Capacity of Entanglement for Non-local Hamiltonian

Entanglement has potential applications in quantum information science ranging from quantum computing, quantum communication and host of other areas such as condensed matter physics, high energy physics and even string theoryKitaev and Preskill (2006); Jiang et al. (2012). It is considered a very useful resource in information processing tasks. Over several years, how to create and quantify entanglement has been a subject of major explorationsHorodecki et al. (2009); Das et al. (2016). Thanks to technological progress, now we can create entanglement between two or more number of particles in quantum optical systems Boto et al. (2000), ion traps Pedrozo-Peñafiel et al. (2020), superconducting systems Lee et al. (2021); Mooney et al. (2019), and NMR setups Doronin et al. (2019). How to create entanglement between more and more number of particles and distribute over long distances still continues to be quite challenging van Leent et al. (2020). Quantum entanglement between two particles can of course be created depending on the choice of the initial state and suitable non-local interaction between them. However, design of suitable interacting Hamiltonian is not always easy. This makes the production of entanglement a non-trivial task. Therefore, it is natural to ask the question, for a given non-local Hamiltonian, what is the best way of exploiting this Hamiltonian to create entanglement. This was addressed in Ref.Dür et al. (2001)

Entanglement entropy is quite a useful diagnostic tool which measures degree of quantum entanglement between subsystems in a many-body quantum systems Laflorencie (2016). A different quantity, called as the capacity of entanglement has been proposed to characterize topologically ordered states in the context of the Kitaev model de Boer et al. (2019). Given a pure bipartite entangled state $\rho_{AB}$, the capacity of entanglement is defined as the second cumulant of the entanglement spectrum. Thus, associated to a reduced density matrix, we can define the capacity of entanglement as the variance of the modular Hamiltonian in the mixed state. If $\{\lambda_{i}\}$’s are the eigenvalues of the reduced density matrix of one of the subsystem, than the entanglement entropy is defined as $S_{EE}=S(\rho_{A})=-\operatorname{tr}(\rho_{A}\log{\rho_{A}})=-\sum_{i}\lambda_{i}\log\lambda_{i}$. Now, the capacity of entanglement $C_{E}$ is defined as the second cumulant of this entanglement spectrumLi and Haldane (2008), i.e., the variance in the entanglement entropy operator. It is similar to the heat capacity of thermal systems and is given by Yao and Qi (2010); Li and Haldane (2008); Schliemann (2011)

$$C_{E}=\sum_{i}\lambda_{i}\log^{2}\lambda_{i}-S^{2}_{EE}.$$

The above quantity can be thought of as the variance of the distribution of $\,-\log\lambda_{i}$ with probability $\lambda_{i}$, and thus it contains information about the width of the eigenvalue distribution of the reduced density matrix. We can gain insight on the whole spectrum by studying upto first two cumulants, i.e., the entanglement entropy and the capacity of entanglement. Defining a modular Hamiltonian as $K_{A}=-\log\rho_{A}$, they are the expectation value and the variance of $K_{A}$. The capacity of entanglement has found useful applications in the conformal and the nonconformal quantum field theories Nandy (2021); Verlinde and Zurek (2020), as well as in models related with the gravitational phase transitions Kawabata et al. (2021a, b); Verlinde and Zurek (2020); Arias et al. (2020); Kawabata et al. (2021c).

In this paper, we address the entanglement capacities for non-local Hamiltonians. To be specific, we answer the following question: Given a non-local Hamiltonian, what is the capacity of entanglement for bipartite systems? We show that the entanglement rate is bounded by the fluctuation in the non-local Hamiltonian and the capacity of entanglement. In addition, the quantum speed limit for creating the entanglement depends inversely on the fluctuation in the non-local Hamiltonian as well as on the time average of the square root of the capacity of entanglement. Thus, the more the capacity of entanglement, the shorter the time duration system may take to produce the desired amount of entanglement. We illustrate the quantum speed limit for general two-qubit non-local Hamiltonian and find that our bound is indeed tight. Furthermore, we discuss the capacity of entanglement for self-inverse Hamiltonains and provide a bound on the rate of capacity of entanglement. Finally, we generalise the capacity of entanglement for bipartite mixed states based on the relative entropy of entanglement measure. This definition reduces to the capacity of entanglement for the pure bipartite states. This will open up its explorations for mixed states in future. We believe that our results can find applications in diverse areas of physics ranging from condensed matter systems to conformal field theories and alike.

