The Underlying Order Induced by Orthogonality and the Quantum Speed Limit

The progress on the experimental manipulation of quantum states and control of quantum dynamics to achieve specific tasks has been based on well-established theoretical elements that have pointed to new applications in quantum information science. One of these elements refers to the time $\tau$ required by an initial pure state to reach (for the first time) an orthogonal, distinguishable state when evolved under a unitary (Hamiltonian) transformation [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]. This orthogonality time $\tau$ represents the time required to perform an elementary computational step, and it establishes a characteristic scale for the dynamical evolution of the system. Its study has therefore both practical and fundamental implications [13, 14, 15].

Specially relevant is the minimal amount of time required to reach orthogonality, setting a lower bound for $\tau$ known as the quantum speed limit (QSL), $\tau_{\textrm{qsl}}$. Fundamental limitations on the QSL have been discovered [16, 3, 17, 18, 19], particularly related to the state’s energy dispersion in the form of the Mandelstam–Tamm (MT) bound or to the state’s mean energy (relative to the minimum energy), as discovered by Margolus and Levitin (ML) [20]. Since then, there has been an impressive progress on the investigations regarding the minimal amount of transformation linking two orthogonal states (see, e.g., the reviews [21, 22]).

In their seminal work, Levitin and Toffoli [6] investigated the tightness of the unified bound that encompasses the MT and the ML bound. They demonstrated that only equally-weighted superpositions of two energy eigenstates saturate both bounds and attain the QSL, whereas for superpositions of three energy eigenstates the unified bound is tight, so $\tau$ may be arbitrarily close, although not equal, to $\tau_{\text{qsl}}$. Further investigations consider more general situations as the case of systems of qubits [23] or mixed quantum states [24, 25]. More recently, an experimental confirmation of the unified bound has been observed using fast matter wave interferometry of a single atom moving in an optical trap [26], and a theoretical analysis of the feasibility for measuring QSLs in ultracold atom experiments was presented by del Campo [27].

Despite the progress boosted by Levitin and Toffoli, a thorough analysis of the necessary and sufficient conditions for a pure state, in a low-dimensional Hilbert space, to evolve towards an orthogonal one in a finite $\tau$ is far from having being completed and is still a pending task. The present paper contributes to this effort by presenting a detailed and comprehensive study of the conditions on both: the expansion coefficients $\{r_{i}\}$ of the state in the energy representation, and the system’s accessible energy levels, that guarantee that a given initial (pure) state in a three-dimensional Hilbert space attains a distinguishable one in a finite amount of time. This analysis is presented in Section II for states that evolve under the action of a time-independent, otherwise arbitrary, Hamiltonian. Starting from the orthogonality condition, we classify the allowed sets $\{r_{1},r_{2},r_{3}\}$ into two main families and establish their precise relation with the energy-level spacing and $\tau$. The relation among the coefficients, the energy-level spacing, and the orthogonality time is afterwards represented in a diagram in which the families are depicted. Furthermore, the set of allowed $r_{i}$s is recognized as a 2-simplex contained in the probability 2-simplex of $\mathbb{R}^{3}$. We further investigate, in Section III, whether the quantum speed limit of the states determined by the coefficients $\{r_{i}\}$ found is given by either the MT or the ML bound. Finally, we present a summary and some concluding remarks in Section IV.

Our results disclose the exact and non-trivial interrelation among the orthogonality time, the Hamiltonian eigenvalues and the parameters that provide the energy distribution in qutrit systems, which have potential applications in the dynamics of neutrino oscillations [28], the laser-driven transformations of an atomic qutrit [29], the quantum Fourier transform of a superconducting qutrit [30] or the transfer of quantum information [31]. They also offer a complete characterization of practical value when attempting to prepare low-dimensional states that evolve towards a distinguishable one, either by specifying the appropriate initial state once a specific Hamiltonian is given or an appropriate transformation whenever the initial state is fixed.

