\langle i|dj\right\rangle\text{,}

(41)

that is,

\left\langle i\right|d\mathrm{H}\left|j\right\rangle+\left\langle i\right|\mathrm{H}\left|dj\right\rangle=E_{j}\left\langle i|dj\right\rangle\text{.}

(42)

Observe that,

\left\langle i\right|\mathrm{H}\left|dj\right\rangle=\left\langle dj\right|\mathrm{H}^{\dagger}\left|i\right\rangle^{\ast}=\left\langle dj\right|\mathrm{H}\left|i\right\rangle^{\ast}=E_{i}\left\langle dj|i\right\rangle^{\ast}=E_{i}\left\langle i|dj\right\rangle\text{.}

(43)

Substituting Eq. (43) into Eq. (42), we get

\left\langle i\right|d\mathrm{H}\left|j\right\rangle+E_{i}\left\langle i|dj\right\rangle=E_{j}\left\langle i|dj\right\rangle\text{,}

(44)

that is,

\left\langle i|dj\right\rangle=\frac{\left\langle i\right|d\mathrm{H}\left|j\right\rangle}{E_{j}-E_{i}}\text{.}

(45)

Eq. (45) is the first piece of relevant information we were looking for. Let us not focus on calculating $dp_{i}^{2}$ . From $p_{i}\overset{\text{def}}{=}e^{-\beta E_{i}}/\mathcal{Z}$ , we get

$\displaystyle dp_{i}$	$\displaystyle=d\left(\frac{e^{-\beta E_{i}}}{\mathcal{Z}}\right)$
	$\displaystyle=\frac{1}{\mathcal{Z}}d\left(e^{-\beta E_{i}}\right)+e^{-\beta E_{i}}d\left(\frac{1}{\mathcal{Z}}\right)$
	$\displaystyle=\frac{1}{\mathcal{Z}}\frac{d}{dE_{i}}\left(e^{-\beta E_{i}}\right)dE_{i}+e^{-\beta E_{i}}\left(-\frac{1}{\mathcal{Z}^{2}}d\mathcal{Z}\right)$
	$\displaystyle=-\beta\frac{e^{-\beta E_{i}}}{\mathcal{Z}}dE_{i}-\frac{e^{-\beta E_{i}}}{\mathcal{Z}}\frac{d\mathcal{Z}}{\mathcal{Z}}$
	$\displaystyle=-\beta\frac{e^{-\beta E_{i}}}{\mathcal{Z}}dE_{i}-\frac{e^{-\beta E_{i}}}{\mathcal{Z}}\sum_{j}\left(\frac{d\mathcal{Z}}{dE_{j}}\frac{dE_{j}}{\mathcal{Z}}\right)\text{,}$	(46)

that is,

dp_{i}=-\beta\frac{e^{-\beta E_{i}}}{\mathcal{Z}}dE_{i}-\frac{e^{-\beta E_{i}}}{\mathcal{Z}}\sum_{j}\left(\frac{d\mathcal{Z}}{dE_{j}}\frac{dE_{j}}{\mathcal{Z}}\right)\text{.}

(47)

At this point, note that

$\displaystyle\sum_{j}\frac{d\mathcal{Z}}{dE_{j}}\frac{dE_{j}}{\mathcal{Z}}$	$\displaystyle=\sum_{j}\frac{d}{dE_{j}}\left(\sum_{k}e^{-\beta E_{k}}\right)\frac{dE_{j}}{\mathcal{Z}}$
	$\displaystyle=-\beta\sum_{j}\frac{e^{-\beta E_{j}}}{\mathcal{Z}}dE_{j}$
	$\displaystyle=-\beta\sum_{j}p_{j}dE_{j}\text{.}$	(48)

Substituting Eq. (48) into Eq. (47), we get

dp_{i}=-\beta p_{i}dE_{i}+\beta p_{i}\sum_{j}p_{j}dE_{j}=-\beta p_{i}\left[dE_{i}-\sum_{j}p_{j}dE_{j}\right]\text{.}

(49)

Eq. (49) is the second piece of relevant information we were looking for. We can now calculate $ds_{\mathrm{Bures}}^{2}\left(\rho\text{, }\rho+d\rho\right)$ in Eq. (37) by means of Eqs. (45) and (49). We obtain,

$\displaystyle\frac{1}{4}\sum_{i}\frac{dp_{i}^{2}}{p_{i}}$	$\displaystyle=\frac{1}{4}\sum_{i}\beta^{2}\frac{p_{i}^{2}}{p_{i}}\left[dE_{i}-\sum_{j}p_{j}dE_{j}\right]^{2}$
	$\displaystyle=\frac{\beta^{2}}{4}\sum_{i}p_{i}\left[dE_{i}-\left\langle dE\right\rangle_{\beta}\right]^{2}$
	$\displaystyle=\frac{\beta^{2}}{4}\left(\left\langle dE^{2}\right\rangle_{\beta}-\left\langle dE\right\rangle_{\beta}^{2}\right)$
	$\displaystyle=\frac{\beta^{2}}{4}\left(\left\langle d\mathrm{H}_{d}^{2}\right\rangle_{\beta}-\left\langle d\mathrm{H}_{d}\right\rangle_{\beta}^{2}\right)\text{,}$	(50)

that is,

\frac{1}{4}\sum_{i}\frac{dp_{i}^{2}}{p_{i}}=\frac{\beta^{2}}{4}\left(\left\langle d\mathrm{H}_{d}^{2}\right\rangle_{\beta}-\left\langle d\mathrm{H}_{d}\right\rangle_{\beta}^{2}\right)\text{.}

(51)

The quantity $d\mathrm{H}_{d}$ in Eq. (51) is defined as

d\mathrm{H}_{d}\overset{\text{def}}{=}\sum_{j}dE_{j}\left|j\right\rangle\left\langle j\right|\text{,}

(52)

and is different from $d\mathrm{H}$ . For clarity, we also observe that

\left\langle d\mathrm{H}_{d}\right\rangle_{\beta}\overset{\text{def}}{=}\sum_{i}p_{i}dE_{i}\text{, and }\left\langle d\mathrm{H}_{d}^{2}\right\rangle_{\beta}\overset{\text{def}}{=}\sum_{i}p_{i}dE_{i}^{2}\text{.}

(53)

Finally, using Eqs. (45) and (51), we get

ds_{\mathrm{Bures}}^{2}\left(\rho\text{, }\rho+d\rho\right)=\frac{\beta^{2}}{4}\left(\left\langle d\mathrm{H}_{d}^{2}\right\rangle_{\beta}-\left\langle d\mathrm{H}_{d}\right\rangle_{\beta}^{2}\right)+\frac{1}{2}\sum_{n\neq m}\left|\frac{\left\langle n|d\mathrm{H}|m\right\rangle}{E_{n}-E_{m}}\right|^{2}\frac{\left(e^{-\beta E_{n}}-e^{-\beta E_{m}}\right)^{2}}{\mathcal{Z}\left(e^{-\beta E_{n}}+e^{-\beta E_{m}}\right)}\text{.}

(54)

The Bures metric $ds_{\mathrm{Bures}}^{2}\left(\rho\text{, }\rho+d\rho\right)$ in Eq. (54) is the Bures metric in Eq. (37) between two mixed thermal states $\rho\left(\beta\text{, }\lambda\right)$ and $\left(\rho+d\rho\right)\left(\beta\text{, }\lambda\right)$ when only changes in $\lambda$ are permitted.

II.3.2 Case: $\beta$ -nonconstant and $\lambda$ -nonconstant

In what follows, we consider the general case where both $\beta$ and the set of $\left\{\lambda\right\}$ can change. The sub-case where $\beta$ changes while the set of $\left\{\lambda\right\}$ is kept constant is then obtained as a special case. For simplicity, let us assume we have two parameters, $\beta$ and a single parameter $\lambda$ that we denote with $h$ (a magnetic field intensity, for instance). In this two-dimensional parametric case, we generally have that

ds_{\mathrm{Bures}}^{2}\left(\beta\text{, }h\right)=\left(\begin{array}[c]{cc}d\beta&dh\end{array}\right)\left(\begin{array}[c]{cc}g_{\beta\beta}&g_{\beta h}\\ g_{h\beta}&g_{hh}\end{array}\right)\left(\begin{array}[c]{c}d\beta\\ dh\end{array}\right)=g_{\beta\beta}d\beta^{2}+g_{hh}dh^{2}+2g_{\beta h}d\beta dh\text{,}

(55)

where we used the fact that $g_{h\beta}=g_{\beta h}$ . From Eq. (55), we note that

ds_{\mathrm{Bures}}^{2}\left(\beta\text{, }h\right)=\left\{\begin{array}[c]{c}g_{\beta\beta}\left(\beta\text{, }h\right)d\beta^{2}\text{, if }h=\text{{const}.}\\ g_{hh}\left(\beta\text{, }h\right)dh^{2}\text{, if }\beta=\text{{const}.}\\ g_{\beta\beta}\left(\beta\text{, }h\right)d\beta^{2}+g_{hh}\left(\beta\text{, }h\right)dh^{2}+2g_{\beta h}\left(\beta\text{, }h\right)d\beta dh\text{, if }\beta\neq\text{{const}. and }h\neq\text{{const}.}\end{array}\right.\text{.}

(56)

Recalling $ds_{\mathrm{Bures}}^{2}\left(\rho\text{, }\rho+d\rho\right)$ in Eq. (37), we start by calculating the expression of $dp_{n}$ with $p_{n}=p_{n}\left(h\text{, }\beta\right)\overset{\text{def}}{=}e^{-\beta E_{n}}/\mathcal{Z}$ . We observe that $dp_{n}$ can be written as,

dp_{n}=\frac{\partial p_{n}}{\partial h}dh+\frac{\partial p_{n}}{\partial\beta}d\beta\text{,}

(57)

where $\partial p_{n}/\partial\beta$ is given by,

$\displaystyle\frac{\partial p_{n}}{\partial\beta}$	$\displaystyle=\frac{\partial}{\partial\beta}\left(\frac{e^{-\beta E_{n}}}{\mathcal{Z}}\right)$
	$\displaystyle=\frac{1}{\mathcal{Z}}\frac{\partial}{\partial\beta}\left(e^{-\beta E_{n}}\right)+e^{-\beta E_{n}}\frac{\partial}{\partial\beta}\left(\frac{1}{\mathcal{Z}}\right)$
	$\displaystyle=-\frac{E_{n}}{\mathcal{Z}}e^{-\beta E_{n}}+e^{-\beta E_{n}}\frac{\partial}{\partial\mathcal{Z}}\left(\frac{1}{\mathcal{Z}}\right)\frac{\partial\mathcal{Z}}{\partial\beta}$
	$\displaystyle=-p_{n}E_{n}-\frac{e^{-\beta E_{n}}}{\mathcal{Z}}\frac{1}{\mathcal{Z}}\frac{\partial\mathcal{Z}}{\partial\beta}$
	$\displaystyle=-p_{n}E_{n}-p_{n}\frac{\partial\ln\mathcal{Z}}{\partial\beta}$
	$\displaystyle=-p_{n}E_{n}+p_{n}\frac{1}{\mathcal{Z}}\sum_{n}E_{n}e^{-\beta E_{n}}$
	$\displaystyle=-p_{n}E_{n}+p_{n}\sum_{n}p_{n}E_{n}$
	$\displaystyle=-p_{n}E_{n}+p_{n}\left\langle\mathrm{H}\right\rangle\text{,}$	(58)

that is,

\frac{\partial p_{n}}{\partial\beta}d\beta=p_{n}\left[\left\langle\mathrm{H}\right\rangle-E_{n}\right]d\beta\text{.}

(59)

Note that the expectation value $\left\langle\mathrm{H}\right\rangle$ in Eq. (59) is defined as $\left\langle\mathrm{H}\right\rangle$ $\overset{\text{def}}{=}\sum_{n}p_{n}E_{n}$ . From Eq. (49), we also have

\frac{\partial p_{n}}{\partial h}dh=-\beta p_{n}\left[\frac{\partial E_{n}}{\partial h}-\sum_{j}p_{j}\frac{\partial E_{j}}{\partial h}\right]dh=\beta p_{n}\left[\left\langle\left(\partial_{h}\mathrm{H}\right)_{d}\right\rangle-\partial_{h}E_{n}\right]dh\text{,}

(60)

where $\left\langle\left(\partial_{h}\mathrm{H}\right)_{d}\right\rangle$ is defined as

\left\langle\left(\partial_{h}\mathrm{H}\right)_{d}\right\rangle\overset{\text{def}}{=}\sum_{j}p_{j}\partial_{h}E_{j}\text{.}

(61)

Using Eqs. (57), (59), and (60), we wish to calculate the term $\left(1/4\right)\sum_{n}dp_{n}^{2}/p_{n}$ in $ds_{\mathrm{Bures}}^{2}\left(\rho\text{, }\rho+d\rho\right)$ in Eq. (37). Let us begin by observing that

dp_{n}^{2}=\left(\frac{\partial p_{n}}{\partial h}dh+\frac{\partial p_{n}}{\partial\beta}d\beta\right)^{2}=\left(\partial_{h}p_{n}dh\right)^{2}+\left(\partial_{\beta}p_{n}d\beta\right)^{2}+2\partial_{\beta}p_{n}\partial_{h}p_{n}d\beta dh\text{.}

(62)

Therefore, we get

	$\displaystyle\frac{1}{4}\sum_{n}\frac{dp_{n}^{2}}{p_{n}}$	$\displaystyle=\frac{1}{4}\sum_{n}\frac{\left(\partial_{h}p_{n}dh\right)^{2}+\left(\partial_{\beta}p_{n}d\beta\right)^{2}+2\partial_{\beta}p_{n}\partial_{h}p_{n}d\beta dh}{p_{n}}$
		$\displaystyle=\frac{1}{4}\sum_{n}\frac{\left(\partial_{h}p_{n}\right)^{2}}{p_{n}}dh^{2}+\frac{1}{4}\sum_{n}\frac{\left(\partial_{\beta}p_{n}\right)^{2}}{p_{n}}d\beta^{2}+\frac{1}{4}\sum_{n}\frac{2\partial_{\beta}p_{n}\partial_{h}p_{n}}{p_{n}}d\beta dh\text{.}$		(63)

First, note that

\frac{1}{4}\sum_{n}\frac{\left(\partial_{\beta}p_{n}\right)^{2}}{p_{n}}d\beta^{2}=\frac{1}{4}\left[\left\langle\mathrm{H}^{2}\right\rangle-\left\langle\mathrm{H}\right\rangle^{2}\right]d\beta^{2}\text{,}

(64)

where $\left\langle\mathrm{H}\right\rangle$ and $\left\langle\mathrm{H}^{2}\right\rangle$ are defined as

\left\langle\mathrm{H}\right\rangle\overset{\text{def}}{=}\sum_{i}p_{i}E_{i}\text{, and }\left\langle\mathrm{H}^{2}\right\rangle\overset{\text{def}}{=}\sum_{i}p_{i}E_{i}^{2}\text{,}

(65)

respectively. Indeed, using Eq. (59), we have

$\displaystyle\sum_{n}\frac{\left(\partial_{\beta}p_{n}\right)^{2}}{p_{n}}$	$\displaystyle=\sum_{n}\frac{p_{n}^{2}\left[\left\langle\mathrm{H}\right\rangle-E_{n}\right]^{2}}{p_{n}}$
	$\displaystyle=\sum_{n}\frac{p_{n}^{2}\left\langle\mathrm{H}\right\rangle^{2}+p_{n}^{2}E_{n}^{2}-2\left\langle\mathrm{H}\right\rangle p_{n}^{2}E_{n}}{p_{n}}$
	$\displaystyle=\left\langle\mathrm{H}\right\rangle^{2}+\left\langle\mathrm{H}^{2}\right\rangle-2\left\langle\mathrm{H}\right\rangle^{2}$
	$\displaystyle=\left\langle\mathrm{H}^{2}\right\rangle-\left\langle\mathrm{H}\right\rangle^{2}\text{.}$	(66)

Second, observe that

\frac{1}{4}\sum_{n}\frac{\left(\partial_{h}p_{n}\right)^{2}}{p_{n}}dh^{2}=\frac{1}{4}\beta^{2}\left\{\left\langle\left[\left(\partial_{h}\mathrm{H}\right)_{d}\right]^{2}\right\rangle-\left\langle\left(\partial_{h}\mathrm{H}\right)_{d}\right\rangle^{2}\right\}dh^{2}\text{,}

(67)

where $\left\langle\left(\partial_{h}\mathrm{H}\right)_{d}\right\rangle$ is given in Eq. (61) and $\left\langle\left[\left(\partial_{h}\mathrm{H}\right)_{d}\right]^{2}\right\rangle$ is defined as

\left\langle\left[\left(\partial_{h}\mathrm{H}\right)_{d}\right]^{2}\right\rangle\overset{\text{def}}{=}\sum_{i}p_{i}\left(\partial_{h}E_{i}\right)^{2}\text{.}

(68)

Indeed, using Eq. (60), we have

$\displaystyle\sum_{n}\frac{\left(\partial_{h}p_{n}\right)^{2}}{p_{n}}$	$\displaystyle=\sum_{n}\frac{\left(\beta p_{n}\right)^{2}\left[\left\langle\left(\partial_{h}\mathrm{H}\right)_{d}\right\rangle-\partial_{h}E_{n}\right]^{2}}{p_{n}}$
	$\displaystyle=\beta^{2}\sum_{n}\left[p_{n}\left\langle\left(\partial_{h}\mathrm{H}\right)_{d}\right\rangle^{2}+p_{n}\left(\partial_{h}E_{n}\right)^{2}-2p_{n}\left\langle\left(\partial_{h}\mathrm{H}\right)_{d}\right\rangle\partial_{h}E_{n}\right]$
	$\displaystyle=\beta^{2}\left\{\left\langle\left(\partial_{h}\mathrm{H}\right)_{d}\right\rangle^{2}+\left\langle\left[\left(\partial_{h}\mathrm{H}\right)_{d}\right]^{2}\right\rangle-2\left\langle\left(\partial_{h}\mathrm{H}\right)_{d}\right\rangle^{2}\right\}$
	$\displaystyle=\beta^{2}\left\{\left\langle\left[\left(\partial_{h}\mathrm{H}\right)_{d}\right]^{2}\right\rangle-\left\langle\left(\partial_{h}\mathrm{H}\right)_{d}\right\rangle^{2}\right\}\text{.}$	(69)

Third, we note that

\frac{1}{4}\sum_{n}\frac{2\partial_{\beta}p_{n}\partial_{h}p_{n}}{p_{n}}d\beta dh=\frac{1}{4}2\beta\left[\left\langle\mathrm{H}\left(\partial_{h}\mathrm{H}\right)_{d}\right\rangle-\left\langle\mathrm{H}\right\rangle\left\langle\left(\partial_{h}\mathrm{H}\right)_{d}\right\rangle\right]d\beta dh\text{.}

(70)

Indeed, using Eqs. (59) and (60), we get

$\displaystyle\sum_{n}\frac{2\partial_{\beta}p_{n}\partial_{h}p_{n}}{p_{n}}$	$\displaystyle=\sum_{n}\frac{2p_{n}\left[\left\langle\mathrm{H}\right\rangle-E_{n}\right]\beta p_{n}\left[\left\langle\left(\partial_{h}\mathrm{H}\right)_{d}\right\rangle-\partial_{h}E_{n}\right]}{p_{n}}$
	$\displaystyle=\sum_{n}2\beta\left[\left\langle\mathrm{H}\right\rangle-E_{n}\right]\left[\left\langle\left(\partial_{h}\mathrm{H}\right)_{d}\right\rangle-\partial_{h}E_{n}\right]p_{n}$
	$\displaystyle=\sum_{n}2\beta\left[\left\langle\mathrm{H}\right\rangle\left\langle\left(\partial_{h}\mathrm{H}\right)_{d}\right\rangle-\left\langle\mathrm{H}\right\rangle\partial_{h}E_{n}-E_{n}\left\langle\left(\partial_{h}\mathrm{H}\right)_{d}\right\rangle+E_{n}\partial_{h}E_{n}\right]p_{n}$
	$\displaystyle=2\beta\left[\left\langle\mathrm{H}\right\rangle\left\langle\left(\partial_{h}\mathrm{H}\right)_{d}\right\rangle-\left\langle\mathrm{H}\right\rangle\left\langle\left(\partial_{h}\mathrm{H}\right)_{d}\right\rangle-\left\langle\mathrm{H}\right\rangle\left\langle\left(\partial_{h}\mathrm{H}\right)_{d}\right\rangle+\left\langle\mathrm{H}\left(\partial_{h}\mathrm{H}\right)_{d}\right\rangle\right]$
	$\displaystyle=2\beta\left[\left\langle\mathrm{H}\left(\partial_{h}\mathrm{H}\right)_{d}\right\rangle-\left\langle\mathrm{H}\right\rangle\left\langle\left(\partial_{h}\mathrm{H}\right)_{d}\right\rangle\right]\text{,}$	(71)

where $\left\langle\mathrm{H}\left(\partial_{h}\mathrm{H}\right)_{d}\right\rangle$ is defined as