The present paper is organised as follows. In Section II, we provide basic definitions and useful relations for the capacity of entanglement for pure bipartite states. In Section III, we discuss the capacity of entanglement for non-local Hamiltonians. In Section IV, we prove that the entanglement rate is bounded by the capacity of entanglement and the speed of quantum evolution under the non-local Hamiltonian. We also provide a quantum speed limit for entanglement production or degradation and discuss how the capacity of entanglement helps in deciding the speed limit. In Section V, we discuss the capacity of entanglement for self-inverse Hamiltonians and provide a bound on the rate of the capacity of entanglement. In Section VI, we generalise the definition of the capacity of entanglement for bipartite mixed states based on the notion of relative entropy of entanglement. Finally, in Section VII, we summarise our findings.

Let $\mathcal{H}$ represent a separable Hilbert space and $\dim(\mathcal{H})$ be the dimension of Hilbert space. Let us consider a bipartite quantum system described by state vector $\ket{\Psi}_{AB}\in{\cal{H}}_{AB}={\cal{H}}_{A}\otimes{\cal{H}}_{B}$ with unit norm. It is possible to express the state vector $\ket{\Psi}_{AB}$ as

$$\ket{\Psi}_{AB}=\sum_{n}\sqrt{\lambda_{n}}\ket{\psi_{n}}_{A}\otimes\ket{\phi_{n}}_{B},$$

(1)

where $\{\ket{\psi_{n}}\}_{A}$ and $\{\ket{\phi_{n}}\}_{B}$ are the Schmidt basis in ${\cal{H}}_{A}$ and ${\cal{H}}_{B}$, respectively and $\{\lambda_{n}\}$ are the non-negative real numbers with $\sum_{n}\lambda_{n}=1$. Eq. (1) is called the Schmidt decomposition of $\ket{\Psi}_{AB}$ and $\lambda_{n}$ are known as the Schmidt coefficients. If the Schmidt decomposition of $\ket{\Psi}_{AB}$ has more than one non-zero Schmidt coefficients then we say that system $A$ and $B$ are “entangled”. If there is only one non-zero Schmidt coefficient then the state is not entangled.

Let ${\cal B({\cal H}}_{AB})$ denotes the algebra of linear operators acting on a finite–dimensional Hilbert space ${\cal H}_{AB}$ of dimension ${\rm dim}({\cal{H}}_{AB})$ and let ${\cal D}({\cal H}_{AB})$ denote the set of density operators for the bipartite system. The density operators are positive operators of unit trace acting on ${\cal H}_{AB}$. For any state $\rho_{AB}\in{\cal D}({\cal H})$, if we can express $\rho_{AB}$ as $\rho_{AB}=\sum_{i}p_{i}{\rho_{i}}^{A}\otimes{\rho_{i}}^{B}$ then it is separable state, otherwise the mixed state is entangled one. Given a density operator $\rho_{AB}$ associated with a bipartite quantum system $AB$, the reduced density matrix for subsystem $A$ (or $B$) is obtained by taking partial trace over subsystem $B$ (or $A$), i.e., $\rho_{A}=\operatorname{tr}_{B}(\rho_{AB})$. A physical quantity of system $A$ represented by a self-adjoint operator ${\cal{O}}_{A}$ on ${\cal{H}}_{A}$ is identified with a self-adjoint operator ${\cal{O}}_{A}\otimes{\cal{I}}_{B}$ on ${\cal{H}}_{AB}$, where ${\cal{I}}_{B}$ is the identity operator on ${\cal{H}}_{B}$. The expectation value of ${\cal{O}}_{A}\otimes{\cal{I}}_{B}$ on state $\rho_{AB}$ is given by $\operatorname{tr}(\rho_{A}{\cal{O}}_{A})$, where $\rho_{A}$ is the reduced density operator of system $A$.