We start by carrying out an analysis in the three-dimensional Hilbert space $\mathcal{H}_{3}$, to determine the conditions that drive a pure state under an arbitrary (time-independent) Hamiltonian evolution into an orthogonal state in a finite time. Our analysis is general enough and does not depend on the peculiarities of the physical system, opening the possibility of analyzing the minimal amount of transformation between orthogonal states for specific transformations of interest in the engineering of quantum computation.

An arbitrary initial pure state $\left|{\psi(0)}\right\rangle$ in $\mathcal{H}_{3}$ can be expanded in the eigenbasis $\{\left|{E_{i}}\right\rangle\}$ ($i=1,2,3$) of a Hamiltonian $\hat{H}$, namely $\hat{H}\left|{E_{i}}\right\rangle=E_{i}\left|{E_{i}}\right\rangle$, as

where $0\leq\theta_{i}\leq 2\pi$, and the triad of coefficients $r_{i}$ forms a probability distribution $\{r_{1},r_{2},r_{3}\}$ for which $0\leq r_{i}\leq 1$ and $\sum_{i=1}^{3}r_{i}=1$.

We consider initial states whose expansion (1) involves non-degenerate eigenstates $\left|{E_{i}}\right\rangle$ and assume that their corresponding eigenvalues $E_{i}$ are ordered according to $E_{1}<E_{2}<E_{3}$. Thus, the evolved state, $\left|{\psi(t)}\right\rangle=e^{-\text{i}\hat{H}t/\hbar}\left|{\psi(0)}\right\rangle$, is given explicitly by

and the overlap $\langle{\psi(0)}|{\psi(t)}\rangle$, which measures how distinguishable the state $\left|{\psi(t)}\right\rangle$ is from the initial one $\left|{\psi(0)}\right\rangle$, reads $\langle{\psi(0)}|{\psi(t)}\rangle=\sum_{i=1}^{3}r_{i}e^{-\text{i}E_{i}t/\hbar}$. The system thus attains an orthogonal (distinguishable) state at time $t=\tau$ whenever

Along with the normalization condition, we have two equations (the real and imaginary parts of (3)) and four unknowns: $r_{1},r_{2},r_{3}$ and $\tau$. In what follows, we identify the families of triads $\{r_{i}\}$ that solve Equation (3) in terms of the orthogonality time $\tau$ and the frequencies $\omega_{ij}\equiv(E_{i}-E_{j})/\hbar$.

It is a well-established fact that, for given energetic resources, the orthogonality time of an initial state cannot be less than the (minimal) time imposed by the so-called quantum speed limit. The QSL establishes a natural and intrinsic time-scale of the system’s dynamics and becomes relevant when characterizing the evolution rate of the system. It is therefore the central quantity of this section, aimed at determining the quantum speed limit of the states associated to the expansion coefficients that belong to the Families I and II.

The fundamental limits on the speed at which a pure quantum state evolves towards an orthogonal one under a (time-independent) Hamiltonian have been advanced in the form of lower bounds, either in terms of the relative mean energy $\mathcal{E}$ (as measured from the lowest eigenvalue that contributes to (1), say $E_{j}$) [16],

with $\tau_{\text{{qsl}}}$ denoting the quantum speed limit. They proved that only for states in an equally-weighted superposition of two energy eigenstates both bounds (ML and MT) coincide and the quantum speed limit is attained, whereas for superpositions of three energy eigenstates the unified bound is tight, so $\tau$ may be arbitrarily close, although not equal, to $\tau_{\text{qsl}}$. This means that for the triads $\{r_{i}\}$ in Family I-qubit $\tau_{\text{qsl}}=\tau$, whereas for those in the Families I-b and II it holds that $\tau_{\text{qsl}}<\tau$. Our aim is thus to discern which of the two bounds, ML or MT, determines the quantum speed limit of the states (1) specified by the elements of these latter families. We do so by analyzing the parameter

which evidently determines $\tau_{\text{qsl}}$ as the Mandelstam–Tamm bound whenever $\alpha<1$, as the Margolus–Levitin bound provided $\alpha>1$ and by either of them (being equal) for $\alpha=1$.