\left\langle\mathrm{H}\left(\partial_{h}\mathrm{H}\right)_{d}\right\rangle\overset{\text{def}}{=}\sum_{i}p_{i}E_{i}\partial_{h}E_{i}\text{.}

(72)

Finally, employing Eqs. (64), (67), and (70), the most general expression of $ds_{\mathrm{Bures}}^{2}\left(\rho\text{, }\rho+d\rho\right)$ in Eq. (37) between two mixed thermal states $\rho\left(\beta\text{, }h\right)$ and $\left(\rho+d\rho\right)\left(\beta\text{, }h\right)$ when either changes in the parameter $\beta$ or $h$ are allotted becomes

$\displaystyle ds_{\mathrm{Bures}}^{2}\left(\rho\text{, }\rho+d\rho\right)$	$\displaystyle=\frac{1}{4}\left[\left\langle\mathrm{H}^{2}\right\rangle-\left\langle\mathrm{H}\right\rangle^{2}\right]d\beta^{2}$
	$\displaystyle+\frac{1}{4}\left\{\beta^{2}\left\{\left\langle\left[\left(\partial_{h}\mathrm{H}\right)_{d}\right]^{2}\right\rangle-\left\langle\left(\partial_{h}\mathrm{H}\right)_{d}\right\rangle^{2}\right\}+2\sum_{n\neq m}\left\|\frac{\left\langle n\|\partial_{h}\mathrm{H}\|m\right\rangle}{E_{n}-E_{m}}\right\|^{2}\frac{\left(e^{-\beta E_{n}}-e^{-\beta E_{m}}\right)^{2}}{\mathcal{Z}\left(e^{-\beta E_{n}}+e^{-\beta E_{m}}\right)}\right\}dh^{2}+$
	$\displaystyle+\frac{1}{4}\left\{2\beta\left[\left\langle\mathrm{H}\left(\partial_{h}\mathrm{H}\right)_{d}\right\rangle-\left\langle\mathrm{H}\right\rangle\left\langle\left(\partial_{h}\mathrm{H}\right)_{d}\right\rangle\right]\right\}d\beta dh\text{.}$	(73)

Note that $ds_{\mathrm{Bures}}^{2}\left(\rho\text{, }\rho+d\rho\right)$ is the sum of two contributions, the classical Fisher-Rao information metric contribution and the non-classical metric contribution expressed in the summation term in the right-hand-side of Eq. (73). For later convenience, we also remark that the quadratic term $\left|\left\langle n|\partial_{h}\mathrm{H}|m\right\rangle\right|^{2}$ in the summation term in the right-hand-side of Eq. (73) is invariant under change of sign of the Hamiltonian of the system. Clearly, from Eq. (73) we find that $ds_{\mathrm{Bures}}^{2}\left(\rho\text{, }\rho+d\rho\right)=(1/4)\left[\left\langle\mathrm{H}^{2}\right\rangle-\left\langle\mathrm{H}\right\rangle^{2}\right]d\beta^{2}$ when $h=$ const. and only $\beta$ can change. If $\beta=$ const., $ds_{\mathrm{Bures}}^{2}\left(\rho\text{, }\rho+d\rho\right)$ in Eq. (73) reduces to Eq. (54). The explicit derivation of Eq. (73) ends our calculation of the Bures metric between neighboring thermal states undergoing temperature and/or magnetic field intensity changes as originally presented by Zanardi and collaborators in Ref. zanardi07 .

III The Sjöqvist metric

In this section, we introduce the Sjöqvist metric erik20 for nondegerante mixed states with an explicit derivation. Assume to consider two rank- $N$ neighboring nondegenerate density operators $\rho\left(t\right)$ and $\rho\left(t+dt\right)$ linked by means of a smooth path $t\mapsto\rho\left(t\right)$ specifying the evolution of a given quantum system. The nondegeneracy property implies that the phase of the eigenvectors represents the gauge freedom in the spectral decomposition of the density operators. As a consequence, there exists a one-to-one correspondence between the set of two orthogonal rays $\left\{e^{i\phi_{k}\left(t\right)}\left|e_{k}\left(t\right)\right\rangle:0\leq\phi_{k}\left(t\right)<2\pi\right\}_{1\leq k\leq N}$ that specify the spectral decomposition along the path $t\mapsto\rho\left(t\right)$ and the rank- $N$ nondegenerate density operator $\rho\left(t\right)$ . Obviously, if some nonzero eigenvalue of $\rho\left(t\right)$ is degenerate, this correspondence would no longer exist. We present next the explicit derivation of the Sjöqvist metric.

III.1 The explicit derivation

Consider two neighboring states $\rho\left(t\right)$ and $\rho\left(t+dt\right)$ with spectral decompositions given by $\mathcal{B}\left(t\right)=\left\{\sqrt{p_{k}\left(t\right)}e^{if_{k}\left(t\right)}\left|n_{k}\left(t\right)\right\rangle\right\}_{1\leq k\leq N}$ and $\mathcal{B}\left(t+dt\right)=\left\{\sqrt{p_{k}\left(t+dt\right)}e^{if_{k}\left(t+dt\right)}\left|n_{k}\left(t+dt\right)\right\rangle\right\}_{1\leq k\leq N}$ , respectively. The quantity $N$ denotes the rank of the nondegenerate density operator $\rho\left(t\right)$ . Consider the infinitesimal distance $d^{2}\left(t\text{, }t+dt\right)$ between $\rho\left(t\right)$ and $\rho\left(t+dt\right)$ defined as

d^{2}\left(t\text{, }t+dt\right)\overset{\text{def}}{=}\sum_{k=1}^{N}\left\|\sqrt{p_{k}\left(t\right)}e^{if_{k}\left(t\right)}\left|n_{k}\left(t\right)\right\rangle-\sqrt{p_{k}\left(t+dt\right)}e^{if_{k}\left(t+dt\right)}\left|n_{k}\left(t+dt\right)\right\rangle\right\|^{2}\text{.}

(74)

The Sjöqvist metric is defined as the minimum of $d^{2}\left(t\text{, }t+dt\right)$ in Eq. (74). Note that the squared norm term $\left\|\sqrt{p_{k}\left(t\right)}e^{if_{k}\left(t\right)}\left|n_{k}\left(t\right)\right\rangle-\sqrt{p_{k}\left(t+dt\right)}e^{if_{k}\left(t+dt\right)}\left|n_{k}\left(t+dt\right)\right\rangle\right\|^{2}$ can be written as

	$\displaystyle\left(\sqrt{p_{k}\left(t\right)}e^{-if_{k}\left(t\right)}\left\langle n_{k}\left(t\right)\right\|-\sqrt{p_{k}\left(t+dt\right)}e^{-if_{k}\left(t+dt\right)}\left\langle n_{k}\left(t+dt\right)\right\|\right)$
	$\displaystyle\left(\sqrt{p_{k}\left(t\right)}e^{if_{k}\left(t\right)}\left\|n_{k}\left(t\right)\right\rangle-\sqrt{p_{k}\left(t+dt\right)}e^{if_{k}\left(t+dt\right)}\left\|n_{k}\left(t+dt\right)\right\rangle\right)$
	$\displaystyle=p_{k}\left(t\right)+p_{k}\left(t+dt\right)-\sqrt{p_{k}\left(t\right)p_{k}\left(t+dt\right)}e^{i\left[f_{k}\left(t+dt\right)-f_{k}\left(t\right)\right]}\left\langle n_{k}\left(t\right)\|n_{k}\left(t+dt\right)\right\rangle+$
	$\displaystyle-\sqrt{p_{k}\left(t\right)p_{k}\left(t+dt\right)}e^{-i\left[f_{k}\left(t+dt\right)-f_{k}\left(t\right)\right]}\left\langle n_{k}\left(t+dt\right)\|n_{k}\left(t\right)\right\rangle\text{,}$		(75)

with $e^{i\left[f_{k}\left(t+dt\right)-f_{k}\left(t\right)\right]}\left\langle n_{k}\left(t\right)|n_{k}\left(t+dt\right)\right\rangle+e^{-i\left[f_{k}\left(t+dt\right)-f_{k}\left(t\right)\right]}\left\langle n_{k}\left(t+dt\right)|n_{k}\left(t\right)\right\rangle$ equal to

	$\displaystyle 2\operatorname{Re}\left\{e^{i\left[f_{k}\left(t+dt\right)-f_{k}\left(t\right)\right]}\left\langle n_{k}\left(t\right)\|n_{k}\left(t+dt\right)\right\rangle\right\}$
	$\displaystyle=2\operatorname{Re}\left\{e^{i\left[f_{k}\left(t+dt\right)-f_{k}\left(t\right)\right]}\left\|\left\langle n_{k}\left(t\right)\|n_{k}\left(t+dt\right)\right\rangle\right\|e^{i\arg\left(\left\langle n_{k}\left(t\right)\|n_{k}\left(t+dt\right)\right\rangle\right)}\right\}$
	$\displaystyle=2\left\|\left\langle n_{k}\left(t\right)\|n_{k}\left(t+dt\right)\right\rangle\right\|\operatorname{Re}\left\{e^{i\left[f_{k}\left(t+dt\right)-f_{k}\left(t\right)\right]}e^{i\arg\left(\left\langle n_{k}\left(t\right)\|n_{k}\left(t+dt\right)\right\rangle\right)}\right\}$
	$\displaystyle=2\left\|\left\langle n_{k}\left(t\right)\|n_{k}\left(t+dt\right)\right\rangle\right\|\cos\left[f_{k}\left(t+dt\right)-f_{k}\left(t\right)+\arg\left(\left\langle n_{k}\left(t\right)\|n_{k}\left(t+dt\right)\right\rangle\right)\right]\text{.}$		(76)

Note that $f_{k}\left(t+dt\right)=f_{k}\left(t\right)+\dot{f}_{k}\left(t\right)dt+O\left(dt^{2}\right)$ and $\left|n_{k}\left(t+dt\right)\right\rangle=\left|n_{k}\left(t\right)\right\rangle+\left|\dot{n}_{k}\left(t\right)\right\rangle dt+O\left(dt^{2}\right)$ . Therefore, we have

\cos\left[f_{k}\left(t+dt\right)-f_{k}\left(t\right)+\arg\left(\left\langle n_{k}\left(t\right)|n_{k}\left(t+dt\right)\right\rangle\right)\right]=\cos\left\{\dot{f}_{k}\left(t\right)dt+\arg\left[1+\left\langle n_{k}\left(t\right)|\dot{n}_{k}\left(t\right)\right\rangle dt\right]+O\left(dt^{2}\right)\right\}\text{.}

(77)

Setting $\dot{f}_{k}\left(t\right)dt+\arg\left[1+\left\langle n_{k}\left(t\right)|\dot{n}_{k}\left(t\right)\right\rangle dt\right]+O\left(dt^{2}\right)\overset{\text{def}}{=}\lambda_{k}\left(t\text{, }t+dt\right)$ , the infinitesimal distance $d^{2}\left(t\text{, }t+dt\right)$ becomes

d^{2}\left(t\text{, }t+dt\right)=2-2\sum_{k=1}^{N}\sqrt{p_{k}\left(t\right)p_{k}\left(t+dt\right)}\left|\left\langle n_{k}\left(t\right)|n_{k}\left(t+dt\right)\right\rangle\right|\cos\left[\lambda_{k}\left(t\text{, }t+dt\right)\right]\text{.}

(78)

Then, the Sjöqvist metric $ds_{\mathrm{Sj\ddot{o}qvist}}^{2}$ is the minimum of $d^{2}\left(t\text{, }t+dt\right)$ , $d_{\min}^{2}\left(t\text{, }t+dt\right)$ , and is obtained when $\lambda_{k}\left(t\text{, }t+dt\right)$ equals zero for any $1\leq k\leq N$ . Its expression is given by,

ds_{\mathrm{Sj\ddot{o}qvist}}^{2}=2-2\sum_{k=1}^{N}\sqrt{p_{k}\left(t\right)p_{k}\left(t+dt\right)}\left|\left\langle n_{k}\left(t\right)|n_{k}\left(t+dt\right)\right\rangle\right|\text{.}

(79)

It is worthwhile emphasizing that the minimum of $d^{2}\left(t\text{, }t+dt\right)$ is achieved by selecting phases $\left\{f_{k}\left(t\right)\text{, }f_{k}\left(t+dt\right)\right\}$ such that

\dot{f}_{k}\left(t\right)dt+\arg\left[1+\left\langle n_{k}\left(t\right)\left|\dot{n}_{k}\left(t\right)\right.\right\rangle dt+O\left(dt^{2}\right)\right]=0

(80)

Observing that $e^{\left\langle n_{k}\left(t\right)\left|\dot{n}_{k}\left(t\right)\right.\right\rangle dt}=1+\left\langle n_{k}\left(t\right)\left|\dot{n}_{k}\left(t\right)\right.\right\rangle dt+O\left(dt^{2}\right)$ is such that $\arg\left[e^{\left\langle n_{k}\left(t\right)\left|\dot{n}_{k}\left(t\right)\right.\right\rangle dt}\right]=-i\left\langle n_{k}\left(t\right)\left|\dot{n}_{k}\left(t\right)\right.\right\rangle dt$ , Eq. (80) can be rewritten to the first order in $dt$ as

\dot{f}_{k}\left(t\right)-i\left\langle n_{k}\left(t\right)\left|\dot{n}_{k}\left(t\right)\right.\right\rangle=0\text{.}

(81)

Eq. (81) denotes the parallel transport condition $\left\langle\psi_{k}\left(t\right)\left|\psi_{k}\left(t\right)\right.\right\rangle=0$ where $\left|\psi_{k}\left(t\right)\right\rangle\overset{\text{def}}{=}e^{if_{k}\left(t\right)}\left|n_{k}\left(t\right)\right\rangle$ is associated with individual pure state paths in the chosen ensemble that defines the mixed state $\rho\left(t\right)$ aharonov87 . To find a more useful expression of $ds_{\mathrm{Sj\ddot{o}qvist}}^{2}$ , let us start by observing that,

\sqrt{p_{k}\left(t\right)p_{k}\left(t+dt\right)}=p_{k}\left(t\right)\sqrt{1+\frac{dp_{k}\left(t\right)}{p_{k}\left(t\right)}}=p_{k}+\frac{1}{2}\dot{p}_{k}dt-\frac{1}{8}\frac{\dot{p}_{k}^{2}}{p_{k}}dt^{2}+O\left(dt^{2}\right)\text{.}

(82)

Furthermore, to the second order in $dt$ , the state $\left|n_{k}\left(t+dt\right)\right\rangle$ can be written as

\left|n_{k}\left(t+dt\right)\right\rangle=\left|n_{k}\left(t\right)\right\rangle+\left|\dot{n}_{k}\left(t\right)\right\rangle dt+\frac{1}{2}\left|\ddot{n}_{k}\left(t\right)\right\rangle dt^{2}+O\left(dt^{2}\right)\text{.}

(83)

Therefore, to the second order in $dt$ , the quantum overlap $\left\langle n_{k}\left(t\right)|n_{k}\left(t+dt\right)\right\rangle$ becomes

\left\langle n_{k}\left(t\right)|n_{k}\left(t+dt\right)\right\rangle=\left\langle n_{k}\left(t\right)|n_{k}\left(t\right)\right\rangle+\left\langle n_{k}\left(t\right)|\dot{n}_{k}\left(t\right)\right\rangle dt+\frac{1}{2}\left\langle n_{k}\left(t\right)|\ddot{n}_{k}\left(t\right)\right\rangle dt^{2}+O\left(dt^{2}\right)

(84)

Let us focus now on calculating $\left|\left\langle n_{k}\left(t\right)|n_{k}\left(t+dt\right)\right\rangle\right|=\sqrt{\left|\left\langle n_{k}\left(t\right)|n_{k}\left(t+dt\right)\right\rangle\right|^{2}}$ , where

\left|\left\langle n_{k}\left(t\right)|n_{k}\left(t+dt\right)\right\rangle\right|^{2}=\left\langle n_{k}\left(t\right)|n_{k}\left(t+dt\right)\right\rangle\left\langle n_{k}\left(t+dt\right)|n_{k}\left(t\right)\right\rangle\text{.}

(85)

Using Eq. (83), Eq. (85) becomes

$\displaystyle\left\langle n_{k}\left(t\right)\|n_{k}\left(t+dt\right)\right\rangle\left\langle n_{k}\left(t+dt\right)\|n_{k}\left(t\right)\right\rangle$	$\displaystyle\approx\left[\left\langle n_{k}\left(t\right)\|n_{k}\left(t\right)\right\rangle+\left\langle n_{k}\left(t\right)\|\dot{n}_{k}\left(t\right)\right\rangle dt+\frac{1}{2}\left\langle n_{k}\left(t\right)\|\ddot{n}_{k}\left(t\right)\right\rangle dt^{2}\right]\cdot$
	$\displaystyle\cdot\left[\left\langle n_{k}\left(t\right)\|n_{k}\left(t\right)\right\rangle+\left\langle\dot{n}_{k}\left(t\right)\|n_{k}\left(t\right)\right\rangle dt+\frac{1}{2}\left\langle\ddot{n}_{k}\left(t\right)\|n_{k}\left(t\right)\right\rangle dt^{2}\right]$
	$\displaystyle=\left[1+\left\langle n_{k}\|\dot{n}_{k}\right\rangle dt+\frac{1}{2}\left\langle n_{k}\|\ddot{n}_{k}\right\rangle dt^{2}\right]\left[1+\left\langle\dot{n}_{k}\|n_{k}\right\rangle dt+\frac{1}{2}\left\langle\ddot{n}_{k}\|n_{k}\right\rangle dt^{2}\right]$
	$\displaystyle\approx 1+\left\langle\dot{n}_{k}\|n_{k}\right\rangle dt+\frac{1}{2}\left\langle\ddot{n}_{k}\|n_{k}\right\rangle dt^{2}+\left\langle n_{k}\|\dot{n}_{k}\right\rangle dt+\left\langle n_{k}\|\dot{n}_{k}\right\rangle\left\langle\dot{n}_{k}\|n_{k}\right\rangle dt^{2}+\frac{1}{2}\left\langle n_{k}\|\ddot{n}_{k}\right\rangle dt^{2}$
	$\displaystyle=1+\left[\left\langle\dot{n}_{k}\|n_{k}\right\rangle+\left\langle n_{k}\|\dot{n}_{k}\right\rangle\right]dt+\left\langle n_{k}\|\dot{n}_{k}\right\rangle\left\langle\dot{n}_{k}\|n_{k}\right\rangle dt^{2}+\frac{1}{2}\left[\left\langle n_{k}\|\ddot{n}_{k}\right\rangle+\left\langle\ddot{n}_{k}\|n_{k}\right\rangle\right]dt^{2}$
	$\displaystyle=1+\left\langle n_{k}\|\dot{n}_{k}\right\rangle\left\langle\dot{n}_{k}\|n_{k}\right\rangle dt^{2}-\left\langle\dot{n}_{k}\|\dot{n}_{k}\right\rangle dt^{2}\text{,}$	(86)

that is,

\left\langle n_{k}\left(t\right)|n_{k}\left(t+dt\right)\right\rangle\left\langle n_{k}\left(t+dt\right)|n_{k}\left(t\right)\right\rangle=1+\left\langle n_{k}|\dot{n}_{k}\right\rangle\left\langle\dot{n}_{k}|n_{k}\right\rangle dt^{2}-\left\langle\dot{n}_{k}|\dot{n}_{k}\right\rangle dt^{2}+O\left(dt^{2}\right)\text{,}

(87)

since $\left\langle n_{k}|n_{k}\right\rangle=1$ implies $\left\langle\dot{n}_{k}|n_{k}\right\rangle+\left\langle n_{k}|\dot{n}_{k}\right\rangle=0$ and $\left\langle n_{k}|\ddot{n}_{k}\right\rangle+\left\langle\ddot{n}_{k}|n_{k}\right\rangle=-2\left\langle\dot{n}_{k}|\dot{n}_{k}\right\rangle$ . Finally, using Eqs. (82) and (87) along with noting that $\sum_{k}\dot{p}_{k}=0$ , the Sjöqvist metric $ds_{\mathrm{Sj\ddot{o}qvist}}^{2}$ in Eq. (79) becomes