The quantum relative entropy between two density operators $\rho$ and $\sigma$ acting on the same Hilbert space $\cal{H}$ is defined as Umegaki (1962)

$$S(\rho\|\sigma):=\begin{cases}\operatorname{tr}(\rho(\ln{\rho}-\ln{\sigma}))&\text{if}\ \operatorname{supp}(\rho)\subseteq\operatorname{supp}(\sigma),\\ +\infty&\text{otherwise},\end{cases}$$

(2)

where $\operatorname{supp}(\rho)$ and $\operatorname{supp}(\sigma)$ are the supports of $\rho$ and $\sigma$, respectively. The quantum relative entropy satisfies important properties: (i) $S(\rho\|\sigma)\geq 0$ and $S(\rho\|\sigma)=0$ iff $\rho=\sigma$, (ii) $\sum_{i}p_{i}S(\rho_{i}\|\sigma_{i})\geq S(\sum_{i}p_{i}\rho_{i}\|\sum_{i}p_{i}\sigma_{i})$ and (iii) $S(\rho\|\sigma)\geq S({\cal E}(\rho)\|{\cal E}(\sigma))$ for any completely positive trace preserving map ${\cal E}$.

Let us consider a composite system $AB$ with pure state $|\Psi\rangle_{AB}$. The amount of entanglement between subsystems $A$ and $B$ can be quantified via the entanglement entropy which is defined as the von Neumann entropy of the reduced density operator $\rho_{A}=\sum_{n}\lambda_{n}\ket{\psi_{n}}_{A}\bra{\psi_{n}}$ (or $\rho_{B}$), i.e.,

$$S_{EE}=S(\rho_{A})=-\operatorname{tr}(\rho_{A}\log\rho_{A})=-\sum_{n}\lambda_{n}\log\lambda_{n}$$

(3)

which is invariant under local unitary transformations on $\rho_{A}$. The von Neumann entropy vanishes when density operator $\rho_{A}$ is a pure state. For a completely mixed density operator, the von Neumann entropy attains its maximum value of $\rm\log d_{A}$, where $\rm d_{A}=dim(\cal{H}_{A})$.

For any density operator $\rho_{A}$ associated with quantum system $A$, we can define a formal “Hamiltonian” $K_{A}$, called the modular Hamiltonian, with respect to which the density operator $\rho_{A}$ is a Gibbs like state (with $\beta=1$)

$$\rho_{A}=\frac{e^{-K_{A}}}{Z},$$

where $Z=\operatorname{tr}(e^{-K_{A}}).$ Note that any density matrix can be written in this form for some choice of Hermitian operator $K_{A}$. With slight adjustments in the above equation, the modular Hamiltonian $K_{A}$ can be written as $K_{A}=-\log\rho_{A}$. In this case, the entanglement entropy of the system is equivalent to the thermodynamic entropy of a system described by Hamiltonian $K_{A}$ (with $\beta=1$). Writing in terms of modular Hamiltonian $K_{A}=-\log\rho_{A}$, the entanglement entropy becomes the expectation value of the modular Hamiltonian

$$S_{EE}=-\operatorname{tr}(\rho_{A}\log\rho_{A})=\operatorname{tr}(\rho_{A}K_{A})=\langle K_{A}\rangle\,.$$

(4)

The capacity of entanglement is another information-theoretic quantity that has gained some interest recentlyCaputa et al. (2022); de Boer et al. (2019). It is defined as the variance of the modular Hamiltonian $K_{A}$de Boer et al. (2019) in the state $\ket{\Psi}_{AB}$ and can be expressed as