In terms of the transition frequencies $\omega_{ij}$ and the coefficients $r_{i}$, $\mathcal{E}$ and $\sigma_{\hat{\mathnormal{H}}}$ are explicitly given by

For the triads in Family I-qubit, Equation (24) reduces to $\mathcal{E}=\sigma_{\hat{\mathnormal{H}}}=\hbar\omega_{ij}/2$; therefore, $\alpha=1$, bounds (22a) and (22b) coincide, and it is verified that $\tau=\tau_{\text{qsl}}$, with $\tau_{\text{qsl}}=\pi/\omega_{ij}$. Notice that, among the three members of Family I-qubit, the one with the lowest quantum speed limit is $\bigl{\{}\frac{1}{2},0,\frac{1}{2}\bigr{\}}$ (represented by the red vertex of $\delta^{2}$ in Figure 2), which gives rise to the fastest qubit $\left|{\psi(0)}\right\rangle=\frac{1}{\sqrt{2}}\Bigl{(}e^{\text{i}\theta_{1}}\left|{E_{1}}\right\rangle+e^{\text{i}\theta_{3}}\left|{E_{3}}\right\rangle\Bigr{)}$ that reaches a distinguishable state at $\tau$ given precisely by Equation (19).

When all three coefficients $r_{i}$ are nonzero, we can rewrite Equation (24) as follows

from which $\alpha$ can be expressed as

This allows the realization of a quantitative analysis of the quantum speed limit of states given rise by Families I and II. For an arbitrary fixed $\{r_{i}\}$, $\alpha$ depends explicitly on the energy separations only through the ratio $\Omega=\omega_{32}/\omega_{21}$. However, for the $\{r_{i}\}$ in Family II, the detailed dependence of $\alpha$ on $\Omega$ becomes more intricate due to Equation (8). In Figure 3, we show a large sample of points of the 2-simplex $\delta^{2}$ that correspond to coefficients for which $\alpha$, computed from (26), is larger than 1 (the Margolus–Levitin subset $\delta^{2}_{\text{ML}}$, colored in magenta), and similarly coefficients for which $\alpha<1$ (the Mandelstam–Tamm subset $\delta^{2}_{\text{MT}}$, colored in cyan), for different values of $\Omega$ in the region $(0,3\pi^{2}]\times(0,18\pi]$ of the space $\omega_{21}\tau$–$\Omega$. The figure reveals the complex spreading of both subsets within the orthogonality simplex $\delta^{2}$, induced by the quantum speed limit; notably, there is no sharp separation between the elements of $\delta^{2}_{\text{MT}}$ and those of $\delta^{2}_{\text{ML}}$. They appear rather mixed, although exhibiting a marked accumulation of cyan points around the edge $\bigl{\{}r,\frac{1}{2}-r,\frac{1}{2}\bigr{\}}$ and the vertex $\bigl{\{}0,\frac{1}{2},\frac{1}{2}\bigr{\}}$. These become diluted towards the opposite vertex $\bigl{\{}\frac{1}{2},\frac{1}{2},0\bigr{\}}$, whereas the magenta points are more dense near the vertices $\bigl{\{}\frac{1}{2},0,\frac{1}{2}\bigr{\}}$ and $\bigl{\{}\frac{1}{2},\frac{1}{2},0\bigr{\}}$.

The case of the triads in Family I-b, represented by the edges of $\delta^{2}$ in Figure 3, allows for a direct and simple analytical analysis for determining the corresponding range of $\alpha$. With this purpose, it is convenient to analyze separately the three possible sets $\{r_{i}=1/2=r_{j}+r_{k}\}$ as follows:

• $\blacksquare$

  $\bigl{\{}\frac{1}{2},r,\frac{1}{2}-r\bigr{\}}$, $0<r<\frac{1}{2}$. These points form the edge that goes from the vertex $\bigl{\{}\frac{1}{2},0,\frac{1}{2}\bigr{\}}$ to $\bigl{\{}\frac{1}{2},\frac{1}{2},0\bigr{\}}$ of the 2-simplex $\delta^{2}$ in Figures 2 and 3, and they correspond to

  $$\alpha=\sqrt{1+\frac{r(1-2r)\Omega^{2}}{\big{[}r+(\frac{1}{2}-r)(1+\Omega)\big{]}^{2}}}>1\quad\textrm{for all}\;\>\Omega,$$

  (27)

  where the inequality follows from the fact that $0<1-2r<1$.