	$\displaystyle ds_{\mathrm{Sj\ddot{o}qvist}}^{2}$	$\displaystyle\approx 2-2\sum_{k=1}^{N}\left(p_{k}+\frac{1}{2}\dot{p}_{k}dt-\frac{1}{8}\frac{\dot{p}_{k}^{2}}{p_{k}}dt^{2}\right)\left(1+\frac{1}{2}\left\langle n_{k}\|\dot{n}_{k}\right\rangle\left\langle\dot{n}_{k}\|n_{k}\right\rangle dt^{2}-\frac{1}{2}\left\langle\dot{n}_{k}\|\dot{n}_{k}\right\rangle dt^{2}\right)$
		$\displaystyle\approx 2-2\sum_{k=1}^{N}p_{k}-\sum_{k=1}^{N}p_{k}\left\langle n_{k}\|\dot{n}_{k}\right\rangle\left\langle\dot{n}_{k}\|n_{k}\right\rangle dt^{2}+\sum_{k=1}^{N}p_{k}\left\langle\dot{n}_{k}\|\dot{n}_{k}\right\rangle dt^{2}+\frac{1}{4}\sum_{k=1}^{N}\frac{\dot{p}_{k}^{2}}{p_{k}}dt^{2}\text{,}$		(88)

that is,

	$\displaystyle ds_{\mathrm{Sj\ddot{o}qvist}}^{2}$	$\displaystyle\approx\frac{1}{4}\sum_{k=1}^{N}\frac{\dot{p}_{k}^{2}}{p_{k}}dt^{2}+\sum_{k=1}^{N}p_{k}\left[\left\langle\dot{n}_{k}\|\dot{n}_{k}\right\rangle-\left\langle n_{k}\|\dot{n}_{k}\right\rangle\left\langle\dot{n}_{k}\|n_{k}\right\rangle\right]dt^{2}$
		$\displaystyle\approx\frac{1}{4}\sum_{k=1}^{N}\frac{\dot{p}_{k}^{2}}{p_{k}}dt^{2}+\sum_{k=1}^{N}p_{k}\left[\left\langle\dot{n}_{k}\|\left(\mathrm{I}-\left\|n_{k}\right\rangle\left\langle n_{k}\right\|\right)\|\dot{n}_{k}\right\rangle\right]dt^{2}\text{,}$		(89)

where I in Eq. (89) denotes the identity operator on the $N$ -dimensional Hilbert space. Finally, neglecting terms that are smaller than $O\left(dt^{2}\right)$ in Eq. (79) and defining $ds_{k}^{2}\overset{\text{def}}{=}\left[\left\langle\dot{n}_{k}|\left(\mathrm{I}-\left|n_{k}\right\rangle\left\langle n_{k}\right|\right)|\dot{n}_{k}\right\rangle\right]dt^{2}$ , the expression of the Sjöqvist metric will be formally taken to be

ds_{\mathrm{Sj\ddot{o}qvist}}^{2}\overset{\text{def}}{=}\frac{1}{4}\sum_{k=1}^{N}\frac{\dot{p}_{k}^{2}}{p_{k}}dt^{2}+\sum_{k=1}^{N}p_{k}ds_{k}^{2}\text{.}

(90)

The derivation of Eq. (90) concludes our explicit calculation of the Sjöqvist metric for nondegerante mixed states. Interestingly, note that $ds_{k}^{2}\overset{\text{def}}{=}\left\langle\dot{n}_{k}\left|\left(\mathrm{I}-\left|n_{k}\right\rangle\left\langle n_{k}\right|\right)\right|\dot{n}_{k}\right\rangle dt^{2}$ in Eq. (90) can be written as $ds_{k}^{2}=\left\langle\nabla n_{k}\left|\nabla n_{k}\right.\right\rangle$ with $\left|\nabla n_{k}\right\rangle\overset{\text{def}}{=}\mathrm{P}_{\bot}^{\left(k\right)}\left|\dot{n}_{k}\right\rangle$ being the covariant derivative of $\left|n_{k}\right\rangle$ and $\mathrm{P}_{\bot}^{\left(k\right)}\overset{\text{def}}{=}\mathrm{I}-\left|n_{k}\right\rangle\left\langle n_{k}\right|$ denoting the projector onto states perpendicular to $\left|n_{k}\right\rangle$ . In analogy to the Bures metric case (see the comment right below Eq. (73)), we stress for later convenience that the quadratic term $ds_{k}^{2}$ does not change under change of sign of the Hamiltonian of the system. The expression of the Sjöqvist metric in Eq. (90) can be viewed as expressed by two contributions, a classical and a nonclassical term. The first term in Eq. (90) is the classical one and is represented by the classical Fisher-Rao information metric between the two probability distributions $\left\{p_{k}\right\}_{1\leq k\leq N}$ and $\left\{p_{k}+dp_{k}\right\}_{1\leq k\leq N}$ . The second term is the nonclassical one and is represented by a weighted average of pure state Fubini-Study metrics along directions specified by state vectors $\left\{\left|n_{k}\right\rangle\right\}_{1\leq k\leq N}$ . We are now ready to introduce our Hamiltonian models.

IV The Hamiltonian Models

In this section, we present two Hamiltonian models. The first Hamiltonian model specifies a spin- $1/2$ particle in a uniform and time-independent external magnetic field oriented along the $z$ -axis. The second Hamiltonian model, instead, describes a superconducting flux qubit. Finally, we construct the two corresponding parametric families of thermal states by bringing these two systems in thermal equilibrium with a reservoir at finite and non-zero temperature $T$ .

Refer to caption — Figure 1: Schematic depiction of a spin qubit (a) and a superconducting flux qubit (b) in thermal equilibrium with a reservoir at non-zero temperature T. The spin qubit in (a) has opposite orientations of the spin along the quantization axis as its two states. The superconducting flux qubit in (b), instead, has circulating currents of opposite sign as its two states.

IV.1 Spin-1/2 qubit Hamiltonian

Consider a spin- $1/2$ particle represented by an electron of $m$ , charge $-e$ with $e\geq 0$ immersed in an external magnetic field $\vec{B}\left(t\right)$ . From a quantum-mechanical perspective, the Hamiltonian of this system can be described the Hermitian operator H $\left(t\right)$ given by $\mathrm{H}\left(t\right)\overset{\text{def}}{=}-\vec{\mu}\mathbf{\cdot}\vec{B}\left(t\right)$ sakurai , with $\vec{\mu}$ denoting the electron magnetic moment operator. The quantity $\vec{\mu}$ is defined as $\vec{\mu}\overset{\text{def}}{=}-\left(e/m\right)\vec{s}$ with $\vec{s}\overset{\text{def}}{=}\left(\hslash/2\right)\vec{\sigma}$ being the spin operator. Clearly, $\hslash\overset{\text{def}}{=}h/(2\pi)$ is the reduced Planck constant and $\vec{\sigma}\overset{\text{def}}{=}\left(\sigma_{x}\text{, }\sigma_{y}\text{, }\sigma_{z}\right)$ is the usual Pauli spin vector operator. Assuming a time-independent magnetic field along the $z$ -direction given by $\vec{B}\left(t\right)=B_{0}\hat{z}$ and introducing the frequency $\omega\overset{\text{def}}{=}(e/m)B_{0}$ , the spin- $1/2$ qubit (SQ) Hamiltonian becomes

\mathrm{H}_{\mathrm{SQ}}\left(\omega\right)\overset{\text{def}}{=}\frac{\hslash\omega}{2}\sigma_{z}\text{,}

(91)

where $\sigma_{z}\overset{\text{def}}{=}\left|\uparrow\right\rangle\left\langle\uparrow\right|-\left|\downarrow\right\rangle\left\langle\downarrow\right|$ with $\left|\uparrow\right\rangle$ and $\left|\downarrow\right\rangle$ denoting the spin-up and the spin-down quantum states, respectively. Observe that with the sign convention used for $\mathrm{H}_{\mathrm{SQ}}\left(\omega\right)$ in Eq. (91), we have that $\left|\downarrow\right\rangle$ ( $\left|\uparrow\right\rangle$ ) denotes the ground (excited) state of the system with energy $-\hslash\omega/2$ ( $+\hslash\omega/2$ ).

IV.2 Superconducting flux qubit Hamiltonian

It is known that a qubit is a two-level (or, a two-state) quantum system and, moreover, it is possible to realize the two levels in a number of ways. For example, the two-levels can be regarded as the spin-up and spin-down of an electron, or as the vertical and horizontal polarization of a single photon. Interestingly, the two-levels of a qubit can be also realized as the supercurrent flowing in an anti-clockwise and clockwise directions in a superconducting loop clarke08 ; devoret13 . A flux qubit is a superconducting loop interrupted by one or three Josephson junctions (i.e., a dissipationless device with a nonlinear inductance). An arbitrary flux qubit can be described as a superposition of two persistent current basis states. The two quantum states are total magnetic flux $\Phi$ pointing up $\left|\uparrow\right\rangle$ and $\Phi$ pointing down $\left\langle\downarrow\right|$ . Alternatively, as previously mentioned, the two-levels of the quantum system can be described as the supercurrent $I_{q}$ circulating in the loop anti-clockwise and $I_{q}$ circulating clockwise. The Hamiltonian of a superconducting flux qubit (SFQ) in persistent current basis $\left\{\left|\uparrow\right\rangle\text{, }\left\langle\downarrow\right|\right\}$ is given by chiorescu03 ; pekola07 ; paauw09 ; pekola16 ,

\mathrm{H}_{\mathrm{SFQ}}\left(\Delta\text{, }\epsilon\right)\overset{\text{def}}{=}-\frac{\hslash}{2}\left(\Delta\sigma_{x}+\epsilon\sigma_{z}\right)\text{.}

(92)

In Eq. (92), $\hslash\overset{\text{def}}{=}h/\left(2\pi\right)$ is the reduced Planck constant, while $\sigma_{x}$ and $\sigma_{z}$ are Pauli matrices. Furthermore, $\hslash\epsilon\overset{\text{def}}{=}2I_{q}\left(\Phi_{e}-\frac{\Phi_{0}}{2}\right)$ is the magnetic energy bias defined in terms of the supercurrent $I_{q}$ , the externally applied magnetic flux $\Phi_{e}$ , and the magnetic flux quantum $\Phi_{0}\overset{\text{def}}{=}h/\left(2e\right)$ with $e$ being the absolute value of the electron charge. Finally, $\hslash\Delta$ is the energy gap at the degeneracy point specified by the relation $\Phi_{e}=\Phi_{0}/2$ (i.e., $\epsilon=0$ ) and represents the minimum splitting of the energy levels of the ground state $\left|g\right\rangle$ and the first excited state $\left|e\right\rangle$ of the superconducting qubit. At the gap, the coherence properties of the qubit are optimal. Away from the degeneracy point, $\epsilon\neq 0$ and the energy-level splitting becomes $\hslash\nu\overset{\text{def}}{=}\hslash\sqrt{\epsilon^{2}+\Delta^{2}}$ , with $\nu$ being the transition angular frequency of the qubit. The energy level splitting $\hslash\Delta$ depends on the critical current of the three Josephson junctions and their capacitance paauw09 . For flux qubits one has $\Delta\sim E_{C}/E_{J}$ with $E_{C}$ and $E_{J}$ denoting the Cooper pair charging energy and the Josephson coupling energy pekola16 , respectively. In summary, a flux qubit can be represented by a double-well potential whose shape (symmetrical versus asymmetrical) can be tuned with the externally applied magnetic flux $\Phi_{e}$ . When $\Phi_{e}=\Phi_{0}/2$ , the double-well is symmetric, the energy eigenstates (i.e., ground state and first excited states $\left|g\right\rangle$ and $\left|e\right\rangle$ , respectively) are symmetric (i.e., $\left|g\right\rangle\overset{\text{def}}{=}\left[\left|\uparrow\right\rangle+\left|\downarrow\right\rangle\right]/\sqrt{2}$ ) and antisymmetric (i.e., $\left|e\right\rangle\overset{\text{def}}{=}\left[\left|\uparrow\right\rangle-\left|\downarrow\right\rangle\right]/\sqrt{2}$ ) superpositions of the two states $\left|\uparrow\right\rangle$ and $\left|\downarrow\right\rangle$ and, finally, the splitting of the energy levels of $\left|g\right\rangle$ and $\left|e\right\rangle$ is $\Delta$ . Instead, when $\Phi_{e}\neq\Phi_{0}/2$ , the double-well is not symmetric, the energy eigenstates are arbitrary superpositions of the basis states $\left|\uparrow\right\rangle$ and $\left|\downarrow\right\rangle$ (i.e., $\alpha\left|\uparrow\right\rangle\pm\beta\left|\downarrow\right\rangle$ with $\left|\alpha\right|^{2}+\left|\beta\right|^{2}=1$ ) and, finally, the energy gap becomes $\hslash\nu\overset{\text{def}}{=}\hslash\sqrt{\epsilon^{2}+\Delta^{2}}$ . For more details on the theory underlying superconducting flux qubits, we refer to Ref. clarke08 .

The transition from (isolated) physical systems specified by pure states evolving according to the Hamiltonians in Eqs. (91) and (92) to the same (open) physical systems described by mixed quantum states can be explained as follows. Assume a quantum system specified by an Hamiltonian $\mathrm{H}$ is in thermal equilibrium with a reservoir at non-zero temperature $T$ . Then, following the principles of quantum statistical mechanics huang87 , the system has temperature $T$ and its state is described by a thermal state strocchi08 specified by a density matrix $\rho$ given by,

\rho\overset{\text{def}}{=}\frac{e^{-\beta\mathrm{H}}}{\mathrm{tr}\left(e^{-\beta\mathrm{H}}\right)}\text{.}

(93)

In Eq. (93), $\beta\overset{\text{def}}{=}\left(k_{B}T\right)^{-1}$ denotes the so-called inverse temperature, while $k_{B}$ is the Boltzmann constant. In what follows, we shall consider two families of mixed quantum thermal states given by

\rho_{\mathrm{SQ}}\left(\beta\text{, }\omega\right)\overset{\text{def}}{=}\frac{e^{-\beta\mathrm{H}_{\mathrm{SQ}}\left(\omega\right)}}{\mathrm{tr}\left(e^{-\beta\mathrm{H}_{\mathrm{SQ}}\left(\omega\right)}\right)}\text{ and, }\rho_{\mathrm{SFQ}}\left(\beta\text{, }\epsilon\right)\overset{\text{def}}{=}\frac{e^{-\beta\mathrm{H}_{\mathrm{SFQ}}\left(\epsilon\right)}}{\mathrm{tr}\left(e^{-\beta\mathrm{H}_{\mathrm{SFQ}}\left(\epsilon\right)}\right)}\text{.}

(94)

Note that in $\rho_{\mathrm{SFQ}}\left(\beta\text{, }\epsilon\right)$ in Eq. (94), we assume that the parameter $\Delta$ is fixed. For a work on how to tune the energy gap $\Delta$ in a flux qubit from an experimental standpoint, we refer to Ref. paauw09 . In Fig. $1$ , we present a schematic depiction of of a spin qubit and a superconducting flux qubit in thermal equilibrium with a reservoir at non-zero temperature $T$ .

V Applications

In this section, we calculate both the Sjöqvist and the Bures metrics for each one of the two distinct families of parametric thermal states mentioned in the previous section. From our comparative investigation, we find that the two metric coincide for the first Hamiltonian model (electron in a constant magnetic field along the $z$ -direction), while they differ for the second Hamiltonian model (superconducting flux qubit).

V.1 Spin qubits

Let us consider a system with an Hamiltonian described by $\mathrm{H}_{\mathrm{SQ}}\left(\omega\right)\overset{\text{def}}{=}\left(\hslash\omega/2\right)\sigma_{z}$ in Eq. (91). Observe that $\mathrm{H}_{\mathrm{SQ}}\left(\omega\right)$ can be recast as

\mathrm{H}_{\mathrm{SQ}}\left(\omega\right)=\sum_{n=0}^{1}E_{n}\left|n\right\rangle\left\langle n\right|=\frac{\hslash\omega}{2}\left|0\right\rangle\left\langle 0\right|-\frac{\hslash\omega}{2}\left|1\right\rangle\left\langle 1\right|\text{,}

(95)

where $E_{0}\overset{\text{def}}{=}\hslash\omega/2$ , $E_{1}\overset{\text{def}}{=}-\hslash\omega/2$ , and $\left\{\left|n\right\rangle\right\}\overset{\text{def}}{=}\left\{\left|0\right\rangle=\left|\uparrow\right\rangle\text{, }\left|1\right\rangle=\left|\downarrow\right\rangle\right\}$ . For clarity, note that $\left|1\right\rangle=\left|\downarrow\right\rangle$ ( $\left|0\right\rangle=\left|\uparrow\right\rangle$ ) denotes here the ground (excited) state corresponding to the lowest (highest) energy level with $E_{1}\overset{\text{def}}{=}-\hslash\omega/2$ ( $E_{0}\overset{\text{def}}{=}\hslash\omega/2$ ). Observe that the thermal state $\rho_{\mathrm{SQ}}$ emerging from the Hamiltonian $\mathrm{H}_{\mathrm{SQ}}$ in Eq. (95) can be written as

\rho_{\mathrm{SQ}}=\rho_{\mathrm{SQ}}\left(\beta\text{, }\omega\right)\overset{\text{def}}{=}\frac{e^{-\beta\mathrm{H}_{\mathrm{SQ}}\left(\omega\right)}}{\mathrm{tr}\left(e^{-\beta\mathrm{H}_{\mathrm{SQ}}\left(\omega\right)}\right)}\text{.}

(96)

The thermal state $\rho_{\mathrm{SQ}}\left(\beta\text{, }\omega\right)$ in Eq. (96) can be rewritten as,

$\displaystyle\rho_{\mathrm{SQ}}\left(\beta\text{, }\omega\right)$	$\displaystyle=\frac{e^{-\beta\frac{\hslash\omega}{2}\sigma_{z}}}{\mathrm{tr}\left(e^{-\beta\frac{\hslash\omega}{2}\sigma_{z}}\right)}$
	$\displaystyle=\frac{\left(\begin{array}[c]{cc}e^{-\beta\frac{\hslash\omega}{2}}&0\\ 0&e^{\beta\frac{\hslash\omega}{2}}\end{array}\right)}{e^{-\beta\frac{\hslash\omega}{2}}+e^{\beta\frac{\hslash\omega}{2}}}$	(99)
	$\displaystyle=\frac{\left(\begin{array}[c]{cc}\cosh\left(\beta\frac{\hslash\omega}{2}\right)-\sinh\left(\beta\frac{\hslash\omega}{2}\right)&0\\ 0&\cosh\left(\beta\frac{\hslash\omega}{2}\right)+\sinh\left(\beta\frac{\hslash\omega}{2}\right)\end{array}\right)}{2\cosh\left(\beta\frac{\hslash\omega}{2}\right)}$	(102)
	$\displaystyle=\frac{1}{2}\left(\begin{array}[c]{cc}1-\tanh\left(\beta\frac{\hslash\omega}{2}\right)&0\\ 0&1+\tanh\left(\beta\frac{\hslash\omega}{2}\right)\end{array}\right)$	(105)
	$\displaystyle=\frac{1}{2}\left[\mathrm{I}-\tanh\left(\beta\frac{\hslash\omega}{2}\right)\sigma_{z}\right]\text{,}$	(106)

that is,

\rho_{\mathrm{SQ}}\left(\beta\text{, }\omega\right)=\frac{1}{2}\left[\mathrm{I}-\tanh\left(\beta\frac{\hslash\omega}{2}\right)\sigma_{z}\right]\text{.}

(107)

In what follows, we shall use $\rho_{\mathrm{SQ}}\left(\beta\text{, }\omega\right)$ in Eq. (107) to calculate the Bures and the Sjöqvist metrics.

V.1.1 The Bures metric

We begin by noticing that $ds_{\mathrm{Bures}}^{2}$ in Eq. (73) becomes in our case

$\displaystyle ds_{\mathrm{Bures}}^{2}$	$\displaystyle=\frac{1}{4}\left[\left\langle\mathrm{H}^{2}\right\rangle-\left\langle\mathrm{H}\right\rangle^{2}\right]d\beta^{2}$
	$\displaystyle+\frac{1}{4}\left\{\beta^{2}\left\{\left\langle\left[\left(\partial_{\omega}\mathrm{H}\right)_{d}\right]^{2}\right\rangle-\left\langle\left(\partial_{\omega}\mathrm{H}\right)_{d}\right\rangle^{2}\right\}+2\sum_{n\neq m}\left\|\frac{\left\langle n\|\partial_{\omega}\mathrm{H}\|m\right\rangle}{E_{n}-E_{m}}\right\|^{2}\frac{\left(e^{-\beta E_{n}}-e^{-\beta E_{m}}\right)^{2}}{\mathcal{Z}\cdot\left(e^{-\beta E_{n}}+e^{-\beta E_{m}}\right)}\right\}d\omega^{2}+$
	$\displaystyle+\frac{1}{4}\left\{2\beta\left[\left\langle\mathrm{H}\left(\partial_{\omega}\mathrm{H}\right)_{d}\right\rangle-\left\langle\mathrm{H}\right\rangle\left\langle\left(\partial_{\omega}\mathrm{H}\right)_{d}\right\rangle\right]\right\}d\beta d\omega\text{,}$	(108)

where, for simplicity of notation, we denote $\mathrm{H}_{\mathrm{SQ}}\left(\omega\right)$ in Eq. (95) with $\mathrm{H}$ . To calculate $ds_{\mathrm{Bures}}^{2}$ in Eq. (73), we perform three distinct calculations. Specifically, we compute the metric tensor components $g_{\beta\beta}$ , $2g_{\beta\omega}$ , and $g_{\omega\omega}$ defined as

g_{\beta\beta}\left(\beta\text{, }\omega\right)\overset{\text{def}}{=}\frac{1}{4}\left[\left\langle\mathrm{H}^{2}\right\rangle-\left\langle\mathrm{H}\right\rangle^{2}\right]\text{, }2g_{\beta\omega}\left(\beta\text{, }\omega\right)\overset{\text{def}}{=}\frac{1}{4}\left\{2\beta\left[\left\langle\mathrm{H}\left(\partial_{\omega}\mathrm{H}\right)_{d}\right\rangle-\left\langle\mathrm{H}\right\rangle\left\langle\left(\partial_{\omega}\mathrm{H}\right)_{d}\right\rangle\right]\right\}\text{, }

(109)

and,

g_{\omega\omega}\left(\beta\text{, }\omega\right)\overset{\text{def}}{=}\frac{1}{4}\left\{\beta^{2}\left\{\left\langle\left[\left(\partial_{\omega}\mathrm{H}\right)_{d}\right]^{2}\right\rangle-\left\langle\left(\partial_{\omega}\mathrm{H}\right)_{d}\right\rangle^{2}\right\}+2\sum_{n\neq m}\left|\frac{\left\langle n|\partial_{\omega}\mathrm{H}|m\right\rangle}{E_{n}-E_{m}}\right|^{2}\frac{\left(e^{-\beta E_{n}}-e^{-\beta E_{m}}\right)^{2}}{\mathcal{Z}\cdot\left(e^{-\beta E_{n}}+e^{-\beta E_{m}}\right)}\right\}\text{,}

(110)

respectively.