	$\displaystyle C_{E}(\rho_{A})$	$\displaystyle=\Delta K_{A}^{2}=\bra{\Psi}(K_{A}\otimes\mathcal{I}_{B})^{2}\ket{\Psi}-\bra{\Psi}(K_{A}\otimes\mathcal{I}_{B})\ket{\Psi}^{2}$
		$\displaystyle=\operatorname{tr}[\rho_{A}(-\log\rho_{A})^{2}]-[\operatorname{tr}(-\rho_{A}\,\log\rho_{A})]^{2}$		(5)
		$\displaystyle=\operatorname{tr}[\rho_{A}K_{A}^{2}]-[\operatorname{tr}(\rho_{A}K_{A})]^{2}$
		$\displaystyle=\langle K_{A}^{2}\rangle-\langle K_{A}\rangle^{2}=\Delta K_{A}^{2}.$		(6)

The capacity of entanglement can also be defined in terms of the variance of the relative surprisal between two density matrices $V(\rho||\sigma)$Boes et al. (2022):

$$V(\rho||\sigma)=\operatorname{tr}(\rho\left(\log\rho-\log\sigma\right)^{2})-\left(D(\rho||\sigma)\right)^{2}.$$

(7)

If one of the density matrices becomes maximally mixed (i.e., either $\rho$ or $\sigma$ becomes I/d), then the variance of the relative surprisal becomes the capacity of entanglement.

As shown in Ref.Pati and Sahu (2007), uncertainty for any observable is a convex function. Given two or more Hermitian operators such as $O_{1}$ and $O_{2}$, the standard deviation or the uncertainty for observables satisfy $\Delta(p_{1}O_{1}+p_{2}O_{2})\leq p_{1}\Delta O_{1}+p_{2}\Delta O_{2}$ for $0\leq p_{i}\leq 1\,(i=1,2)$ with $\Delta O_{i}=\sqrt{\langle O_{i}^{2}\rangle-\langle O_{i}\rangle^{2}}$. This shows that adding two or more observables always reduces the uncertainty. If we define the standard deviation in the modular Hamiltonian as uncertainty in the entanglement operator, then for any two modular Hamiltonian $K_{1}$ and $K_{2}$, we will have

$$\Delta(\sum_{i}p_{i}K_{i})\leq\sum_{i}p_{i}\Delta K_{i},$$

(8)

where $K_{i}=-\ln{\rho_{i}}$. This property has an interesting implication when we have a modular Hamiltonian undergoing some variation. Suppose, we allow a variation in the modular Hamiltonian as $K\rightarrow K^{\prime}=K+xV$, where $V$ is the additional term in the modular Hamiltonian and $x$ is a real parameter. Then, the following relation holds true, i.e., $\Delta K^{\prime}\leq\Delta K+x\Delta V$.

For the sake of completeness, we mention the following properties which are applicable for $C_{E}$ on account of having similar form as the relative surprisal between two density matrices:

1. 1.

   Additivity under tensor product:

   $$C_{E}(\rho_{A}\otimes\rho_{B})=C_{E}(\rho_{A})+C_{E}(\rho_{B}).$$

2. 2.

   Positivity : $C_{E}(\rho)\geq 0$.

3. 3.

   Uniform Continuity:

   $$|C_{E}(\rho)-C_{E}(\rho^{{}^{\prime}})|^{2}\leq\xi\log^{2}{d}\cdot D(\rho,\rho^{{}^{\prime}})$$

   for $\xi$ some constant and $l_{1}$ trace norm between states $D(\rho,\rho^{{}^{\prime}})$.

4. 4.

   $C_{E}(\rho)=0$ if and only if all non-zero eigenvalues of $\rho$ are the same. Such states are termed as flat states. Examples include any pure state or maximally mixed state.

5. 5.

   Corrections to subadditivity:

   $$C_{E}(\rho)\leq C_{E}(\rho_{1})+C_{E}(\rho_{2})+\chi\log^{2}{d}\cdot f(I_{\rho})$$

   for any bipartite state $\rho$ with marginal states $\rho_{1}$, $\rho_{2}$ and mutual information $I_{\rho}$, with constant $\chi$ and $f(x)=\max(x^{1/4},x^{2})$.