• $\blacksquare$

  $\bigl{\{}r,\frac{1}{2}-r,\frac{1}{2}\bigr{\}}$, $0<r<\frac{1}{2}$. These points form the edge that goes from the vertex $\bigl{\{}\frac{1}{2},0,\frac{1}{2}\bigr{\}}$ to $\bigl{\{}0,\frac{1}{2},\frac{1}{2}\bigr{\}}$ in Figures 2 and 3, and give

  $$\alpha=\sqrt{1-\frac{(1-2r)(1-r+\Omega)}{\big{[}1-r+\frac{1}{2}\Omega\big{]}^{2}}}<1\quad\textrm{for all}\;\>\Omega.$$

  (28)

• $\blacksquare$

  $\bigl{\{}r,\frac{1}{2},\frac{1}{2}-r\bigr{\}}$, $0<r<\frac{1}{2}$. This set of points lies along the edge that goes from the vertex $\bigl{\{}0,\frac{1}{2},\frac{1}{2}\bigr{\}}$ to $\bigl{\{}\frac{1}{2},\frac{1}{2},0\bigr{\}}$ in Figures 2 and 3, and their corresponding $\alpha$ reads

  $$\alpha=\sqrt{1+\frac{(1-2r)(1+\Omega)\big{[}r(1+\Omega)-1\big{]}}{\big{[}1-r+\Omega\bigl{(}\frac{1}{2}-r\bigr{)}\big{]}^{2}}}.$$

  (29)

  Unlike the previous cases, here, the range of $\alpha$ depends on the range of $\Omega$. For $\Omega\leq 1$ we have $\alpha<1$, whereas for $\Omega>1$ we find three cases:

  	$\displaystyle\alpha>1$	$\displaystyle\;\;\text{for}\;\;\frac{1}{1+\Omega}<r<\frac{1}{2},$		(30a)
  	$\displaystyle\alpha=1$	$\displaystyle\;\;\text{for}\;\;r=\frac{1}{1+\Omega},$		(30b)
  	$\displaystyle\alpha<1$	$\displaystyle\;\;\text{for}\;\;0<r<\frac{1}{1+\Omega}.$		(30c)

  In this way, for a given $\Omega$, the edge under consideration is divided into two segments, one cyan and one magenta, separated by the point at $r=(1+\Omega)^{-1}$. Notice that such segments are not appreciated in Figure 3, since (as explained above) this figure considers a sample of different values of $\Omega$ and not a single one.

Finally, Table 1 contains a practical summary of our main results, which completely characterize all qutrits that evolve towards an orthogonal state under a time-independent, otherwise arbitrary, Hamiltonian.

In this paper, we completely determine and organize the set of coefficients $\{r_{i}\}$ that provide the energy distribution and give rise to initial states (1) that reach an orthogonal state in a finite time $\tau$, when evolving under an arbitrary time-independent Hamiltonian. A geometric organization of such coefficients is established, both in the solution diagram in the space $\omega_{21}\tau$-$\Omega$ and in the 2-simplex in $\mathbb{R}^{3}$. In the first case, the sets $\{r_{i}\}$ are represented by points according to the allowed values of $\tau$ and the corresponding energy-levels separation, and characteristic regions whose shape resembles zebra-like stripes emerge. The interior of each of these regions is filled with triads of Family II ($r_{i}\neq 0,1/2$ for all $i$), whereas the borders correspond to triads of Family I: continuous borders represent coefficients associated to qubit states ($r_{i}=0$ for some $i$), and their intersections represent elements of Subfamily I-b ($r_{i}=1/2$ for some $i$, $r_{j}\neq 0$ for all $j$). In the second geometric representation, the $\{r_{i}\}$s are organized in the central 2-simplex $\delta^{2}$, contained in the 2-simplex $\Delta$ of $\mathbb{R}^{3}$. In this case as well, the elements in Family II fill the interior (satisfying $0<r_{i}<1/2)$, whereas elements in Family I lie along the borders (edges and vertices) of $\delta^{2}$.