First sub-calculation

Let us begin with calculating $(1/4)\left[\left\langle\mathrm{H}^{2}\right\rangle-\left\langle\mathrm{H}\right\rangle^{2}\right]d\beta^{2}$ . Observe that the expectation value $\left\langle\mathrm{H}^{2}\right\rangle$ of $\mathrm{H}^{2}$ is given by,

$\displaystyle\left\langle\mathrm{H}^{2}\right\rangle$	$\displaystyle=\mathrm{tr}\left(\mathrm{H}^{2}\rho\right)=\sum_{i=0}^{1}p_{i}E_{i}=p_{0}E_{0}+p_{1}E_{1}$
	$\displaystyle=\frac{e^{-\beta E_{0}}}{\mathcal{Z}}\left(\frac{\hslash\omega}{2}\right)^{2}+\frac{e^{-\beta E_{1}}}{\mathcal{Z}}\left(-\frac{\hslash\omega}{2}\right)^{2}$
	$\displaystyle=\frac{\hslash^{2}\omega^{2}}{4}\left(\frac{e^{-\beta\frac{\hslash\omega}{2}}}{\mathcal{Z}}+\frac{e^{\beta\frac{\hslash\omega}{2}}}{\mathcal{Z}}\right)$
	$\displaystyle=\frac{\hslash^{2}\omega^{2}}{4}\text{,}$	(111)

that is,

\left\langle\mathrm{H}^{2}\right\rangle=\frac{\hslash^{2}\omega^{2}}{4}\text{,}

(112)

where the partition function is $\mathcal{Z}\overset{\text{def}}{=}e^{-\beta\frac{\hslash\omega}{2}}+e^{\beta\frac{\hslash\omega}{2}}=2\cosh\left(\beta\frac{\hslash\omega}{2}\right)$ . Furthermore, we note that the expectation value $\left\langle\mathrm{H}\right\rangle$ of the Hamiltonian is

	$\displaystyle\left\langle\mathrm{H}\right\rangle$	$\displaystyle=\mathrm{tr}\left(\rho\mathrm{H}\right)=\sum_{i=0}^{1}p_{i}E_{i}=p_{0}E_{0}+p_{1}E_{1}=\frac{e^{-\beta E_{0}}}{\mathcal{Z}}\frac{\hslash\omega}{2}-\frac{e^{-\beta E_{1}}}{\mathcal{Z}}\frac{\hslash\omega}{2}$
		$\displaystyle=\frac{e^{-\beta E_{0}}-e^{-\beta E_{1}}}{\mathcal{Z}}\frac{\hslash\omega}{2}=\frac{e^{-\beta\frac{\hslash\omega}{2}}-e^{\beta\frac{\hslash\omega}{2}}}{e^{-\beta\frac{\hslash\omega}{2}}+e^{\beta\frac{\hslash\omega}{2}}}\frac{\hslash\omega}{2}=-\frac{2\sinh\left(\beta\frac{\hslash\omega}{2}\right)}{2\cosh\left(\beta\frac{\hslash\omega}{2}\right)}\frac{\hslash\omega}{2}=-\frac{\hslash\omega}{2}\tanh\left(\beta\frac{\hslash\omega}{2}\right)\text{,}$		(113)

that is,

\left\langle\mathrm{H}\right\rangle=-\frac{\hslash\omega}{2}\tanh\left(\beta\frac{\hslash\omega}{2}\right)\text{.}

(114)

Therefore, using Eqs. (112) and (114), $(1/4)\left[\left\langle\mathrm{H}^{2}\right\rangle-\left\langle\mathrm{H}\right\rangle^{2}\right]d\beta^{2}$ becomes

g_{\beta\beta}\left(\beta\text{, }\omega\right)d\beta^{2}\overset{\text{def}}{=}\frac{1}{4}\left[\left\langle\mathrm{H}^{2}\right\rangle-\left\langle\mathrm{H}\right\rangle^{2}\right]d\beta^{2}=\frac{\hslash^{2}}{16}\omega^{2}\left[1-\tanh^{2}\left(\beta\frac{\hslash\omega}{2}\right)\right]d\beta^{2}\text{.}

(115)

For completeness, we remark that $1-\tanh^{2}\left[\beta\left(\hslash\omega/2\right)\right]$ in Eq. (115) can also be expressed as $1$ / $\cosh^{2}\left[\beta\left(\hslash\omega/2\right)\right]$ . The calculation of $g_{\beta\beta}\left(\beta\text{, }\omega\right)$ in Eq. (115) ends our first sub-calculation.

Second sub-calculation

Let us focus on the second term in Eq. (109). We start by noting that $\left\langle\mathrm{H}\left(\partial_{\omega}\mathrm{H}\right)_{d}\right\rangle$ is given by

$\displaystyle\left\langle\mathrm{H}\left(\partial_{\omega}\mathrm{H}\right)_{d}\right\rangle$	$\displaystyle=\sum_{i=0}^{1}p_{i}E_{i}\partial_{\omega}E_{i}=p_{0}E_{0}\partial_{\omega}E_{0}+p_{1}E_{1}\partial_{\omega}E_{1}$
	$\displaystyle=p_{0}\frac{\hslash\omega}{2}\partial_{\omega}\left(\frac{\hslash\omega}{2}\right)+p_{1}\left(-\frac{\hslash\omega}{2}\right)\partial_{\omega}\left(-\frac{\hslash\omega}{2}\right)$
	$\displaystyle=\frac{\hslash^{2}}{4}\omega p_{0}+\frac{\hslash^{2}}{4}\omega p_{1}=\frac{\hslash^{2}}{4}\omega\left(p_{0}+p_{1}\right)=\frac{\hslash^{2}}{4}\omega\text{,}$	(116)

that is,

\left\langle\mathrm{H}\left(\partial_{\omega}\mathrm{H}\right)_{d}\right\rangle=\frac{\hslash^{2}}{4}\omega\text{.}

(117)

Moreover, $\left\langle\left(\partial_{\omega}\mathrm{H}\right)_{d}\right\rangle$ can be expressed as

$\displaystyle\left\langle\left(\partial_{\omega}\mathrm{H}\right)_{d}\right\rangle$	$\displaystyle=\sum_{i=0}^{1}p_{i}\partial_{\omega}E_{i}=p_{0}\partial_{\omega}E_{0}+p_{1}\partial_{\omega}E_{1}=\frac{\hslash}{2}p_{0}-\frac{\hslash}{2}p_{1}$
	$\displaystyle=\frac{\hslash}{2}\frac{e^{-\beta E_{0}}-e^{-\beta E_{1}}}{\mathcal{Z}}=\frac{\hslash}{2}\frac{e^{-\beta\frac{\hslash\omega}{2}}-e^{\beta\frac{\hslash\omega}{2}}}{e^{-\beta\frac{\hslash\omega}{2}}+e^{\beta\frac{\hslash\omega}{2}}}=-\frac{\hslash}{2}\frac{2\sinh\left(\beta\frac{\hslash\omega}{2}\right)}{2\cosh\left(\beta\frac{\hslash\omega}{2}\right)}$
	$\displaystyle=-\frac{\hslash}{2}\tanh\left(\beta\frac{\hslash\omega}{2}\right)\text{,}$	(118)

that is,

\left\langle\left(\partial_{\omega}\mathrm{H}\right)_{d}\right\rangle=-\frac{\hslash}{2}\tanh\left(\beta\frac{\hslash\omega}{2}\right)\text{.}

(119)

Therefore, using Eqs. (114), (117), and (119), we obtain

2g_{\beta\omega}\left(\beta\text{, }\omega\right)d\beta d\omega\overset{\text{def}}{=}\frac{1}{4}\left\{2\beta\left[\left\langle\mathrm{H}\left(\partial_{\omega}\mathrm{H}\right)_{d}\right\rangle-\left\langle\mathrm{H}\right\rangle\left\langle\left(\partial_{\omega}\mathrm{H}\right)_{d}\right\rangle\right]\right\}d\beta d\omega=\frac{\hslash^{2}}{8}\beta\omega\left[1-\tanh^{2}\left(\beta\frac{\hslash\omega}{2}\right)\right]d\beta d\omega\text{.}

(120)

The calculation of $2g_{\beta\omega}\left(\beta\text{, }\omega\right)$ in Eq. (120) ends our second sub-calculation.

Third sub-calculation

Let us now calculate the term in Eq. (110). Recall from Eq. (119) that $\left\langle\left(\partial_{\omega}\mathrm{H}\right)_{d}\right\rangle=-\frac{\hslash}{2}\tanh\left(\beta\frac{\hslash\omega}{2}\right)$ . Therefore, we have

\left\langle\left(\partial_{\omega}\mathrm{H}\right)_{d}\right\rangle^{2}=\frac{\hslash^{2}}{4}\tanh^{2}\left(\beta\frac{\hslash\omega}{2}\right)\text{.}

(121)

Moreover, we note that $\left\langle\left[\left(\partial_{\omega}\mathrm{H}\right)_{d}\right]^{2}\right\rangle$ can be rewritten as

$\displaystyle\left\langle\left[\left(\partial_{\omega}\mathrm{H}\right)_{d}\right]^{2}\right\rangle$	$\displaystyle=\sum_{i=0}^{1}p_{i}\left(\partial_{\omega}E_{i}\right)^{2}=p_{0}\left(\partial_{\omega}E_{0}\right)^{2}+p_{1}\left(\partial_{\omega}E_{1}\right)^{2}$
	$\displaystyle=p_{0}\left[\partial_{\omega}\left(-\frac{\hslash\omega}{2}\right)\right]^{2}+p_{1}\left[\partial_{\omega}\left(\frac{\hslash\omega}{2}\right)\right]^{2}$
	$\displaystyle=\frac{\hslash^{2}}{4}p_{0}+\frac{\hslash^{2}}{4}p_{1}=\frac{\hslash^{2}}{4}\left(p_{0}+p_{1}\right)=\frac{\hslash^{2}}{4}\text{,}$	(122)

that is,

\left\langle\left[\left(\partial_{\omega}\mathrm{H}\right)_{d}\right]^{2}\right\rangle=\frac{\hslash^{2}}{4}\text{.}

(123)

Finally, note that

	$\displaystyle 2\sum_{n\neq m}\left\|\frac{\left\langle n\|\partial_{\omega}\mathrm{H}\|m\right\rangle}{E_{n}-E_{m}}\right\|^{2}\frac{\left(e^{-\beta E_{n}}-e^{-\beta E_{m}}\right)^{2}}{\mathcal{Z}\cdot\left(e^{-\beta E_{n}}+e^{-\beta E_{m}}\right)}$	$\displaystyle=\frac{2}{\mathcal{Z}}\left\|\frac{\left\langle 0\|\partial_{\omega}\mathrm{H}\|1\right\rangle}{E_{0}-E_{1}}\right\|^{2}\frac{e^{-\beta E_{0}}-e^{-\beta E_{1}}}{e^{-\beta E_{0}}+e^{-\beta E_{1}}}+\frac{2}{\mathcal{Z}}\left\|\frac{\left\langle 1\|\partial_{\omega}\mathrm{H}\|0\right\rangle}{E_{1}-E_{0}}\right\|^{2}\frac{e^{-\beta E_{1}}-e^{-\beta E_{0}}}{e^{-\beta E_{1}}+e^{-\beta E_{0}}}$
		$\displaystyle=0\text{,}$		(124)

since $\left\langle 0|\partial_{\omega}\mathrm{H}|1\right\rangle=\left\langle 1|\partial_{\omega}\mathrm{H}|0\right\rangle=0$ as a consequence of the fact that $\mathrm{H}=\mathrm{H}_{\mathrm{SQ}}\left(\omega\right)$ in Eq. (91) is diagonal. Therefore, using Eqs. (121), (123), and (124), we finally get that Eq. (110) becomes

g_{\omega\omega}\left(\beta\text{, }\omega\right)d\omega^{2}=\frac{\hslash^{2}}{16}\beta^{2}\left[1-\tanh^{2}\left(\beta\frac{\hslash\omega}{2}\right)\right]d\omega^{2}\text{.}

(125)

The calculation of $g_{\omega\omega}\left(\beta\text{, }\omega\right)$ in Eq. (125) ends our third sub-calculation.

In conclusion, exploiting Eqs. (115), (120), and (125), the Bures metric $ds_{\mathrm{Bures}}^{2}=g_{\beta\beta}\left(\beta\text{, }\omega\right)d\beta^{2}+g_{\omega\omega}\left(\beta\text{, }\omega\right)d\omega^{2}+2g_{\beta\omega}\left(\beta\text{, }\omega\right)d\beta d\omega$ in Eq. (108) becomes

ds_{\mathrm{Bures}}^{2}=\frac{\hslash^{2}\omega^{2}}{16}\left[1-\tanh^{2}\left(\beta\frac{\hslash\omega}{2}\right)\right]d\beta^{2}+\frac{\hslash^{2}\beta^{2}}{16}\left[1-\tanh^{2}\left(\beta\frac{\hslash\omega}{2}\right)\right]d\omega^{2}+\frac{\hslash^{2}\beta\omega}{8}\left[1-\tanh^{2}\left(\beta\frac{\hslash\omega}{2}\right)\right]d\beta d\omega\text{.}

(126)

Using Einstein’s summation convention, $ds_{\mathrm{Bures}}^{2}=g_{ij}^{\left(\mathrm{Bures}\right)}\left(\beta\text{, }\omega\right)d\theta^{i}d\theta^{j}$ with $\theta^{1}\overset{\text{def}}{=}\beta$ and $\theta^{2}\overset{\text{def}}{=}\omega$ . Finally, using Eq. (126), the Bures metric metric tensor $g_{ij}^{\left(\mathrm{Bures}\right)}\left(\beta\text{, }\omega\right)$ becomes

g_{ij}^{\left(\mathrm{Bures}\right)}\left(\beta\text{, }\omega\right)=\frac{\hslash^{2}}{16}\left[1-\tanh^{2}\left(\beta\frac{\hslash\omega}{2}\right)\right]\left(\begin{array}[c]{cc}\omega^{2}&\beta\omega\\ \beta\omega&\beta^{2}\end{array}\right)\text{,}

(127)

with $1\leq i$ , $j\leq 2$ . Note that $g_{ij}^{\left(\mathrm{Bures}\right)}\left(\beta\text{, }\omega\right)$ in Eq. (127) equals the classical Fisher-Rao metric since there is no non-classical contribution in this case. The derivation of $g_{ij}^{\left(\mathrm{Bures}\right)}\left(\beta\text{, }\omega\right)$ in Eq. (127) ends our calculation of the Bures metric tensor for spin qubits. Interestingly, we observe that setting $k_{B}=1$ , $\beta=t^{-1}$ , and $\omega_{z}=t$ , our Eq. (127) reduces to the last relation obtained by Zanardi and collaborators in Ref. zanardi07 .

V.1.2 The Sjöqvist metric

Given the expression of $\rho_{\mathrm{SQ}}\left(\beta\text{, }\omega\right)$ in Eq. (107), we can proceed with the calculation of the Sjöqvist metric given by

ds_{\mathrm{Sj\ddot{o}qvist}}^{2}=\frac{1}{4}\sum_{k=0}^{1}\frac{dp_{k}^{2}}{p_{k}}+\sum_{k=0}^{1}p_{k}\left\langle dn_{k}|\left(\mathrm{I}-\left|n_{k}\right\rangle\left\langle n_{k}\right|\right)|dn_{k}\right\rangle\text{.}

(128)

In our case, we note that the probabilities $p_{0}$ and $p_{1}$ are given by

p_{0}=p_{0}\left(\beta\text{, }\omega\right)\overset{\text{def}}{=}\frac{1-\tanh\left(\beta\frac{\hslash\omega}{2}\right)}{2}\text{, and }p_{1}=p_{1}\left(\beta\text{, }\omega\right)\overset{\text{def}}{=}\frac{1+\tanh\left(\beta\frac{\hslash\omega}{2}\right)}{2}\text{,}

(129)

respectively. Furthermore, the states $\left|n_{0}\right\rangle$ and $\left|n_{1}\right\rangle$ are

\left|n_{0}\right\rangle\overset{\text{def}}{=}\left|0\right\rangle\text{, and }\left|n_{1}\right\rangle\overset{\text{def}}{=}\left|1\right\rangle\text{.}

(130)

Observe that since $n_{k}=n_{k}\left(\beta\text{, }\omega\right)$ , we have that $dn_{k}\overset{\text{def}}{=}\frac{\partial n_{k}}{\partial\beta}d\beta+\frac{\partial n_{k}}{\partial\omega}d\omega$ . In our case, we get from Eq. (130) that $\left|dn_{k}\right\rangle=\left|0\right\rangle$ . From Eq. (128), $ds_{\mathrm{Sj\ddot{o}qvist}}^{2}$ reduces to

ds_{\mathrm{Sj\ddot{o}qvist}}^{2}=\frac{1}{4}\sum_{k=0}^{1}\frac{dp_{k}^{2}}{p_{k}}=\frac{1}{4}\left(\frac{dp_{0}^{2}}{p_{0}}+\frac{dp_{1}^{2}}{p_{1}}\right)\text{,}

(131)

where the differentials $dp_{0}$ and $dp_{1}$ are given by

dp_{0}\overset{\text{def}}{=}\frac{\partial p_{0}}{\partial\beta}d\beta+\frac{\partial p_{0}}{\partial\omega}d\omega\text{, and }dp_{1}\overset{\text{def}}{=}\frac{\partial p_{1}}{\partial\beta}d\beta+\frac{\partial p_{1}}{\partial\omega}d\omega\text{,}

(132)

respectively. Therefore, substituting Eq. (132) into Eq. (131), we get

$\displaystyle ds_{\mathrm{Sj\ddot{o}qvist}}^{2}$	$\displaystyle=\frac{1}{4}\frac{\left(\partial_{\beta}p_{0}d\beta+\partial_{\omega}p_{0}d\omega\right)^{2}}{p_{0}}+\frac{1}{4}\frac{\left(\partial_{\beta}p_{1}d\beta+\partial_{\omega}p_{1}d\omega\right)^{2}}{p_{1}}$
	$\displaystyle=\frac{\left(\partial_{\beta}p_{0}\right)^{2}d\beta^{2}+\left(\partial_{\omega}p_{0}\right)^{2}d\omega^{2}+2\partial_{\beta}p_{0}\partial_{\omega}p_{0}d\beta d\omega}{4p_{0}}+\frac{\left(\partial_{\beta}p_{1}\right)^{2}d\beta^{2}+\left(\partial_{\omega}p_{1}\right)^{2}d\omega^{2}+2\partial_{\beta}p_{1}\partial_{\omega}p_{1}d\beta d\omega}{4p_{1}}$
	$\displaystyle=\left[\frac{\left(\partial_{\beta}p_{0}\right)^{2}}{4p_{0}}+\frac{\left(\partial_{\beta}p_{1}\right)^{2}}{4p_{1}}\right]d\beta^{2}+\left[\frac{\left(\partial_{\omega}p_{0}\right)^{2}}{4p_{0}}+\frac{\left(\partial_{\omega}p_{1}\right)^{2}}{4p_{1}}\right]d\omega^{2}+\left[\frac{2\partial_{\beta}p_{0}\partial_{\omega}p_{0}}{4p_{0}}+\frac{2\partial_{\beta}p_{1}\partial_{\omega}p_{1}}{4p_{1}}\right]d\beta d\omega\text{,}$	(133)

that is,

ds_{\mathrm{Sj\ddot{o}qvist}}^{2}=\left[\frac{\left(\partial_{\beta}p_{0}\right)^{2}}{4p_{0}}+\frac{\left(\partial_{\beta}p_{1}\right)^{2}}{4p_{1}}\right]d\beta^{2}+\left[\frac{\left(\partial_{\omega}p_{0}\right)^{2}}{4p_{0}}+\frac{\left(\partial_{\omega}p_{1}\right)^{2}}{4p_{1}}\right]d\omega^{2}+\left[\frac{2\partial_{\beta}p_{0}\partial_{\omega}p_{0}}{4p_{0}}+\frac{2\partial_{\beta}p_{1}\partial_{\omega}p_{1}}{4p_{1}}\right]d\beta d\omega\text{.}

(134)

From Eq. (129), we observe that

	$\displaystyle\partial_{\beta}p_{0}$	$\displaystyle=-\frac{\hslash\omega}{4}\left[1-\tanh^{2}\left(\beta\frac{\hslash\omega}{2}\right)\right]\text{, }\partial_{\omega}p_{0}=-\frac{\hslash\beta}{4}\left[1-\tanh^{2}\left(\beta\frac{\hslash\omega}{2}\right)\right]\text{,}$
	$\displaystyle\partial_{\beta}p_{1}$	$\displaystyle=\frac{\hslash\omega}{4}\left[1-\tanh^{2}\left(\beta\frac{\hslash\omega}{2}\right)\right]\text{, }\partial_{\omega}p_{1}=\frac{\hslash\beta}{4}\left[1-\tanh^{2}\left(\beta\frac{\hslash\omega}{2}\right)\right]\text{.}$		(135)

Finally, substituting Eq. (135) into Eq. (134), we obtain

ds_{\mathrm{Sj\ddot{o}qvist}}^{2}=\frac{\hslash^{2}\omega^{2}}{16}\left[1-\tanh^{2}\left(\beta\frac{\hslash\omega}{2}\right)\right]d\beta^{2}+\frac{\hslash^{2}\beta^{2}}{16}\left[1-\tanh^{2}\left(\beta\frac{\hslash\omega}{2}\right)\right]d\omega^{2}+\frac{\hslash^{2}\beta\omega}{8}\left[1-\tanh^{2}\left(\beta\frac{\hslash\omega}{2}\right)\right]d\beta d\omega\text{.}

(136)

Using Einstein’s summation convention, $ds_{\mathrm{Sj\ddot{o}qvist}}^{2}=g_{ij}^{\left(\mathrm{Sj\ddot{o}qvist}\right)}\left(\beta\text{, }\omega\right)d\theta^{i}d\theta^{j}$ with $\theta^{1}\overset{\text{def}}{=}\beta$ and $\theta^{2}\overset{\text{def}}{=}\omega$ . Finally, using Eq. (136), the Sjöqvist metric metric tensor becomes

g_{ij}^{\left(\mathrm{Sj\ddot{o}qvist}\right)}\left(\beta\text{, }\omega\right)=\frac{\hslash^{2}}{16}\left[1-\tanh^{2}\left(\beta\frac{\hslash\omega}{2}\right)\right]\left(\begin{array}[c]{cc}\omega^{2}&\beta\omega\\ \beta\omega&\beta^{2}\end{array}\right)\text{,}

(137)

with $1\leq i$ , $j\leq 2$ . Note that $g_{ij}^{\left(\mathrm{Sj\ddot{o}qvist}\right)}\left(\beta\text{, }\omega\right)$ in Eq. (127) is equal to the classical Fisher-Rao metric since the non-classical contribution is absent in this case. The derivation of $g_{ij}^{\left(\mathrm{Sj\ddot{o}qvist}\right)}\left(\beta\text{, }\omega\right)$ in Eq. (137) ends our calculation of the Sjöqvist metric tensor for spin qubits.