6. 6.

   For fixed dimensions $d\geq 2$, the state $\rho_{d}$ with maximal variance has the spectrum

   $$spec(\rho_{d})=\Big{(}1-r,\frac{r}{d-1},\cdots,\frac{r}{d-1}\Big{)}$$

   with $r$ being the unique solution to

   $$(1-2r)\ln\big{(}\frac{1-r}{r}(d-1)\big{)}=2$$

   We get $\frac{1}{4}\log^{2}(d-1)<C_{E}(\rho_{d})<\frac{1}{4}\log^{2}(d-1)+\frac{1}{\ln^{2}(2)}$, and for the limit of large $d$, $r\approx\frac{1}{2}$.

For further details and proofs regarding the above properties, readers are advised to go through Ref. Boes et al. (2022).

The dynamics of entanglement under two-qubit nonlocal Hamiltonian has been adressed in Ref.Dür et al. (2001). In this section, we address the following question: What is the capacity of entanglement for arbitrary two-qubit non-local Hamiltonian? Further, we also discuss about the rate of the capacity of entanglement for the non-local Hamiltonian. For any two-qubit system, the non-local Hamiltonian can be expressed as (except for trivial constants)

$$H=\vec{\alpha}\cdot\vec{\sigma}^{A}\otimes\mathcal{I}_{B}+\mathcal{I}_{A}\otimes\vec{\beta}\cdot\vec{\sigma}^{B}+\sum_{i,j=1}^{3}\gamma_{ij}\sigma_{i}^{A}\otimes\sigma_{j}^{B},$$

(9)

where $\vec{\alpha},\vec{\beta}$ are real vectors, $\gamma$ is a real matrix and, $\mathcal{I}_{A}$ and $\mathcal{I}_{B}$ are identity operator acting on $\mathcal{H}_{A}$ and $\mathcal{H}_{B}$. The above Hamiltonian can be rewritten in one of the two standard forms under the action of local unitaries acting on each qubits Bennett et al. (2002); Dür et al. (2001). This is given by

$$H^{\pm}=\mu_{1}\sigma_{1}^{A}\otimes\sigma_{1}^{B}\pm\mu_{2}\sigma_{2}^{A}\otimes\sigma_{2}^{B}+\mu_{3}\sigma_{3}^{A}\otimes\sigma_{3}^{B},$$

(10)

where $\mu_{1}\geq\mu_{2}\geq\mu_{3}\geq 0$ are the singular values of matrix $\gamma$Dür et al. (2001). Using the Schmidt-decomposition, any two qubit pure state can be written as

$$\ket{\Psi}_{AB}=\sqrt{p}\ket{\phi}\ket{\chi}+\sqrt{1-p}\ket{\phi^{\perp}}\ket{\chi^{\perp}}.$$

(11)

We can utilize the form of Hamiltonian in Eq. (10) and choosing $H^{+}$ (i.e. assuming $\det(\gamma)\geq 0$) to evolve the state in Eq. (11) without loosing any generality Dür et al. (2001). To further showcase a specific example, let us choose $\ket{\phi}=\ket{0}$ and $\ket{\chi}=\ket{0}$. Thus, the state at time $t=0$ takes the form

$$\ket{\Psi(0)}_{AB}=\sqrt{p}\ket{0}\ket{0}+\sqrt{1-p}\ket{1}\ket{1}.$$

(12)

Under the action of the non-local Hamiltonian, the joint state at time $t$ can be written as ($\hbar=1$)

$\displaystyle\ket{\Psi(t)}_{AB}=e^{-iHt}\ket{\Psi}_{AB}=\alpha(t)\ket{0}\ket{0}+\beta(t)\ket{1}\ket{1},$