As is clear from the results in Figure 1, the minimum orthogonality time $\tau_{\text{min}}$ (19) of the qubit corresponding to the triad $R_{13}\equiv\bigl{\{}\frac{1}{2},0,\frac{1}{2}\bigr{\}}$ (dotted red line) bounds from below the orthogonality time of any other qubit or qutrit with the same energetic resources. The other qubits that attain an orthogonal state (representing equally weighted superpositions of eigenstates with energies $E_{1}$ and $E_{2}$ and $E_{2}$ and $E_{3}$, respectively) correspond to the triads $R_{12}\equiv\bigl{\{}\frac{1}{2},\frac{1}{2},0\bigr{\}}$ and $R_{23}\equiv\bigl{\{}0,\frac{1}{2},\frac{1}{2}\bigr{\}}$ and have orthogonality times that are bounded according to: $\tau_{\text{min}}<\tau_{R_{12}}\leq\tau_{R_{23}}$ whenever $\omega_{21}\geq\omega_{32}$ and $\tau_{\text{min}}<\tau_{R_{23}}\leq\tau_{R_{12}}$ provided $\omega_{32}\geq\omega_{21}$ (see Equations (9) and (11) with $l=0$). In the latter case, there exist qutrits whose orthogonality times may acquire any value within the interval $(\tau_{\text{min}},\tau_{R_{23}})$, whose length diminishes as $\omega_{32}/\omega_{21}$ increases. Such qutrits are represented in the blue shaded region of the $l=0$ zebra stripe in Figure 1.

We furthermore analyze the quantum speed limit of the pure states given rise by the points within the 2-simplex $\delta^{2}$, by constructing a map where the states whose orthogonality time is bounded by the Margolus–Levitin bound are distinguished from those for which the Mandelstam–Tamm bound limits the time evolution towards orthogonality.

Our investigation constitutes a comprehensive analysis of the exact solutions of the orthogonality condition $\langle{\psi(0)}|{\psi(\tau)}\rangle=0$ in three-level systems. It reveals a rich underlying geometric structure and hierarchy of the allowed parameters that conform the energy distributions, while allowing for the establishment of limiting values for the orthogonality time in terms of the energy levels spacing. Further, knowledge of the expansion coefficients and their relation with $\tau$ provides central tools to analyze the detailed dynamics of qutrits evolving towards orthogonality and explore its relation with other relevant features, such as the amount of entanglement in composite three-level systems, which strongly depends on the sets $\{r_{i}\}$.

We consider that our study contributes from both theoretical and practical points of view to the efforts ultimately aimed at establishing the conditions under which a $N$-level state transforms into a distinguishable one under a unitary transformation, irrespective of the peculiarities of the physical system. In this sense, it should be stressed that, although we assume a Hamiltonian evolution, the analysis just performed can be immediately extended to more general continuous unitary transformations $e^{-\text{i}\hat{G}\gamma}$, with $\hat{G}$ an appropriate ($\gamma$-independent) generator and $\gamma$ the corresponding evolution parameter. Our findings are thus applicable to any pure state (of single or composite systems) that can be expressed as a superposition of any three, non-degenerate, eigenvectors of $\hat{G}$ and throw light onto the $\gamma$-distribution and the amount of evolution $\gamma^{\bot}$ required to reach an orthogonal state under the transformation governed by $\hat{G}$. The results here presented thus acquire relevance in the realm of the dynamics of low-dimensional states, as well as in the field of quantum information processing, in which knowing and approximating to the fundamental limits of a quantum dynamical process improves quantum control technologies.