Recalling the general expressions of the Bures and Sjöqvist metrics in Eqs. (37) and (90) and, moreover, from our first set of explicit calculations, a few remarks are in order. First, both metrics have a classical and a non-classical contribution. Second, the classical Fisher-Rao metric contribution is related to changes $dp_{n}=\partial_{\beta}p_{n}d\beta+\partial_{h}p_{n}dh$ in the probabilities $p_{n}\left(\beta\text{, }h\right)\propto e^{-\beta E_{n}\left(h\right)}$ with $\left\{E_{n}\left(h\right)\right\}$ being the eigenvalues of the Hamiltonian. Finally, the non-classical contribution in the two metrics is linked to changes $\left|dn\right\rangle=\partial_{h}\left|n\right\rangle dh=\left|\partial_{h}n\right\rangle dh$ in the eigenvectors $\left\{\left|n\left(h\right)\right\rangle\right\}$ of the Hamiltonian. In our first Hamiltonian model, H $\propto\sigma_{z}$ is diagonal and, thus, its eigenvectors do not depend on any parameter. Therefore, we found that both the Bures and Sjöqvist metrics reduce to the classical Fisher-Rao metric. However, one expects that if H is not proportional to the Pauli matrix operator $\sigma_{z}$ , non-classical contributions do not vanish any longer and the two metrics may yield different quantum (i.e., non-classical) metric contributions. Indeed, if one considers a spin qubit Hamiltonian specified by a magnetic field with an orientation that is not constrained to be along the $z$ -axis, the Bures and Sjöqvist metrics happen to be different. In particular, for a time-independent and uniform magnetic field given by $\vec{B}=B_{x}\hat{x}+B_{z}\hat{z}$ , the spin qubit Hamiltonian becomes H ${}_{\mathrm{SQ}}\left(\omega_{x}\text{, }\omega_{z}\right)\overset{\text{def}}{=}\left(\hslash/2\right)(\omega_{x}\sigma_{x}+\omega_{z}\sigma_{z})$ . Assuming $\omega_{x}$ -fixed $\neq 0$ , tuning only the parameters $\beta$ and $\omega_{z}$ , and repeating our metric calculations, it can be shown that the Bures and Sjöqvist metric tensor components $g_{ij}^{\mathrm{Bures}}\left(\beta\text{, }\omega_{z}\right)$ and $g_{ij}^{\mathrm{Sj\ddot{o}qvist}}\left(\beta\text{, }\omega_{z}\right)$ are

g_{ij}^{\mathrm{Bures}}\left(\beta\text{, }\omega_{z}\right)=\frac{\hslash^{2}}{16}\left[1-\tanh^{2}\left(\beta\frac{\hslash\sqrt{\omega_{x}^{2}+\omega_{z}^{2}}}{2}\right)\right]\left(\begin{array}[c]{cc}\omega_{x}^{2}+\omega_{z}^{2}&\beta\omega_{z}\\ \beta\omega_{z}&\beta^{2}\frac{\omega_{z}^{2}}{\omega_{x}^{2}+\omega_{z}^{2}}+\frac{4}{\hslash^{2}}\frac{\omega_{x}^{2}}{\left(\omega_{x}^{2}+\omega_{z}^{2}\right)^{2}}\frac{\tanh^{2}\left(\beta\frac{\hslash\sqrt{\omega_{x}^{2}+\omega_{z}^{2}}}{2}\right)}{1-\tanh^{2}\left(\beta\frac{\hslash\sqrt{\omega_{x}^{2}+\omega_{z}^{2}}}{2}\right)}\end{array}\right)\text{,}

(138)

and,

g_{ij}^{\mathrm{Sj\ddot{o}qvist}}\left(\beta\text{, }\omega_{z}\right)=\frac{\hslash^{2}}{16}\left[1-\tanh^{2}\left(\beta\frac{\hslash\sqrt{\omega_{x}^{2}+\omega_{z}^{2}}}{2}\right)\right]\left(\begin{array}[c]{cc}\omega_{x}^{2}+\omega_{z}^{2}&\beta\omega_{z}\\ \beta\omega_{z}&\beta^{2}\frac{\omega_{z}^{2}}{\omega_{x}^{2}+\omega_{z}^{2}}+\frac{4}{\hslash^{2}}\frac{\omega_{x}^{2}}{\left(\omega_{x}^{2}+\omega_{z}^{2}\right)^{2}}\frac{1}{1-\tanh^{2}\left(\beta\frac{\hslash\sqrt{\omega_{x}^{2}+\omega_{z}^{2}}}{2}\right)}\end{array}\right)\text{,}

(139)

respectively. For completeness, we remark that useful calculation techniques to arrive at expressions as in Eqs. (138) and (139) will be performed in the next subsection where H ${}_{\mathrm{SQ}}\left(\omega_{x}\text{, }\omega_{z}\right)$ will be replaced by the superconducting flux qubit Hamiltonian $\mathrm{H}_{\mathrm{SFQ}}\left(\Delta\text{, }\epsilon\right)\overset{\text{def}}{=}\left(-\hslash/2\right)\left(\Delta\sigma_{x}+\epsilon\sigma_{z}\right)$ . Returning to our considerations, recall that for any $x\in\mathbb{R}$ , we have

\frac{\tanh^{2}\left(x\right)}{1-\tanh^{2}\left(x\right)}=\sinh^{2}\left(x\right)\text{, and }\frac{1}{1-\tanh^{2}\left(x\right)}=\cosh^{2}\left(x\right)\text{.}

(140)

Then, using Eqs. (138) and (139), we obtain

0\leq\frac{g_{\omega_{z}\omega_{z}}^{\mathrm{nc}\text{, {Bures}}}\left(\beta\text{, }\omega_{z}\right)}{g_{\omega_{z}\omega_{z}}^{\mathrm{nc}\text{, }\mathrm{Sj\ddot{o}qvist}}\left(\beta\text{, }\omega_{z}\right)}=\frac{\sinh^{2}\left(\beta\frac{\hslash\sqrt{\omega_{x}^{2}+\omega_{z}^{2}}}{2}\right)}{\cosh^{2}\left(\beta\frac{\hslash\sqrt{\omega_{x}^{2}+\omega_{z}^{2}}}{2}\right)}=\tanh^{2}\left(\beta\frac{\hslash\sqrt{\omega_{x}^{2}+\omega_{z}^{2}}}{2}\right)\leq 1\text{,}

(141)

with $g_{\omega_{z}\omega_{z}}^{\mathrm{nc}\text{, {Bures}}}\left(\beta\text{, }\omega_{z}\right)$ and $g_{\omega_{z}\omega_{z}}^{\mathrm{nc}\text{, }\mathrm{Sj\ddot{o}qvist}}\left(\beta\text{, }\omega_{z}\right)$ denoting the non-classical contributions in the Bures and Sjöqvist metric cases, respectively. From Eqs. (138) and (139), we conclude that the introduction of a nonvanishing component of the magnetic field along the $x$ -direction introduces a visible non-commutative probabilistic structure in the quantum mechanics of the system characterized by a non-classical scenario with $\left[\rho\text{, }\rho+d\rho\right]\neq 0$ ). In such a case, the Bures and the Sjöqvist metrics exhibit a different behavior as evident from their nonclassical metric tensor components (i.e., $g_{\omega_{z}\omega_{z}}^{\mathrm{nc}}\left(\beta\text{, }\omega_{z}\right)$ ) in Eq. (141).

V.2 Superconducting flux qubits

Let us consider a system with an Hamiltonian described by $\mathrm{H}_{\mathrm{SFQ}}\left(\Delta\text{, }\epsilon\right)\overset{\text{def}}{=}\left(-\hslash/2\right)\left(\Delta\sigma_{x}+\epsilon\sigma_{z}\right)$ in Eq. (92). The thermal state $\rho_{\mathrm{SFQ}}\left(\beta\text{, }\epsilon\right)$ corresponding to $\mathrm{H}_{\mathrm{SFQ}}\left(\Delta\text{, }\epsilon\right)$ with $\Delta$ assumed to be constant is given by

\rho_{\mathrm{SFQ}}\left(\beta\text{, }\epsilon\right)\overset{\text{def}}{=}\frac{e^{-\beta\mathrm{H}_{\mathrm{SFQ}}\left(\Delta\text{, }\epsilon\right)}}{\mathrm{tr}\left(e^{-\beta\mathrm{H}_{\mathrm{SFQ}}\left(\Delta\text{, }\epsilon\right)}\right)}\text{.}

(142)

Observe that $\mathrm{H}_{\mathrm{SFQ}}\left(\Delta\text{, }\epsilon\right)$ is diagonalizable and can be recast as $\mathrm{H}_{\mathrm{SFQ}}=M_{\mathrm{H}_{\mathrm{SFQ}}}\mathrm{H}_{\mathrm{SFQ}}^{\left(\mathrm{diagonal}\right)}M_{\mathrm{H}_{\mathrm{SFQ}}}^{-1}$ where $M_{\mathrm{H}_{\mathrm{SFQ}}}$ and $M_{\mathrm{H}_{\mathrm{SFQ}}}^{-1}$ are the eigenvector matrix and its inverse, respectively. Therefore, after some algebra, $\rho_{\mathrm{SFQ}}\left(\beta\text{, }\epsilon\right)$ in Eq. (142) can be rewritten as

$\displaystyle\rho_{\mathrm{SFQ}}\left(\beta\text{, }\epsilon\right)$	$\displaystyle=\frac{e^{-\beta\mathrm{H}_{\mathrm{SFQ}}\left(\Delta\text{, }\epsilon\right)}}{\mathrm{tr}(e^{-\beta\mathrm{H}_{\mathrm{SFQ}}\left(\Delta\text{, }\epsilon\right)})}=\frac{e^{-\beta M_{\mathrm{H}_{\mathrm{SFQ}}}\mathrm{H}_{\mathrm{SFQ}}^{\left(\mathrm{diagonal}\right)}M_{\mathrm{H}_{\mathrm{SFQ}}}^{-1}}}{\mathrm{tr}(e^{-\beta M_{\mathrm{H}_{\mathrm{SFQ}}}\mathrm{H}_{\mathrm{SFQ}}^{\left(\mathrm{diagonal}\right)}M_{\mathrm{H}_{\mathrm{SFQ}}}^{-1}})}$
	$\displaystyle=\frac{M_{\mathrm{H}_{\mathrm{SFQ}}}e^{-\beta\mathrm{H}_{\mathrm{SFQ}}^{\left(\mathrm{diagonal}\right)}}M_{\mathrm{H}_{\mathrm{SFQ}}}^{-1}}{\mathrm{tr}(M_{\mathrm{H}_{\mathrm{SFQ}}}e^{-\beta\mathrm{H}_{\mathrm{SFQ}}^{\left(\mathrm{diagonal}\right)}}M_{\mathrm{H}_{\mathrm{SFQ}}}^{-1})}$
	$\displaystyle=M_{\mathrm{H}_{\mathrm{SFQ}}}\frac{e^{-\beta\mathrm{H}_{\mathrm{SFQ}}^{\left(\mathrm{diagonal}\right)}}}{\mathrm{tr}(e^{-\beta\mathrm{H}_{\mathrm{SFQ}}^{\left(\mathrm{diagonal}\right)}})}M_{\mathrm{H}_{\mathrm{SFQ}}}^{-1}\text{,}$	(143)

that is,

\rho_{\mathrm{SFQ}}\left(\beta\text{, }\epsilon\right)=M_{\mathrm{H}_{\mathrm{SFQ}}}\frac{e^{-\beta\mathrm{H}_{\mathrm{SFQ}}^{\left(\mathrm{diagonal}\right)}}}{\mathrm{tr}(e^{-\beta\mathrm{H}_{\mathrm{SFQ}}^{\left(\mathrm{diagonal}\right)}})}M_{\mathrm{H}_{\mathrm{SFQ}}}^{-1}=M_{\mathrm{H}_{\mathrm{SFQ}}}\rho_{\mathrm{SFQ}}^{\left(\mathrm{diagonal}\right)}\left(\beta\text{, }\epsilon\right)M_{\mathrm{H}_{\mathrm{SFQ}}}^{-1}\text{.}

(144)

The quantity $\mathrm{H}_{\mathrm{SFQ}}^{\left(\mathrm{diagonal}\right)}$ in Eq. (144) is defined as,

\mathrm{H}_{\mathrm{SFQ}}^{\left(\mathrm{diagonal}\right)}\overset{\text{def}}{=}E_{0}\left|n_{1}\right\rangle\left\langle n_{1}\right|+E_{1}\left|n_{0}\right\rangle\left\langle n_{0}\right|\text{.}

(145)

The the eigenvalues $E_{0}$ and $E_{1}$ are given by $E_{0}\overset{\text{def}}{=}-\left(\hslash/2\right)\nu$ and $E_{1}\overset{\text{def}}{=}+\left(\hslash/2\right)\nu$ , respectively, with $\nu\overset{\text{def}}{=}\sqrt{\Delta^{2}+\epsilon^{2}}$ . For later use, it is convenient to introduce the notation $\tilde{E}_{0}\overset{\text{def}}{=}E_{1}$ and $\tilde{E}_{1}\overset{\text{def}}{=}E_{0}$ so that $\mathrm{H}_{\mathrm{SFQ}}^{\left(\mathrm{diagonal}\right)}\overset{\text{def}}{=}\tilde{E}_{0}\left|n_{0}\right\rangle\left\langle n_{0}\right|+\tilde{E}_{1}\left|n_{1}\right\rangle\left\langle n_{1}\right|$ . The two orthonormal eigenvectors corresponding to $E_{0}$ and $E_{1}$ are $\left|n_{1}\right\rangle$ and $\left|n_{0}\right\rangle$ , respectively. They are given by

\left|n_{0}\right\rangle\overset{\text{def}}{=}\frac{1}{\sqrt{2}}\left(\begin{array}[c]{c}\frac{\epsilon-\sqrt{\epsilon^{2}+\Delta^{2}}}{\sqrt{\epsilon^{2}+\Delta^{2}-\epsilon\sqrt{\epsilon^{2}+\Delta^{2}}}}\\ \frac{\Delta}{\sqrt{\epsilon^{2}+\Delta^{2}-\epsilon\sqrt{\epsilon^{2}+\Delta^{2}}}}\end{array}\right)\text{ and, }\left|n_{1}\right\rangle\overset{\text{def}}{=}\frac{1}{\sqrt{2}}\left(\begin{array}[c]{c}\frac{\epsilon+\sqrt{\epsilon^{2}+\Delta^{2}}}{\sqrt{\epsilon^{2}+\Delta^{2}+\epsilon\sqrt{\epsilon^{2}+\Delta^{2}}}}\\ \frac{\Delta}{\sqrt{\epsilon^{2}+\Delta^{2}+\epsilon\sqrt{\epsilon^{2}+\Delta^{2}}}}\end{array}\right)\text{,}

(146)

respectively. A suitable choice for the eigenvector matrix $M_{\mathrm{H}_{\mathrm{SFQ}}}$ and its inverse $M_{\mathrm{H}_{\mathrm{SFQ}}}^{-1}$ in Eq. (144) can be expressed as

M_{\mathrm{H}_{\mathrm{SFQ}}}\overset{\text{def}}{=}\left(\begin{array}[c]{cc}\frac{\epsilon+\nu}{\Delta}&\frac{\epsilon-\nu}{\Delta}\\ 1&1\end{array}\right)\text{ and, }M_{\mathrm{H}_{\mathrm{SFQ}}}^{-1}\overset{\text{def}}{=}\left(\begin{array}[c]{cc}\frac{\Delta}{2\nu}&\frac{\nu-\epsilon}{2\nu}\\ -\frac{\Delta}{2\nu}&\frac{\nu+\epsilon}{2\nu}\end{array}\right)\text{,}

(147)

respectively. Using Eqs. (145) and (147), $\rho_{\mathrm{SFQ}}\left(\beta\text{, }\epsilon\right)$ in Eq. (144) becomes

\rho_{\mathrm{SFQ}}\left(\beta\text{, }\epsilon\right)=\frac{1}{2}\left(\begin{array}[c]{cc}1+\frac{\epsilon}{\sqrt{\epsilon^{2}+\Delta^{2}}}\tanh\left(\beta\hslash\frac{\sqrt{\epsilon^{2}+\Delta^{2}}}{2}\right)&\frac{\Delta}{\sqrt{\epsilon^{2}+\Delta^{2}}}\tanh\left(\beta\hslash\frac{\sqrt{\epsilon^{2}+\Delta^{2}}}{2}\right)\\ \frac{\Delta}{\sqrt{\epsilon^{2}+\Delta^{2}}}\tanh\left(\beta\hslash\frac{\sqrt{\epsilon^{2}+\Delta^{2}}}{2}\right)&1-\frac{\epsilon}{\sqrt{\epsilon^{2}+\Delta^{2}}}\tanh\left(\beta\hslash\frac{\sqrt{\epsilon^{2}+\Delta^{2}}}{2}\right)\end{array}\right)\text{,}

(148)

that is,

\rho_{\mathrm{SFQ}}\left(\beta\text{, }\epsilon\right)=\frac{1}{2}\left[\mathrm{I}+\left(\frac{\Delta}{\sqrt{\epsilon^{2}+\Delta^{2}}}\sigma_{x}+\frac{\epsilon}{\sqrt{\epsilon^{2}+\Delta^{2}}}\sigma_{z}\right)\tanh\left(\beta\hslash\frac{\sqrt{\epsilon^{2}+\Delta^{2}}}{2}\right)\right]\text{.}

(149)

For completeness, we note here that the spectral decomposition of $\rho_{\mathrm{SFQ}}\left(\beta\text{, }\epsilon\right)=M_{\mathrm{H}_{\mathrm{SFQ}}}\rho_{\mathrm{SFQ}}^{\left(\mathrm{diagonal}\right)}\left(\beta\text{, }\epsilon\right)M_{\mathrm{H}_{\mathrm{SFQ}}}^{-1}$ in Eq. (149) is given by $\rho_{\mathrm{SFQ}}\overset{\text{def}}{=}p_{0}\left|n_{0}\right\rangle\left\langle n_{0}\right|+p_{1}\left|n_{1}\right\rangle\left\langle n_{1}\right|$ . The probabilities $p_{0}$ and $p_{1}$ are

p_{0}\overset{\text{def}}{=}\frac{e^{-\beta\tilde{E}_{0}}}{\mathcal{Z}}=\frac{1}{2}\left[1-\tanh\left(\beta\hslash\frac{\sqrt{\epsilon^{2}+\Delta^{2}}}{2}\right)\right]\text{, and }p_{1}\overset{\text{def}}{=}\frac{e^{-\beta\tilde{E}_{1}}}{\mathcal{Z}}=\frac{1}{2}\left[1+\tanh\left(\beta\hslash\frac{\sqrt{\epsilon^{2}+\Delta^{2}}}{2}\right)\right]\text{,}

(150)

respectively, with $\mathcal{Z}\overset{\text{def}}{\mathcal{=}}e^{-\beta E_{0}}+e^{-\beta E_{1}}=e^{-\beta\tilde{E}_{0}}+e^{-\beta\tilde{E}_{1}}=2\cosh(\beta\hslash\frac{\sqrt{\epsilon^{2}+\Delta^{2}}}{2})$ denoting the partition function of the system. In what follows, we shall use $\rho_{\mathrm{SFQ}}\left(\beta\text{, }\epsilon\right)$ in Eq. (149) to calculate the Bures and the Sjöqvist metrics.