(13)

where $\alpha(t)=e^{-i\mu_{3}t}\left(\sqrt{p}\cos(\theta t)-i\sqrt{1-p}\sin(\theta t)\right)$ , $\beta(t)=e^{-i\mu_{3}t}\left(\sqrt{1-p}\cos(\theta t)-i\sqrt{p}\sin(\theta t)\right)$ and $\theta=(\mu_{1}-\mu_{2})$. To evaluate the capacity of entanglement, we would require the reduced density matrix of the two qubit evolved state, $\rho_{A}(t)=\operatorname{tr}_{B}(\rho_{AB}(t))$, which is given by

$\displaystyle\rho_{A}(t)$

$\displaystyle=\lambda_{1}(t)\ket{0}\bra{0}+\lambda_{2}(t)\ket{1}\bra{1},$

(14)

where $\lambda_{1}(t)=\left|\alpha(t)^{2}\right|$ and $\lambda_{2}(t)=\left|\beta(t)^{2}\right|$ with

	$\displaystyle\lambda_{1}(t)$	$\displaystyle=\frac{1}{2}\left[1-(1-2p)\cos\left(2\theta t\right)\right],$
	$\displaystyle\lambda_{2}(t)$	$\displaystyle=\frac{1}{2}\left[1+(1-2p)\cos\left(2\theta t\right)\right].$

The capacity of entanglement at a later time t can be calculated from the variance of modular Hamiltonian $K_{A}$. This is given by

	$\displaystyle C_{E}(t)$	$\displaystyle=\operatorname{tr}(\rho_{A}(t)(-\log\rho_{A}(t))^{2})-(\operatorname{tr}(-\rho_{A}(t)\log\rho_{A}(t)))^{2}\,,$
		$\displaystyle=\sum_{i=1}^{2}\lambda_{i}(t)\log^{2}\lambda_{i}(t)-\left(-\sum_{i=1}^{2}\lambda_{i}(t)\log\lambda_{i}(t)\right)^{2}.$		(15)

In order to quantify the entanglement production, we can define the entanglement rate $\Gamma$ as defined in Ref.Dür et al. (2001), i.e.,

$$\Gamma(t)=\frac{{\rm d}S_{EE}(t)}{{\rm d}t}=\frac{{\rm d}S_{EE}(t)}{{\rm d}p}\frac{{\rm d}p}{{\rm d}t}.$$

(16)

The assertion is that this quantity depends upon the entanglement $S_{EE}$ which depends upon some parameter $p$ and the rate of the Schmidt coefficient. The condition(s) to obtain a maximal entanglement rate are of interest for which two things are of significance. First, for a given value of $S_{EE}$ of two-qubit system, we find $|\Psi_{E}\rangle$, for which the interaction produces maximum rate $\Gamma_{E}$ and, the maximal achievable entanglement rate $\Gamma_{max}=\max_{E}\Gamma_{E}$ with corresponding state $|\Psi_{max}\rangle$.

Let us evaluate objects defined above for an arbitrary Hamiltonian $H$. Using the Schmidt-decomposition of the state $|\Psi(t)\rangle$

$$\ket{\Psi}_{AB}=\sqrt{p}|\phi\rangle|\chi\rangle+\sqrt{1-p}|\phi^{\perp}\rangle|\chi^{\perp}\rangle\,,$$

(17)

where $\langle\phi|\phi^{\perp}\rangle=0=\langle\chi|\chi^{\perp}\rangle$ and $p\leq\frac{1}{2}$. The entanglement measure $S_{EE}$, must depend only on the Schmidt coefficient $p$, given the fact that it must be invariant under local unitary operations. If we choose entropy of entanglement as $S_{EE}$, the entropy of reduced density operator of one of the qubits is given by

$$S_{EE}(p)=-p\,\log_{2}(p)-(1-p)\,\log_{2}(1-p).$$

(18)

Operationally, $S_{EE}$ quantifies the amount of EPR entanglement contained asymptotically in a pure state $|\Psi\rangle_{AB}$, thus $S_{EE}$ gives a ratio of maximally entangled EPR state $|\Psi^{-}\rangle_{AB}=\frac{1}{\sqrt{2}}(\ket{0}\ket{0}-\ket{1}\ket{1})$ which can be distilled from $|\Psi\rangle_{AB}$.

Considering the infinitesimal time evolution of Schmidt coefficient of two qubit state, we get

$$|\Psi(t+\delta t)\rangle=e^{iH\delta t}|\Psi(t)\rangle\,\approx\,(1-iH\delta t)|\Psi(t)\rangle.$$

The time evolution of the reduced state for the subsystem $A$ is given by

$$\rho_{A}(t+\delta t)=\rho_{A}(t)-i\delta t\,\operatorname{tr}_{B}([H,|\Psi(t)\rangle\langle\Psi(t)|]).$$

(19)

Starting from $\rho_{A}|\phi\rangle=p|\phi\rangle$, then using the Schrödinger equation, we have

$$\frac{{\rm d}p}{\rm dt}=2\sqrt{p(1-p)}\,{\rm Im}(\langle\phi,\chi|H|\phi^{\perp},\chi^{\perp}\rangle).$$

(20)

As $\Gamma$ is to be maximized, we can choose

$$\Gamma=f(p)|h(H,\phi,\chi)|,$$

where

$$f(p)=2\sqrt{p(1-p)}S_{EE}^{{}^{\prime}}(p)\,\text{and}\,\,h(H,\phi,\chi)=\langle\phi,\chi|H|\phi^{\perp},\chi^{\perp}\rangle.$$

Note that fixing $S_{EE}$, means fixing $p$ and so the maximum entropy corresponds to a state with some fixed $|\phi\rangle$ and $|\chi\rangle$. For any value $S_{EE}$ of entanglement, the state $|\phi\rangle$ and $|\chi\rangle$ for which maximum entanglement rate $\Gamma_{E}$ is obtained does not depend on $S_{EE}$, but only on the form of Hamiltonian $H$.
Let $h_{max}$ be the maximum value of $|h|$, i.e.,

$$h_{max}=\max_{||\phi||,||\chi||=1}\big{|}\langle\phi,\chi|H|\phi^{\perp},\chi^{\perp}\rangle\big{|}.$$

(21)

Now, we need to drive the two qubit state with local operators so that for all time the corresponding state is the one with maximum rate and we would then know how capacity of entanglement evolves with time.

Evaluating the capacity for entanglement for general pure bipartite states in the Schmidt-decomposed form as in Eq. (17) and using the modular Hamiltonian, we can express it as

	$\displaystyle C_{E}(\Psi_{AB})$	$\displaystyle=\rm tr(\rho_{A}(\log\rho_{A})^{2})-(tr(\rho_{A}\log\rho_{A}))^{2}$
		$\displaystyle=p(1-p)\big{(}\log\big{(}\frac{p}{1-p}\big{)}\big{)}^{2}.$		(22)

We can define the rate of capacity of entanglement as

$$\frac{{\rm d}C_{E}}{{\rm d}t}=\frac{{\rm d}C_{E}}{{\rm d}p}\frac{{\rm d}p}{{\rm d}t},$$

where

$$\frac{{\rm d}C_{E}}{{\rm d}p}=(1-2p)\big{(}\log\frac{p}{1-p}\big{)}^{2}+2\log\frac{p}{1-p}\\$$

which diverges for $p\to\{0\}\cup\{1\}$.

Let ${\Gamma_{C}}$ denote the rate of capacity of entanglement, i.e, $\Gamma_{C}:=\frac{{\rm d}C_{E}(t)}{{\rm d}t}$. From the earlier result, using the transformed Hamiltonian, we have

	$\displaystyle{\Gamma_{C}}=\hskip 2.84544pt$	$\displaystyle 2\sqrt{p(1-p)}\Big{(}(1-2p)\big{(}\log\frac{p}{1-p}\big{)}^{2}$
		$\displaystyle+2\,\log\frac{p}{1-p}\Big{)}\,|h(H,\phi,\chi)|.$		(23)

Thus, it will not diverge with this form for $p=0\,\text{or}\,1$.