V.2.1 The Bures metric

For simplicity of notation, we replace $\mathrm{H}_{\mathrm{SFQ}}$ with $\mathrm{H}$ in the forthcoming calculation. We begin by noting that, in our case, the general expression of the Bures metric $ds_{\mathrm{Bures}}^{2}$ in Eq. (73) becomes

$\displaystyle ds_{\mathrm{Bures}}^{2}$	$\displaystyle=\frac{1}{4}\left[\left\langle\mathrm{H}^{2}\right\rangle-\left\langle\mathrm{H}\right\rangle^{2}\right]d\beta^{2}$
	$\displaystyle+\frac{1}{4}\left\{\beta^{2}\left\{\left\langle\left[\left(\partial_{\epsilon}\mathrm{H}\right)_{d}\right]^{2}\right\rangle-\left\langle\left(\partial_{\epsilon}\mathrm{H}\right)_{d}\right\rangle^{2}\right\}+2\sum_{n\neq m}\left\|\frac{\left\langle n\|\partial_{\epsilon}\mathrm{H}\|m\right\rangle}{\tilde{E}_{n}-\tilde{E}_{m}}\right\|^{2}\frac{\left(e^{-\beta\tilde{E}_{n}}-e^{-\beta\tilde{E}_{m}}\right)^{2}}{\mathcal{Z}\cdot\left(e^{-\beta\tilde{E}_{n}}+e^{-\beta\tilde{E}_{m}}\right)}\right\}d\epsilon^{2}+$
	$\displaystyle+\frac{1}{4}\left\{2\beta\left[\left\langle\mathrm{H}\left(\partial_{\epsilon}\mathrm{H}\right)_{d}\right\rangle-\left\langle\mathrm{H}\right\rangle\left\langle\left(\partial_{\epsilon}\mathrm{H}\right)_{d}\right\rangle\right]\right\}d\beta d\epsilon\text{.}$	(151)

As previously pointed out in this manuscript, $ds_{\mathrm{Bures}}^{2}$ is the sum of two contributions, the classical Fisher-Rao information metric contribution and the non-classical metric contribution described in the summation term in the right-hand-side of Eq. (151). In what follows, we shall the that the presence of nonvanishing terms $\left|\left\langle n|\partial_{\epsilon}\mathrm{H}|m\right\rangle\right|^{2}$ leads to the existence of a non-classical contribution in $ds_{\mathrm{Bures}}^{2}$ . Following our previous line of reasoning, we partition our calculation in three parts. In particular, since $ds_{\mathrm{Bures}}^{2}=g_{\beta\beta}\left(\beta\text{, }\epsilon\right)d\beta^{2}+g_{\epsilon\epsilon}\left(\beta\text{, }\epsilon\right)d\epsilon^{2}+2g_{\beta\epsilon}\left(\beta\text{, }\epsilon\right)d\beta d\epsilon$ , we focus on computing $g_{\beta\beta}\left(\beta\text{, }\epsilon\right)$ , $2g_{\beta\epsilon}\left(\beta\text{, }\epsilon\right)$ , and $g_{\epsilon\epsilon}\left(\beta\text{, }\epsilon\right)$ .

V.2.2 First sub-calculation

We proceed here with the first sub-calculation. We recall that $g_{\beta\beta}\left(\beta\text{, }\epsilon\right)d\beta^{2}=(1/4)\left[\left\langle\mathrm{H}^{2}\right\rangle-\left\langle\mathrm{H}\right\rangle^{2}\right]d\beta^{2}$ . Note that $\left\langle\mathrm{H}^{2}\right\rangle$ and $\left\langle\mathrm{H}\right\rangle^{2}$ are given by,

\left\langle\mathrm{H}^{2}\right\rangle=\mathrm{tr}\left(\mathrm{H}^{2}\rho\right)=\frac{\hslash^{2}}{4}\left(\epsilon^{2}+\Delta^{2}\right)\text{,}

(152)

and,

\left\langle\mathrm{H}\right\rangle^{2}=\left[\mathrm{tr}\left(\mathrm{H}\rho\right)\right]^{2}=\frac{\hslash^{2}}{4}\left(\epsilon^{2}+\Delta^{2}\right)\tanh^{2}\left(\beta\hslash\frac{\sqrt{\epsilon^{2}+\Delta^{2}}}{2}\right)\text{,}

(153)

respectively. Therefore, using Eqs. (152) and (153), $g_{\beta\beta}\left(\beta\text{, }\epsilon\right)d\beta^{2}$ becomes

g_{\beta\beta}\left(\beta\text{, }\epsilon\right)d\beta^{2}=\frac{\hslash^{2}}{16}\left(\epsilon^{2}+\Delta^{2}\right)\left[1-\tanh^{2}\left(\beta\hslash\frac{\sqrt{\epsilon^{2}+\Delta^{2}}}{2}\right)\right]d\beta^{2}\text{.}

(154)

Our fist sub-calculation ends with the derivation of Eq. (154).

V.2.3 Second sub-calculation

In our second calculation, we focus on calculating the term $2g_{\beta\epsilon}\left(\beta\text{, }\epsilon\right)d\beta d\epsilon$ defined as

2g_{\beta\epsilon}\left(\beta\text{, }\epsilon\right)d\beta d\epsilon\overset{\text{def}}{=}\frac{1}{4}\left\{2\beta\left[\left\langle\mathrm{H}\left(\partial_{\epsilon}\mathrm{H}\right)_{d}\right\rangle-\left\langle\mathrm{H}\right\rangle\left\langle\left(\partial_{\epsilon}\mathrm{H}\right)_{d}\right\rangle\right]\right\}d\beta d\epsilon\text{.}

(155)

Note that $\left\langle\mathrm{H}\left(\partial_{\epsilon}\mathrm{H}\right)_{d}\right\rangle$ can be recast as

$\displaystyle\left\langle\mathrm{H}\left(\partial_{\epsilon}\mathrm{H}\right)_{d}\right\rangle$	$\displaystyle=\sum_{i=0}^{1}p_{i}\tilde{E}_{i}\partial_{\epsilon}\tilde{E}_{i}=p_{0}\tilde{E}_{0}\partial_{\epsilon}\tilde{E}_{0}+p_{1}\tilde{E}_{1}\partial_{\epsilon}\tilde{E}_{1}$
	$\displaystyle=p_{0}\left(\frac{\hslash}{2}\sqrt{\epsilon^{2}+\Delta^{2}}\right)\partial_{\epsilon}\left(\frac{\hslash}{2}\sqrt{\epsilon^{2}+\Delta^{2}}\right)+p_{1}\left(-\frac{\hslash}{2}\sqrt{\epsilon^{2}+\Delta^{2}}\right)\partial_{\epsilon}\left(-\frac{\hslash}{2}\sqrt{\epsilon^{2}+\Delta^{2}}\right)$
	$\displaystyle=\left(p_{0}+p_{1}\right)\left(\frac{\hslash}{2}\sqrt{\epsilon^{2}+\Delta^{2}}\right)\partial_{\epsilon}\left(\frac{\hslash}{2}\sqrt{\epsilon^{2}+\Delta^{2}}\right)$
	$\displaystyle=\left(\frac{\hslash}{2}\sqrt{\epsilon^{2}+\Delta^{2}}\right)\partial_{\epsilon}\left(\frac{\hslash}{2}\sqrt{\epsilon^{2}+\Delta^{2}}\right)$
	$\displaystyle=\frac{\hslash^{2}}{4}\epsilon\text{,}$	(156)

that is,

\left\langle\mathrm{H}\left(\partial_{\epsilon}\mathrm{H}\right)_{d}\right\rangle=\frac{\hslash^{2}}{4}\epsilon\text{.}

(157)

We also note that the expectation value $\left\langle\mathrm{H}\right\rangle$ of the Hamiltonian $\mathrm{H}$ equals

\left\langle\mathrm{H}\right\rangle=-\frac{\hslash}{2}\sqrt{\epsilon^{2}+\Delta^{2}}\tanh\left(\beta\hslash\frac{\sqrt{\epsilon^{2}+\Delta^{2}}}{2}\right)\text{.}

(158)

Finally, the quantity $\left\langle\left(\partial_{\epsilon}\mathrm{H}\right)_{d}\right\rangle$ can be rewritten as

$\displaystyle\left\langle\left(\partial_{\epsilon}\mathrm{H}\right)_{d}\right\rangle$	$\displaystyle=\sum_{i=0}^{1}p_{i}\partial_{\epsilon}\tilde{E}_{i}=p_{0}\partial_{\epsilon}\tilde{E}_{0}+p_{1}\partial_{\epsilon}\tilde{E}_{1}$
	$\displaystyle=p_{0}\partial_{\epsilon}\left(\frac{\hslash}{2}\sqrt{\epsilon^{2}+\Delta^{2}}\right)+p_{1}\partial_{\epsilon}\left(-\frac{\hslash}{2}\sqrt{\epsilon^{2}+\Delta^{2}}\right)$
	$\displaystyle=\frac{e^{-\beta\hslash\frac{\sqrt{\epsilon^{2}+\Delta^{2}}}{2}}}{\mathcal{Z}}\left(\frac{\hslash}{2}\right)\frac{\epsilon}{\sqrt{\epsilon^{2}+\Delta^{2}}}+\frac{e^{\beta\hslash\frac{\sqrt{\epsilon^{2}+\Delta^{2}}}{2}}}{\mathcal{Z}}\left(-\frac{\hslash}{2}\right)\frac{\epsilon}{\sqrt{\epsilon^{2}+\Delta^{2}}}$
	$\displaystyle=-\frac{\hslash}{2}\frac{2\sinh\left(\beta\hslash\frac{\sqrt{\epsilon^{2}+\Delta^{2}}}{2}\right)}{2\cosh\left(\beta\hslash\frac{\sqrt{\epsilon^{2}+\Delta^{2}}}{2}\right)}\frac{\epsilon}{\sqrt{\epsilon^{2}+\Delta^{2}}}$
	$\displaystyle=-\frac{\hslash}{2}\frac{\epsilon}{\sqrt{\epsilon^{2}+\Delta^{2}}}\tanh\left(\beta\hslash\frac{\sqrt{\epsilon^{2}+\Delta^{2}}}{2}\right)\text{,}$	(159)

that is,

\left\langle\left(\partial_{\epsilon}\mathrm{H}\right)_{d}\right\rangle=-\frac{\hslash}{2}\frac{\epsilon}{\sqrt{\epsilon^{2}+\Delta^{2}}}\tanh\left(\beta\hslash\frac{\sqrt{\epsilon^{2}+\Delta^{2}}}{2}\right)\text{.}

(160)

Finally, using Eqs. (157), (158), and (160), $2g_{\beta\epsilon}\left(\beta\text{, }\epsilon\right)d\beta d\epsilon$ in Eq. (155) becomes

$\displaystyle 2g_{\beta\epsilon}\left(\beta\text{, }\epsilon\right)d\beta d\epsilon$	$\displaystyle=\frac{1}{4}\left\{2\beta\left[\begin{array}[c]{c}\frac{\hslash^{2}}{4}\epsilon+\frac{\hslash}{2}\sqrt{\epsilon^{2}+\Delta^{2}}\tanh\left(\beta\hslash\frac{\sqrt{\epsilon^{2}+\Delta^{2}}}{2}\right)\cdot\\ \left(-\frac{\hslash}{2}\frac{\epsilon}{\sqrt{\epsilon^{2}+\Delta^{2}}}\tanh\left(\beta\hslash\frac{\sqrt{\epsilon^{2}+\Delta^{2}}}{2}\right)\right)\end{array}\right]\right\}d\beta d\epsilon$	(163)
	$\displaystyle=\frac{1}{4}\left\{2\beta\left[\frac{\hslash^{2}}{4}\epsilon-\frac{\hslash^{2}}{4}\epsilon\tanh^{2}\left(\beta\hslash\frac{\sqrt{\epsilon^{2}+\Delta^{2}}}{2}\right)\right]\right\}d\beta d\epsilon$
	$\displaystyle=\frac{\hslash^{2}}{8}\beta\epsilon\left[1-\tanh^{2}\left(\beta\hslash\frac{\sqrt{\epsilon^{2}+\Delta^{2}}}{2}\right)\right]d\beta d\epsilon\text{,}$	(164)

that is,

2g_{\beta\epsilon}\left(\beta\text{, }\epsilon\right)d\beta d\epsilon=\frac{\hslash^{2}}{8}\beta\epsilon\left[1-\tanh^{2}\left(\beta\hslash\frac{\sqrt{\epsilon^{2}+\Delta^{2}}}{2}\right)\right]d\beta d\epsilon\text{.}

(165)

Our second sub-calculation ends with the derivation of Eq. (165).

V.2.4 Third sub-calculation

In what follows, we focus on the calculation of $g_{\epsilon\epsilon}\left(\beta\text{, }\epsilon\right)d\epsilon^{2}$ ,

\ g_{\epsilon\epsilon}\left(\beta\text{, }\epsilon\right)d\epsilon^{2}\ \overset{\text{def}}{=}\frac{1}{4}\left\{\beta^{2}\left\{\left\langle\left[\left(\partial_{\epsilon}\mathrm{H}\right)_{d}\right]^{2}\right\rangle-\left\langle\left(\partial_{\epsilon}\mathrm{H}\right)_{d}\right\rangle^{2}\right\}+2\sum_{n\neq m}\left|\frac{\left\langle n|\partial_{\epsilon}\mathrm{H}|m\right\rangle}{\tilde{E}_{n}-\tilde{E}_{m}}\right|^{2}\frac{\left(e^{-\beta\tilde{E}_{n}}-e^{-\beta\tilde{E}_{m}}\right)^{2}}{\mathcal{Z}\cdot\left(e^{-\beta\tilde{E}_{n}}+e^{-\beta\tilde{E}_{m}}\right)}\right\}d\epsilon^{2}\text{.}

(166)

Let us recall that $\left\langle\left(\partial_{\epsilon}\mathrm{H}\right)_{d}\right\rangle$ is given in Eq. (160). Therefore, we get

\left\langle\left(\partial_{\epsilon}\mathrm{H}\right)_{d}\right\rangle^{2}=\frac{\hslash^{2}}{4}\frac{\epsilon^{2}}{\epsilon^{2}+\Delta^{2}}\tanh^{2}\left(\beta\hslash\frac{\sqrt{\epsilon^{2}+\Delta^{2}}}{2}\right)\text{.}

(167)

Moreover, $\left\langle\left(\partial_{\epsilon}\mathrm{H}\right)_{d}^{2}\right\rangle$ is given by

$\displaystyle\left\langle\left(\partial_{\epsilon}\mathrm{H}\right)_{d}^{2}\right\rangle$	$\displaystyle=\sum_{i=0}^{1}p_{i}\left(\partial_{\epsilon}\tilde{E}_{i}\right)^{2}=p_{0}\left(\partial_{\epsilon}\tilde{E}_{0}\right)^{2}+p_{1}\left(\partial_{\epsilon}\tilde{E}_{1}\right)^{2}$
	$\displaystyle=p_{0}\frac{\hslash^{2}}{4}\frac{\epsilon^{2}}{\epsilon^{2}+\Delta^{2}}+p_{1}\frac{\hslash^{2}}{4}\frac{\epsilon^{2}}{\epsilon^{2}+\Delta^{2}}$
	$\displaystyle=\left(p_{0}+p_{1}\right)\frac{\hslash^{2}}{4}\frac{\epsilon^{2}}{\epsilon^{2}+\Delta^{2}}$
	$\displaystyle=\frac{\hslash^{2}}{4}\frac{\epsilon^{2}}{\epsilon^{2}+\Delta^{2}}\text{,}$	(168)

that is,

\left\langle\left(\partial_{\epsilon}\mathrm{H}\right)_{d}^{2}\right\rangle=\frac{\hslash^{2}}{4}\frac{\epsilon^{2}}{\epsilon^{2}+\Delta^{2}}\text{.}

(169)

Finally, let us focus on the term in Eq. (166) given by

$\displaystyle 2\sum_{n\neq m}\left\|\frac{\left\langle n\|\partial_{\epsilon}\mathrm{H}\|m\right\rangle}{\tilde{E}_{n}-\tilde{E}_{m}}\right\|^{2}\frac{\left(e^{-\beta\tilde{E}_{n}}-e^{-\beta\tilde{E}_{m}}\right)^{2}}{\mathcal{Z}\cdot\left(e^{-\beta\tilde{E}_{n}}+e^{-\beta\tilde{E}_{m}}\right)}$	$\displaystyle=\frac{2}{\mathcal{Z}}\left\|\frac{\left\langle n_{0}\|\partial_{\epsilon}\mathrm{H}\|n_{1}\right\rangle}{\tilde{E}_{0}-\tilde{E}_{1}}\right\|^{2}\frac{\left(e^{-\beta\tilde{E}_{0}}-e^{-\beta\tilde{E}_{1}}\right)^{2}}{\left(e^{-\beta\tilde{E}_{0}}+e^{-\beta\tilde{E}_{1}}\right)}+$
	$\displaystyle+\frac{2}{\mathcal{Z}}\left\|\frac{\left\langle n_{1}\|\partial_{\epsilon}\mathrm{H}\|n_{0}\right\rangle}{\tilde{E}_{1}-\tilde{E}_{0}}\right\|^{2}\frac{\left(e^{-\beta\tilde{E}_{1}}-e^{-\beta\tilde{E}_{0}}\right)^{2}}{\left(e^{-\beta\tilde{E}_{0}}+e^{-\beta\tilde{E}_{1}}\right)}$
	$\displaystyle=\frac{2}{\mathcal{Z}}\frac{\left(e^{-\beta\tilde{E}_{0}}-e^{-\beta\tilde{E}_{1}}\right)^{2}}{\left(e^{-\beta\tilde{E}_{0}}+e^{-\beta\tilde{E}_{1}}\right)}\frac{\left(\left\|\left\langle n_{0}\|\partial_{\epsilon}\mathrm{H}\|n_{1}\right\rangle\right\|^{2}+\left\|\left\langle n_{1}\|\partial_{\epsilon}\mathrm{H}\|n_{0}\right\rangle\right\|^{2}\right)}{\left\|\tilde{E}_{0}-\tilde{E}_{1}\right\|^{2}}$
	$\displaystyle=2\frac{\left(e^{-\beta\tilde{E}_{0}}-e^{-\beta\tilde{E}_{1}}\right)^{2}}{\left(e^{-\beta\tilde{E}_{0}}+e^{-\beta\tilde{E}_{1}}\right)^{2}}\frac{\frac{\hslash^{2}}{4}\frac{\Delta^{2}}{\Delta^{2}+\epsilon^{2}}+\frac{\hslash^{2}}{4}\frac{\Delta^{2}}{\Delta^{2}+\epsilon^{2}}}{\hslash^{2}\left(\epsilon^{2}+\Delta^{2}\right)}$
	$\displaystyle=\frac{\Delta^{2}}{\left(\Delta^{2}+\epsilon^{2}\right)^{2}}\tanh^{2}\left(\beta\hslash\frac{\sqrt{\epsilon^{2}+\Delta^{2}}}{2}\right)\text{,}$	(170)

that is,

2\sum_{n\neq m}\left|\frac{\left\langle n|\partial_{\epsilon}\mathrm{H}|m\right\rangle}{\tilde{E}_{n}-\tilde{E}_{m}}\right|^{2}\frac{\left(e^{-\beta\tilde{E}_{n}}-e^{-\beta\tilde{E}_{m}}\right)^{2}}{\mathcal{Z}\cdot\left(e^{-\beta\tilde{E}_{n}}+e^{-\beta\tilde{E}_{m}}\right)}d\epsilon^{2}=\frac{\Delta^{2}}{\left(\Delta^{2}+\epsilon^{2}\right)^{2}}\tanh^{2}\left(\beta\hslash\frac{\sqrt{\epsilon^{2}+\Delta^{2}}}{2}\right)d\epsilon^{2}\text{.}

(171)

For clarity, note that $\partial_{\epsilon}\mathrm{H}$ in Eq. (171) equals $\left(-\hslash/2\right)\sigma_{z}$ in the standard computational basis $\left\{\left|0\right\rangle\text{, }\left|1\right\rangle\right\}$ . Therefore, combining Eqs. (167), (169), and (171) we get that $g_{\epsilon\epsilon}\left(\beta\text{, }\epsilon\right)d\epsilon^{2}$ in Eq. (166) equals

g_{\epsilon\epsilon}\left(\beta\text{, }\epsilon\right)d\epsilon^{2}=\frac{\hslash^{2}}{16}\left\{\frac{4\Delta^{2}}{\hslash^{2}\left(\Delta^{2}+\epsilon^{2}\right)^{2}}\tanh^{2}\left(\beta\hslash\frac{\sqrt{\epsilon^{2}+\Delta^{2}}}{2}\right)+\frac{\beta^{2}\epsilon^{2}}{\epsilon^{2}+\Delta^{2}}\left[1-\tanh^{2}\left(\beta\hslash\frac{\sqrt{\epsilon^{2}+\Delta^{2}}}{2}\right)\right]\right\}d\epsilon^{2}\text{.}

(172)

Then, using Eqs. (154), (165), and (172), $ds_{\mathrm{Bures}}^{2}$ in Eq. (151) becomes

$\displaystyle ds_{\mathrm{Bures}}^{2}$	$\displaystyle=\frac{\hslash^{2}}{16}\left(\epsilon^{2}+\Delta^{2}\right)\left[1-\tanh^{2}\left(\beta\hslash\frac{\sqrt{\epsilon^{2}+\Delta^{2}}}{2}\right)\right]d\beta^{2}+$
	$\displaystyle+\frac{\hslash^{2}}{8}\beta\epsilon\left[1-\tanh^{2}\left(\beta\hslash\frac{\sqrt{\epsilon^{2}+\Delta^{2}}}{2}\right)\right]d\beta d\epsilon+$
	$\displaystyle+\frac{\hslash^{2}}{16}\left\{\frac{4\Delta^{2}}{\hslash^{2}\left(\Delta^{2}+\epsilon^{2}\right)^{2}}\tanh^{2}\left(\beta\hslash\frac{\sqrt{\epsilon^{2}+\Delta^{2}}}{2}\right)+\frac{\beta^{2}\epsilon^{2}}{\epsilon^{2}+\Delta^{2}}\left[1-\tanh^{2}\left(\beta\hslash\frac{\sqrt{\epsilon^{2}+\Delta^{2}}}{2}\right)\right]\right\}d\epsilon^{2}\text{.}$	(173)

Finally, using Eq. (173), the Bures metric tensor in the case of a superconducting flux qubit becomes

g_{ij}^{\left(\mathrm{Bures}\right)}\left(\beta\text{, }\epsilon\right)=\frac{\hslash^{2}}{16}\left[1-\tanh^{2}\left(\beta\hslash\frac{\sqrt{\epsilon^{2}+\Delta^{2}}}{2}\right)\right]\left(\begin{array}[c]{cc}\epsilon^{2}+\Delta^{2}&\beta\epsilon\\ \beta\epsilon&\frac{\beta^{2}\epsilon^{2}}{\epsilon^{2}+\Delta^{2}}+\frac{4\Delta^{2}}{\hslash^{2}\left(\Delta^{2}+\epsilon^{2}\right)^{2}}\frac{\tanh^{2}\left(\beta\hslash\frac{\sqrt{\epsilon^{2}+\Delta^{2}}}{2}\right)}{1-\tanh^{2}\left(\beta\hslash\frac{\sqrt{\epsilon^{2}+\Delta^{2}}}{2}\right)}\end{array}\right)\text{,}

(174)

with $1\leq i$ , $j\leq 2$ . The derivation of Eqs. (173) and (174) completes our calculation of the Bures metric structure for a superconducting flux qubit.

V.2.5 The Sjöqvist metric

Let us observe that the Sjöqvist metric in Eq. (90) can be rewritten in our case as

ds_{\mathrm{Sj\ddot{o}qvist}}^{2}\overset{\text{def}}{=}\frac{1}{4}\sum_{k=0}^{1}\frac{dp_{k}^{2}}{p_{k}}+\sum_{k=0}^{1}p_{k}ds_{k}^{2}\text{,}

(175)

where $ds_{k}^{2}\overset{\text{def}}{=}\left[\left\langle dn_{k}|\left(\mathrm{I}-\left|n_{k}\right\rangle\left\langle n_{k}\right|\right)|dn_{k}\right\rangle\right]$ and $\left\langle n_{k}\left|n_{k^{\prime}}\right.\right\rangle=\delta_{kk^{\prime}}$ . From Eq. (146), the states $\left|dn_{0}\right\rangle$ and $\left|dn_{1}\right\rangle$ become

\left|dn_{0}\right\rangle\overset{\text{def}}{=}\frac{1}{\sqrt{2}}\left(\begin{array}[c]{c}\frac{1}{2}\frac{\Delta^{2}}{\sqrt{\epsilon^{2}+\Delta^{2}}}\frac{\sqrt{\epsilon^{2}+\Delta^{2}}-\epsilon}{\left(\epsilon^{2}+\Delta^{2}-\epsilon\sqrt{\epsilon^{2}+\Delta^{2}}\right)^{3/2}}d\epsilon\\ \frac{1}{2}\frac{\Delta^{2}}{\sqrt{\epsilon^{2}+\Delta^{2}}}\frac{\left(\sqrt{\epsilon^{2}+\Delta^{2}}-\epsilon\right)^{2}}{\left(\epsilon^{2}+\Delta^{2}-\epsilon\sqrt{\epsilon^{2}+\Delta^{2}}\right)^{3/2}}d\epsilon\end{array}\right)\text{, and }\left|dn_{1}\right\rangle\overset{\text{def}}{=}\frac{1}{\sqrt{2}}\left(\begin{array}[c]{c}\frac{1}{2}\frac{\Delta^{2}}{\sqrt{\epsilon^{2}+\Delta^{2}}}\frac{\epsilon+\sqrt{\epsilon^{2}+\Delta^{2}}}{\left(\epsilon^{2}+\Delta^{2}+\epsilon\sqrt{\epsilon^{2}+\Delta^{2}}\right)^{3/2}}d\epsilon\\ -\frac{1}{2}\frac{\Delta^{2}}{\sqrt{\epsilon^{2}+\Delta^{2}}}\frac{\left(\epsilon+\sqrt{\epsilon^{2}+\Delta^{2}}\right)^{2}}{\left(\epsilon^{2}+\Delta^{2}+\epsilon\sqrt{\epsilon^{2}+\Delta^{2}}\right)^{3/2}}d\epsilon\end{array}\right)\text{,}

(176)

respectively. Eqs. (146) and (176) will be used to calculate the nonclassical contribution that appears in the Sjöqvist metric in Eq. (175). In what follows, however, let us consider the classical contribution $\left[ds_{\mathrm{Sj\ddot{o}qvist}}^{2}\right]^{\left(\mathrm{classical}\right)}$ in Eq. (175). We note that $\left[ds_{\mathrm{Sj\ddot{o}qvist}}^{2}\right]^{\left(\mathrm{classical}\right)}$ equals

	$\displaystyle\frac{1}{4}\sum_{k=0}^{1}\frac{dp_{k}^{2}}{p_{k}}$	$\displaystyle=\frac{1}{4}\frac{dp_{0}^{2}}{p_{0}}+\frac{1}{4}\frac{dp_{1}^{2}}{p_{1}}$
		$\displaystyle=\left[\frac{\left(\partial_{\beta}p_{0}\right)^{2}}{4p_{0}}+\frac{\left(\partial_{\beta}p_{1}\right)^{2}}{4p_{1}}\right]d\beta^{2}+\left[\frac{\left(\partial_{\epsilon}p_{0}\right)^{2}}{4p_{0}}+\frac{\left(\partial_{\epsilon}p_{1}\right)^{2}}{4p_{1}}\right]d\epsilon^{2}+\left[\frac{2\partial_{\beta}p_{0}\partial_{\epsilon}p_{0}}{4p_{0}}+\frac{2\partial_{\beta}p_{1}\partial_{\epsilon}p_{1}}{4p_{1}}\right]d\beta d\epsilon\text{.}$		(177)

Using Eq. (150), $\left[ds_{\mathrm{Sj\ddot{o}qvist}}^{2}\right]^{\left(\mathrm{classical}\right)}$ in Eq. (177) reduces to

$\displaystyle\left[ds_{\mathrm{Sj\ddot{o}qvist}}^{2}\right]^{\left(\mathrm{classical}\right)}$	$\displaystyle=\frac{1}{4}\sum_{k=0}^{1}\frac{dp_{k}^{2}}{p_{k}}$
	$\displaystyle=\frac{\hslash^{2}}{16}\left(\epsilon^{2}+\Delta^{2}\right)\left[1-\tanh^{2}\left(\beta\hslash\frac{\sqrt{\epsilon^{2}+\Delta^{2}}}{2}\right)\right]d\beta^{2}+$
	$\displaystyle+\frac{\hslash^{2}}{16}\frac{\beta^{2}\epsilon^{2}}{\epsilon^{2}+\Delta^{2}}\left[1-\tanh^{2}\left(\beta\hslash\frac{\sqrt{\epsilon^{2}+\Delta^{2}}}{2}\right)\right]d\epsilon^{2}+$
	$\displaystyle+\frac{\hslash^{2}}{8}\beta\epsilon\left[1-\tanh^{2}\left(\beta\hslash\frac{\sqrt{\epsilon^{2}+\Delta^{2}}}{2}\right)\right]d\beta d\epsilon\text{.}$	(178)

We can now return our focus on the nonclassical contribution $\left[ds_{\mathrm{Sj\ddot{o}qvist}}^{2}\right]^{\left(\text{{quantum}}\right)}$ that specifies the Sjöqvist metric. We have

	$\displaystyle\left[ds_{\mathrm{Sj\ddot{o}qvist}}^{2}\right]^{\left(\text{{quantum}}\right)}$	$\displaystyle=\sum_{k=0}^{1}p_{k}\left\langle dn_{k}\|\left(\mathrm{I}-\left\|n_{k}\right\rangle\left\langle n_{k}\right\|\right)\|dn_{k}\right\rangle$
		$\displaystyle=p_{0}\left\langle dn_{0}\|dn_{0}\right\rangle-p_{0}\left\|\left\langle dn_{0}\|n_{0}\right\rangle\right\|^{2}+p_{1}\left\langle dn_{1}\|dn_{1}\right\rangle-p_{1}\left\|\left\langle dn_{1}\|n_{1}\right\rangle\right\|^{2}\text{.}$		(179)

A simple check allows us to verify that $\left\langle dn_{0}|n_{0}\right\rangle=0$ and $\left\langle dn_{1}|n_{1}\right\rangle=0$ . Therefore, $\left[ds_{\mathrm{Sj\ddot{o}qvist}}^{2}\right]^{\left(\text{{quantum}}\right)}$ becomes

$\displaystyle\left[ds_{\mathrm{Sj\ddot{o}qvist}}^{2}\right]^{\left(\text{{quantum}}\right)}$	$\displaystyle=p_{0}\left\langle dn_{0}\|dn_{0}\right\rangle+p_{1}\left\langle dn_{1}\|dn_{1}\right\rangle$
	$\displaystyle=p_{0}\frac{1}{8}\frac{\Delta^{2}}{\Delta^{2}+\epsilon^{2}}\left(\epsilon-\sqrt{\Delta^{2}+\epsilon^{2}}\right)^{2}\frac{2\Delta^{2}+2\epsilon^{2}-2\epsilon\sqrt{\Delta^{2}+\epsilon^{2}}}{\left(\Delta^{2}+\epsilon^{2}-\epsilon\sqrt{\Delta^{2}+\epsilon^{2}}\right)^{3}}d\epsilon^{2}+$
	$\displaystyle+p_{1}\frac{1}{8}\frac{\Delta^{2}}{\Delta^{2}+\epsilon^{2}}\left(\epsilon+\sqrt{\Delta^{2}+\epsilon^{2}}\right)^{2}\frac{2\Delta^{2}+2\epsilon^{2}+2\epsilon\sqrt{\Delta^{2}+\epsilon^{2}}}{\left(\Delta^{2}+\epsilon^{2}+\epsilon\sqrt{\Delta^{2}+\epsilon^{2}}\right)^{3}}d\epsilon^{2}$
	$\displaystyle=\frac{1}{4}\frac{\Delta^{2}}{\left(\Delta^{2}+\epsilon^{2}\right)^{2}}d\epsilon^{2}$
	$\displaystyle=\frac{\hslash^{2}}{16}\frac{4\Delta^{2}}{\hslash^{2}\left(\Delta^{2}+\epsilon^{2}\right)^{2}}d\epsilon^{2}\text{,}$	(180)

that is,

\left[ds_{\mathrm{Sj\ddot{o}qvist}}^{2}\right]^{\left(\text{{quantum}}\right)}=\frac{\hslash^{2}}{16}\frac{4\Delta^{2}}{\hslash^{2}\left(\Delta^{2}+\epsilon^{2}\right)^{2}}d\epsilon^{2}\text{.}

(181)

Finally, combining Eqs. (178) and (181), the Sjöqvist metric $ds_{\mathrm{Sj\ddot{o}qvist}}^{2}$ becomes

$\displaystyle ds_{\mathrm{Sj\ddot{o}qvist}}^{2}$	$\displaystyle=\frac{\hslash^{2}}{16}\left(\epsilon^{2}+\Delta^{2}\right)\left[1-\tanh^{2}\left(\beta\hslash\frac{\sqrt{\epsilon^{2}+\Delta^{2}}}{2}\right)\right]d\beta^{2}+$
	$\displaystyle+\frac{\hslash^{2}}{8}\beta\epsilon\left[1-\tanh^{2}\left(\beta\hslash\frac{\sqrt{\epsilon^{2}+\Delta^{2}}}{2}\right)\right]d\beta d\epsilon+$
	$\displaystyle+\frac{\hslash^{2}}{16}\left\{\frac{4\Delta^{2}}{\hslash^{2}\left(\Delta^{2}+\epsilon^{2}\right)^{2}}+\frac{\beta^{2}\epsilon^{2}}{\epsilon^{2}+\Delta^{2}}\left[1-\tanh^{2}\left(\beta\hslash\frac{\sqrt{\epsilon^{2}+\Delta^{2}}}{2}\right)\right]\right\}d\epsilon^{2}\text{.}$	(182)

The metric tensor $g_{ij}^{\left(\mathrm{Sj\ddot{o}qvist}\right)}\left(\beta\text{, }\epsilon\right)$ from Eq. (182) is given by

g_{ij}^{\left(\mathrm{Sj\ddot{o}qvist}\right)}\left(\beta\text{, }\epsilon\right)=\frac{\hslash^{2}}{16}\left[1-\tanh^{2}\left(\beta\hslash\frac{\sqrt{\epsilon^{2}+\Delta^{2}}}{2}\right)\right]\left(\begin{array}[c]{cc}\epsilon^{2}+\Delta^{2}&\beta\epsilon\\ \beta\epsilon&\frac{\beta^{2}\epsilon^{2}}{\epsilon^{2}+\Delta^{2}}+\frac{4\Delta^{2}}{\hslash^{2}\left(\Delta^{2}+\epsilon^{2}\right)^{2}}\frac{1}{1-\tanh^{2}\left(\beta\hslash\frac{\sqrt{\epsilon^{2}+\Delta^{2}}}{2}\right)}\end{array}\right)\text{,}

(183)

with $1\leq i$ , $j\leq 2$ . The derivation of Eqs. (182) and (183) completes our calculation of the Sjöqvist metric structure for superconducting flux qubits.

VI Considerations from the comparative analysis

In this section, we discuss the outcomes of the comparative analysis carried out in Section V concerning the Bures and Sjöqvist metrics for spin qubits and superconducting flux qubits Hamiltonian models presented in Section IV.

VI.1 The asymptotic limit of $\beta$ approaching infinity

We begin by discussing the asymptotic limit of $\beta$ approaching infinity. In the case of a spin qubit with Hamiltonian H ${}_{\mathrm{SQ}}\left(\omega\right)$ in Eq. (91), the density matrix $\rho_{\mathrm{SQ}}\left(\beta\text{, }\omega\right)$ in Eq. (107) approaches $\rho_{\mathrm{SQ}}^{\beta\rightarrow\infty}\left(\omega\right)\overset{\text{def}}{=}\left|1\right\rangle\left\langle 1\right|$ as $\beta\rightarrow\infty$ . Observe that $\left|1\right\rangle$ denotes here the ground state, the state of lowest energy $-\hslash\omega/2$ . Since $\rho_{\mathrm{SQ}}^{\beta\rightarrow\infty}\left(\omega\right)$ is a constant in $\omega$ , the Bures and Sjöqvist metrics in Eqs. (126) and (136), respectively, both vanish in this limiting scenario. In this regard, the case of the superconducting flux qubit specified by the Hamiltonian H ${}_{\mathrm{SFQ}}\left(\Delta\text{, }\epsilon\right)$ in Eq. (92) is more interesting. Indeed, in this case the density matrix $\rho_{\mathrm{SFQ}}\left(\beta\text{, }\epsilon\right)$ in Eq. (149) approaches $\rho_{\mathrm{SFQ}}^{\beta\rightarrow\infty}\left(\epsilon\right)$ when $\beta$ approaches infinity. The quantity $\rho_{\mathrm{SFQ}}^{\beta\rightarrow\infty}\left(\epsilon\right)$ represents a pure (ground) state of lowest energy $\left(-\hslash/2\right)\sqrt{\Delta^{2}+\epsilon^{2}}$ and is given by

\rho_{\mathrm{SFQ}}^{\beta\rightarrow\infty}\left(\epsilon\right)=\frac{1}{2}\left(\begin{array}[c]{cc}1+\frac{\epsilon}{\sqrt{\epsilon^{2}+\Delta^{2}}}&\frac{\Delta}{\sqrt{\epsilon^{2}+\Delta^{2}}}\\ \frac{\Delta}{\sqrt{\epsilon^{2}+\Delta^{2}}}&1-\frac{\epsilon}{\sqrt{\epsilon^{2}+\Delta^{2}}}\end{array}\right)\text{,}

(184)

with $\rho_{\mathrm{SFQ}}^{\beta\rightarrow\infty}\left(\epsilon\right)=(\rho_{\mathrm{SFQ}}^{\beta\rightarrow\infty}\left(\epsilon\right))^{2}$ and tr $\left(\rho_{\mathrm{SFQ}}^{\beta\rightarrow\infty}\left(\epsilon\right)\right)=1$ . Furthermore, when $\beta\rightarrow\infty$ , the Bures and Sjöqvist metrics in Eqs. (173) and (182), respectively, reduce to the same expression

ds_{\mathrm{Bures}}^{2}\overset{\beta\rightarrow\infty}{\rightarrow}\frac{1}{4}\frac{\Delta^{2}}{\left(\Delta^{2}+\epsilon^{2}\right)^{2}}d\epsilon^{2}\text{, and }ds_{\mathrm{Sj\ddot{o}qvist}}^{2}\overset{\beta\rightarrow\infty}{\rightarrow}\frac{1}{4}\frac{\Delta^{2}}{\left(\Delta^{2}+\epsilon^{2}\right)^{2}}d\epsilon^{2}\text{.}

(185)

The limiting expressions assumed by the Bures and Sjöqvist metrics in Eq. (185) are, modulo an unimportant constant factor, the Fubini-Study metric $ds_{\mathrm{FS}}^{2}$ for pure states. Indeed, we have

ds_{\mathrm{FS}}^{2}\overset{\text{def}}{=}\mathrm{tr}\left[\left(\frac{\partial\rho_{\mathrm{SFQ}}^{\beta\rightarrow\infty}\left(\epsilon\right)}{\partial\epsilon}\right)^{2}\right]d\epsilon^{2}=\frac{1}{2}\frac{\Delta^{2}}{\left(\Delta^{2}+\epsilon^{2}\right)^{2}}d\epsilon^{2}\text{.}

(186)

In the next subsection, we discuss the discrepancy in the Bures (Eqs. (126) and (173)) and Sjöqvist (Eqs. (136) and (182)) metrics emerging from the different nature of the nonclassical contributions in the two metrics.

VI.2 The metrics discrepancy

We begin by noting that in the case of the spin qubit Hamiltonian model in Eq. (91), there is no discrepancy since the Bures and the Sjöqvist metrics in Eqs. (126) and (136), respectively, coincide. Indeed, in this case, both metrics reduce to the classical Fisher-Rao information metric. The nonclassical/quantum terms vanish in both metrics due to the commutativity of $\rho_{\mathrm{SQ}}\left(\beta\text{, }\omega\right)$ and $\left(\rho_{\mathrm{SQ}}+d\rho_{\mathrm{SQ}}\right)\left(\beta\text{, }\omega\right)$ , with $\rho_{\mathrm{SQ}}\left(\beta\text{, }\omega\right)$ in Eq. (107). In the case of the superconducting flux qubit Hamiltonian model in Eq. (92), instead, the nonclassical/quantum terms vanish in neither the Bures nor the Sjöqvist metrics due to the non-commutativity of $\rho_{\mathrm{SFQ}}\left(\beta\text{, }\epsilon\right)$ and $\left(\rho_{\mathrm{SFQ}}+d\rho_{\mathrm{SFQ}}\right)\left(\beta\text{, }\epsilon\right)$ , with $\rho_{\mathrm{SQ}}\left(\beta\text{, }\epsilon\right)$ in Eq. (149). However, these nonclassical contributions differ in the two metrics and this leads to a discrepancy in the Bures and Sjöqvist metrics in Eqs. (173) and (182), respectively. More specifically, we have

ds_{\mathrm{Sj\ddot{o}qvist}}^{2}-ds_{\mathrm{Bures}}^{2}=\Delta g_{\epsilon\epsilon}^{\mathrm{nc}}\left(\beta\text{, }\epsilon\right)d\epsilon^{2}\geq 0\text{,}

(187)

for any $\beta$ and $\epsilon$ . Note that $\Delta g_{\epsilon\epsilon}^{\mathrm{nc}}\left(\beta\text{, }\epsilon\right)\overset{\text{def}}{=}g_{\epsilon\epsilon}^{\mathrm{nc,}\text{ }\mathrm{Sj\ddot{o}qvist}}\left(\beta\text{, }\epsilon\right)-g_{\epsilon\epsilon}^{\mathrm{nc,}\text{ }\mathrm{Bures}}\left(\beta\text{, }\epsilon\right)$ is the difference between the nonclassical (nc) contributions in the metric components $g_{\epsilon\epsilon}\left(\beta\text{, }\epsilon\right)\overset{\text{def}}{=}g_{\epsilon\epsilon}^{\mathrm{c}}\left(\beta\text{, }\epsilon\right)+g_{\epsilon\epsilon}^{\mathrm{nc}}\left(\beta\text{, }\epsilon\right)$ and is given by

\Delta g_{\epsilon\epsilon}^{\mathrm{nc}}\left(\beta\text{, }\epsilon\right)\overset{\text{def}}{=}\frac{1}{4}\frac{\Delta^{2}}{\left(\Delta^{2}+\epsilon^{2}\right)^{2}}\left[1-\tanh^{2}\left(\beta\hslash\frac{\sqrt{\epsilon^{2}+\Delta^{2}}}{2}\right)\right]\text{,}

(188)

with $0\leq$ $\tanh^{2}\left(x\right)\leq 1$ for any $x\in\mathbb{R}$ . To be crystal clear, it is useful to view the metric tensor $g_{ij}\left(\beta\text{, }\epsilon\right)$ with $1\leq i$ , $j\leq 2$ (i.e., $\theta^{1}\overset{\text{def}}{=}\beta$ and $\theta^{2}\overset{\text{def}}{=}\epsilon$ ) recast as

g_{ij}\left(\beta\text{, }\epsilon\right)=\left(\begin{array}[c]{cc}g_{\beta\beta}\left(\beta\text{, }\epsilon\right)&g_{\beta\epsilon}\left(\beta\text{, }\epsilon\right)\\ g_{\epsilon\beta}\left(\beta\text{, }\epsilon\right)&g_{\epsilon\epsilon}^{\mathrm{c}}\left(\beta\text{, }\epsilon\right)+g_{\epsilon\epsilon}^{\mathrm{nc}}\left(\beta\text{, }\epsilon\right)\end{array}\right)\text{.}

(189)

The discrepancy between the Bures and Sjöqvist metrics arises only because $g_{\epsilon\epsilon}^{\mathrm{nc,}\text{ }\mathrm{Sj\ddot{o}qvist}}\left(\beta\text{, }\epsilon\right)\neq g_{\epsilon\epsilon}^{\mathrm{nc,}\text{ }\mathrm{Bures}}\left(\beta\text{, }\epsilon\right)$ . However, the metric discrepancy $\Delta g_{\epsilon\epsilon}^{\mathrm{nc}}\left(\beta\text{, }\epsilon\right)$ in Eq. (188) vanishes in the asymptotic limit of $\beta$ approaching infinity. In Fig. $1$ , we plot the discrepancy between the Bures and the Sjöqvist metrics for the superconducting flux qubit Hamiltonian model. In Table I, instead, we summarize the links between the Bures and the Sjöqvist metrics.

Description of quantum states	Quantum states	Bures metric	Sjöqvist metric
Pure	$\rho=\rho^{2}$	Fubini-Study metric	Fubini-Study metric
Mixed, classical scenario	$\rho\neq\rho^{2}$ , $\left[\rho\text{, }\rho+d\rho\right]=0$	Fisher-Rao metric	Fisher-Rao metric
Mixed, nonclassical scenario	$\rho\neq\rho^{2}$ , $\left[\rho\text{, }\rho+d\rho\right]\neq 0$	$ds_{\mathrm{Bures}}^{2}\neq ds_{\mathrm{Sj\ddot{o}qvist}}^{2}$	$ds_{\mathrm{Sj\ddot{o}qvist}}^{2}\neq ds_{\mathrm{Bures}}^{2}$

Table 1: Relation between the Bures and the Sjöqvist metrics. The Bures and the Sjöqvist metrics are identical when considering pure quantum states

\left(\rho=\rho^{2}\right)

or mixed quantum states

\left(\rho\neq\rho^{2}\right)

for which the non-commutative probabilistic structure underlying quantum theory is not visible (i.e., in the classical scenario with

\left[\rho\text{, }\rho+d\rho\right]=0

). In particular, in the former and latter cases, they becomes the Fubini-Study and the Fisher-Rao information metrics, respectively. However, the Bures and the Sjöqvist metrics are generally distinct when considering mixed quantum states

\left(\rho\neq\rho^{2}\right)

for which the non-commutative probabilistic structure of quantum mechanics is visible (i.e., in the nonclassical scenario with

\left[\rho\text{, }\rho+d\rho\right]\neq 0

VII Conclusive Remarks

In this paper, building on our recent scientific effort in Ref. cafaroprd22 , we presented an explicit analysis of the Bures and Sjöqvist metrics over the manifolds of thermal states for the spin qubit (Eq. (91)) and the superconducting flux qubit Hamiltonian (Eq. (92)) models. We observed that while both metrics (Eqs. (126) and (136)) reduce to the Fubini-Study metric in the (zero-temperature) asymptotic limiting case of the inverse temperature $\beta$ approaching infinity for both Hamiltonian models, the two metrics are generally different. We observed this different behavior in the case of the superconducting flux Hamiltonian model. In general, we note that the two metrics (Eqs. (173) and (182)) seem to differ when nonclassical behavior is present since they quantify noncommutativity of neighboring mixed quantum states in different manners (Eqs. (187) and (188)). In summary, we reach (see Table I) the conclusion that for pure quantum states $\left(\rho=\rho^{2}\right)$ and for mixed quantum states $\left(\rho\neq\rho^{2}\right)$ for which the non-commutative probabilistic structure underlying quantum theory is not visible (i.e., in the classical scenario with $\left[\rho\text{, }\rho+d\rho\right]=0$ ), the Bures and the Sjöqvist metrics are the same (Eqs. (127) and (137)). Indeed, in the former and latter cases, they reduce to the Fubini-Study and Fisher-Rao information metrics, respectively. Instead, when investigating mixed quantum states $\left(\rho\neq\rho^{2}\right)$ for which the non-commutative probabilistic structure of quantum mechanics is visible (i.e., in the non-classical scenario with $\left[\rho\text{, }\rho+d\rho\right]\neq 0$ ), the Bures and the Sjöqvist metrics exhibit a different behavior (Eqs. (138) and (139); Eqs. (174) and (183)).

Our main conclusions can be outlined as follows:

[i]

We presented an explicit derivation of Bures metric for arbitrary density matrices in Hübner’s form (Eq. (24)) and in Zanardi’s form (Eq. (37)). Moreover, we presented a clear derivation of Zanardi’s form of the Bures metric suitable for the special class of thermal states (Eq. (73)). Finally, we reported an explicit derivation of the Sjöqvist metric for nondegenerate density matrices (Eq. (90)).
[ii]

Using our explicit derivations outlined in [i], we performed detailed analytical calculations yielding the expressions of the Bures (Eqs. (126) and (173)) and Sjöqvist (Eqs. (136) and (182)) metrics on manifolds of thermal states (Eqs. (107) and (149)) that correspond to a spin qubit (Eq. (91)) and a superconducting flux qubit (Eq. (92)) Hamiltonian models.
[iii]

In the absence of nonclassical features in which the neighboring density matrices $\rho$ and $d\rho$ commute, the Bures and the Sjöqvist metrics lead to and identical metric expression exemplified by the classical Fisher-Rao metric tensor. We have explicitly verified this similarity in the case of a manifold of thermal states for spin qubits in the presence of a constant magnetic field along the quantization $z$ -axis.
[iv]

In general, the Bures and the Sjöqvist metrics are expected to yield different expressions. Indeed, the Bures and Sjöqvist metrics seem to quantify the noncommutativity of neighboring mixed states $\rho$ and $d\rho$ in different manners, in general. We have explicitly verified this difference in the case of a manifold of thermal states for superconducting flux qubits (see Fig. $2$ ).
[v]

In the asymptotic limit of $\beta\rightarrow\infty$ , both Bures and Sjöqvist metric tensors reduce to the same limiting value (Eq. (185)) specified by, modulo an unimportant constant factor, the Fubini-Study metric tensor (Eq. (186)) for the zero-temperature pure states.
[vi]

In the superconducting flux qubit Hamiltonian model considered here, we observe that the difference $ds_{\mathrm{Sj\ddot{o}qvist}}^{2}-ds_{\mathrm{Bures}}^{2}$ is a positive quantity that depends solely on the difference between the nonclassical nature of the metric tensor component $g_{\epsilon\epsilon}\left(\beta\text{, }\epsilon\right)$ . Indeed, we have shown that $ds_{\mathrm{Sj\ddot{o}qvist}}^{2}-ds_{\mathrm{Bures}}^{2}=\Delta g_{\epsilon\epsilon}^{\mathrm{nc}}\left(\beta\text{, }\epsilon\right)d\epsilon^{2}\geq 0$ (Eqs. (187) and (188)).
[vii]

The existence of nonclassical contributions in the Bures and Sjöqvist metrics is related to the presence of non-vanishing quadratic terms like $\left|\left\langle n\left|\partial_{h}\mathrm{H}\right|m\right\rangle\right|^{2}$ and $\left|\left\langle dn\left|n\right.\right\rangle\right|^{2}$ , respectively. The former term is related to modulus squared of suitable quantum overlaps defined in terms of parametric variations in the Hamiltonian of the system. The latter term, instead, is specified by the modulus squared of suitable quantum overlaps characterized by parametric variations of the eigenstates of the Hamiltonian of the system. It is not unreasonable to expect a formal connection between these two types of terms causing the noncommutativity between $\rho$ and $\rho+d\rho$ (see, for instance, Eq. (15.30) in Ref. karol06 ) and find a deeper relation between the Bures and Sjöqvist metrics for the class of thermal quantum states. Indeed, for a more quantitative discussion on the link between these two terms, see Ref. alsing23 .
[viii]

The differential $d\rho\left(\beta\text{, }h\right)\overset{\text{def}}{=}\partial_{\beta}\rho d\beta+\partial_{h}\rho dh$ depends both on eigenvalues and eigenvectors parametric variations. However, the noncommutativity between $\rho$ and $d\rho$ is related to that part of $d\rho$ that emerges from the eigenvectors parametric variations. These changes, in turn, can be related to the existence of a nonvanishing commutator between the Hamiltonian of the system and the density matrix specifying the thermal state. Indeed, in the two main examples studied in this paper, we have $\left[\mathrm{H}_{\mathrm{SQ}}\left(\omega\right)\text{, }\rho_{\mathrm{SQ}}\left(\beta\text{, }\omega\right)\right]=0$ and $\left[\mathrm{H}_{\mathrm{SFQ}}\left(\Delta\text{, }\epsilon\right)\text{, }\rho_{\mathrm{SFQ}}\left(\beta\text{, }\epsilon\right)\right]\neq 0$ , respectively. In the former case, unlike the latter case, there is no contribution to $d\rho$ arising from a variation in the eigenvectors of the Hamiltonian.

For the set of pure states, the scenario is rather unambiguous. The Fubini–Study metric represents the only natural option for a measure that characterizes “random states”. Alternatively, for mixed-state density matrices, the geometry of the state space is more complicated karol06 ; brody19 . There is a collection of distinct metrics that can be used, each of them with different physical inspiration, benefits and disadvantages that can rest on the peculiar application one might be interested in examining. Specifically, both simple and complicated geometric quantities (i.e., for instance, path, path length, volume, curvature, and complexity) seem to depend on the measure selected on the space of mixed states that specify the quantum system being investigated karol99 ; cafaroprd22 . Therefore, our work in this paper can be particularly important in offering an explicit comparative study between the (emerging) Sjöqvist interferometric geometry and the (established) Bures geometry for mixed quantum states. Gladly, the importance of the existence of this kind of comparative investigation was lately emphasized in Refs. mera22 and cafaroprd22 too.

From a mathematics standpoint, it would be interesting to formally prove (or, alternatively, disprove with an explicit counterexample) the monotonicity petz96a ; petz99 of the Sjöqvist metric in an arbitrarily finite-dimensional space of mixed quantum states. From a physics perspective that relies on empirical evidence, instead, it would be very important to understand what the physical significance of employing either metric is mera22 ; cafaroprd22 .

In conclusion, despite its relatively limited scope, we hope this work will inspire either young or senior scientists to pursue ways to deepen our understanding (both mathematically and physically) of this fascinating connection among information geometry, statistical physics, and quantum mechanics cafaropre20 ; gassner21 ; hasegawa21 ; miller20 ; cc ; saito20 ; ito20 .

Acknowledgements.

C.C. is grateful to the United States Air Force Research Laboratory (AFRL) Summer Faculty Fellowship Program for providing support for this work. C. C. acknowledges helpful discussions with Orlando Luongo, Cosmo Lupo, Stefano Mancini, and Hernando Quevedo. P.M.A. acknowledges support from the Air Force Office of Scientific Research (AFOSR). Any opinions, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the Air Force Research Laboratory (AFRL).

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$\displaystyle\left\langle i\left\|\dot{A}\left(0\right)A\left(0\right)+A\left(0\right)\dot{A}\left(0\right)\right\|j\right\rangle$	$\displaystyle=\left\langle i\left\|\dot{A}\left(0\right)\rho+\rho\dot{A}\left(0\right)\right\|j\right\rangle$
	$\displaystyle={\displaystyle\sum\limits_{k}}\lambda_{k}\left\langle i\left\|\dot{A}\left(0\right)\right\|k\right\rangle\left\langle k\left\|j\right.\right\rangle+{\displaystyle\sum\limits_{k}}\lambda_{k}\left\langle i\left\|k\right.\right\rangle\left\langle k\left\|\dot{A}\left(0\right)\right\|j\right\rangle$
	$\displaystyle=\lambda_{j}\left\langle i\left\|\dot{A}\left(0\right)\right\|j\right\rangle+\lambda_{i}\left\langle i\left\|\dot{A}\left(0\right)\right\|j\right\rangle$
	$\displaystyle=\left(\lambda_{i}+\lambda_{j}\right)\left\langle i\left\|\dot{A}\left(0\right)\right\|j\right\rangle\text{,}$	(18)

$\displaystyle\left\langle i\left\|\rho^{1/2}d\rho\rho^{1/2}\right\|j\right\rangle$	$\displaystyle=\left\langle i\left\|\left(\sum_{k}\sqrt{\lambda_{k}}\left\|k\right\rangle\left\langle k\right\|\right)d\rho\left(\sum_{m}\sqrt{\lambda_{m}}\left\|m\right\rangle\left\langle m\right\|\right)\right\|\right\rangle$
	$\displaystyle=\sum_{k,m}\sqrt{\lambda_{k}}\sqrt{\lambda_{m}}\left\langle i\left\|k\right.\right\rangle\left\langle k\left\|d\rho\right\|m\right\rangle\left\langle m\left\|j\right.\right\rangle$
	$\displaystyle=\sum_{k,m}\sqrt{\lambda_{k}}\sqrt{\lambda_{m}}\delta_{ik}\delta_{mj}\left\langle k\left\|d\rho\right\|m\right\rangle$
	$\displaystyle=\sqrt{\lambda_{i}}\sqrt{\lambda_{j}}\left\langle i\left\|d\rho\right\|j\right\rangle\text{,}$	(20)

$\displaystyle\left\langle i\left\|d\rho\right\|j\right\rangle$	$\displaystyle=\left\langle i\left\|\left(\sum_{n}\left[dp_{n}\left\|n\right\rangle\left\langle n\right\|+p_{n}\left\|dn\right\rangle\left\langle n\right\|+p_{n}\left\|n\right\rangle\left\langle dn\right\|\right]\right)\right\|j\right\rangle$
	$\displaystyle=\sum_{n}dp_{n}\left\langle i\|n\right\rangle\left\langle n\|j\right\rangle+p_{n}\left\langle i\|dn\right\rangle\left\langle n\|j\right\rangle+p_{n}\left\langle i\|n\right\rangle\left\langle dn\|j\right\rangle$
	$\displaystyle=\sum_{n}dp_{n}\delta_{in}\delta_{nj}+p_{n}\left\langle i\|dn\right\rangle\delta_{nj}+p_{n}\delta_{in}\left\langle dn\|j\right\rangle$
	$\displaystyle=dp_{i}\delta_{ij}+p_{j}\left\langle i\|dj\right\rangle+p_{i}\left\langle di\|j\right\rangle\text{,}$	(28)

$\displaystyle\left\|\left\langle m\|d\rho\|n\right\rangle\right\|^{2}$	$\displaystyle=\left\langle m\|d\rho\|n\right\rangle\left\langle m\|d\rho\|n\right\rangle^{\ast}$
	$\displaystyle=\left\langle m\|d\rho\|n\right\rangle\left\langle n\|d\rho\|m\right\rangle$
	$\displaystyle=\left[\delta_{mn}dp_{m}+\left(p_{n}-p_{m}\right)\left\langle m\|dn\right\rangle\right]\left[\delta_{nm}dp_{n}+\left(p_{m}-p_{n}\right)\left\langle n\|dm\right\rangle\right]$
	$\displaystyle=\delta_{mn}dp_{m}\delta_{nm}dp_{n}+\delta_{mn}dp_{m}\left(p_{m}-p_{n}\right)\left\langle n\|dm\right\rangle+\left(p_{n}-p_{m}\right)\left\langle m\|dn\right\rangle\delta_{nm}dp_{n}+$
	$\displaystyle+\left(p_{n}-p_{m}\right)\left\langle m\|dn\right\rangle\left(p_{m}-p_{n}\right)\left\langle n\|dm\right\rangle$
	$\displaystyle=\delta_{nm}dp_{n}^{2}-\left(p_{n}-p_{m}\right)^{2}\left\langle m\|dn\right\rangle\left\langle n\|dm\right\rangle\text{,}$	(32)

	$\displaystyle\left\|\left\langle m\|d\rho\|n\right\rangle\right\|^{2}$	$\displaystyle=\delta_{nm}dp_{n}^{2}+\left(p_{n}-p_{m}\right)^{2}\left\langle dm\|n\right\rangle\left\langle n\|dm\right\rangle$
		$\displaystyle=\delta_{nm}dp_{n}^{2}+\left(p_{n}-p_{m}\right)^{2}\left\|\left\langle n\|dm\right\rangle\right\|^{2}\text{,}$		(34)

Bures and Sjöqvist Metrics over Thermal State Manifolds for Spin Qubits and Superconducting Flux Qubits

Abstract

pacs:

I Introduction

II The Bures metric

II.1 The explicit derivation of Hübner’s general expression

II.2 The explicit derivation of Zanardi’s general expression

II.3 The explicit derivation of Zanardi’s expression for thermal states

II.3.1 Case: β\beta-constant and λ\lambda-nonconstant

II.3.2 Case: β\beta-nonconstant and λ\lambda-nonconstant

III The Sjöqvist metric

III.1 The explicit derivation

IV The Hamiltonian Models

IV.1 Spin-1/2 qubit Hamiltonian

IV.2 Superconducting flux qubit Hamiltonian

V Applications

V.1 Spin qubits

V.1.1 The Bures metric

First sub-calculation

Second sub-calculation

Third sub-calculation

V.1.2 The Sjöqvist metric

V.2 Superconducting flux qubits

V.2.1 The Bures metric

V.2.2 First sub-calculation

V.2.3 Second sub-calculation

V.2.4 Third sub-calculation

V.2.5 The Sjöqvist metric

VI Considerations from the comparative analysis

VI.1 The asymptotic limit of β\beta approaching infinity

VI.2 The metrics discrepancy

VII Conclusive Remarks

Acknowledgements.

References

II.3.1 Case: $\beta$ -constant and $\lambda$ -nonconstant

II.3.2 Case: $\beta$ -nonconstant and $\lambda$ -nonconstant

VI.1 The asymptotic limit of $\beta$ approaching infinity