It should be clear that local terms corresponding to $\vec{\alpha},\vec{\beta}$ in Eq. (9) give no contribution to $h_{max}$ with the given Schmidt-decomposed form of the bipartite state. Trying to determine $h_{max}$ in terms of $\mu_{1,2,3}$, we get

$$h(H,\phi,\chi)=\sum_{k=1}^{3}\mu_{k}\bra{\phi}\sigma_{k}^{A}\ket{\phi^{\perp}}\bra{\chi}\sigma_{k}^{B}\ket{\chi^{\perp}}.$$

(24)

The maximum is reached for when $\ket{\chi}=\ket{\phi^{\perp}}$. Further utilizing completeness condition $\ket{\phi}\bra{\phi^{\perp}}+\ket{\chi}\bra{\chi^{\perp}}=I$, we get the expression

$$h(H,\phi)=\sum_{k=1}^{3}\mu_{k}-\sum_{k=1}^{3}\mu_{k}\bra{\phi}\sigma_{k}\ket{\phi}^{2}.$$

(25)

It can be further inferred from $\mu_{1}\geq\mu_{2}\geq\mu_{3}$ that maximum value is reached for when $\ket{\phi}=\ket{0}$ or $\ket{1}$, which gives us

$$h_{max}=\mu_{1}+\mu_{2}.$$

(26)

Thus, the state that provides maximum rate of capacity of entanglement and the corresponding rate are given by

$$\ket{\Psi_{E}}=\sqrt{p}\ket{01}+i\sqrt{1-p}\ket{10},$$

(27)

	$\displaystyle{\Gamma_{C}}_{max}=\hskip 2.84544pt$	$\displaystyle\frac{{\rm d}C_{E}}{{\rm d}t}\Big{|}_{max}$
	$\displaystyle=\hskip 2.84544pt$	$\displaystyle 2(\mu_{1}+\mu_{2})\sqrt{p(1-p)}$
		$\displaystyle\Big{[}(1-2p)\big{(}\log\frac{p}{1-p}\big{)}^{2}+2\log\frac{p}{1-p}\Big{]}.$		(28)

The maximum rate ${\Gamma_{C}}_{max}$ is obtained here for $p_{0}\simeq 0.0045$ which maximizes $f(p)$ to $f(p_{0})\simeq 1.2108$ for the corresponding $\ket{\Psi_{max}}$. The capacity of entanglement for this maximum rate is ${C_{E}}(p_{0})\simeq 0.1306$.

It has been shown that if we can allow local operations which can entangle each qubit with local ancilla, that can increase the $\Gamma_{max}$ for certain kinds of Hamiltonian Dür et al. (2001). We shall begin by generalizing the formulas for multilevel systems which contains the ancillas and the qubits. Consider a state $\ket{\Psi}_{AB}$ with the Schmidt-decomposition $\ket{\Psi}_{AB}=\sum_{n=1}^{N}\sqrt{\lambda_{n}}\ket{\phi_{n}}\ket{\chi_{n}}$. Again, the capacity of entanglement only depends on the Schmidt coefficients $\lambda_{n}\geq 0$. Using the definition of capacity of entanglement rate in Eq. (16), we have

	$\displaystyle\tilde{\Gamma}_{C}$	$\displaystyle=\frac{{\rm d}C_{E}}{{\rm d}t}=\sum_{n=1}^{N}\frac{\partial C_{E}}{\partial\lambda_{n}}\frac{{\rm d}\lambda_{n}}{{\rm d}t},$
		$\displaystyle=\frac{1}{N}\sum_{n,m=1}^{N}\Big{[}\frac{\partial C_{E}}{\partial\lambda_{n}}-\frac{\partial C_{E}}{\partial\lambda_{m}}\Big{]}\frac{{\rm d}\lambda_{n}}{{\rm d}t}.$		(29)

Using the Schrödinger equation, we find

$$\frac{{\rm d}\lambda_{n}}{{\rm d}t}=2\sum_{m=1}^{N}\sqrt{\lambda_{n}\lambda_{m}}\hskip 2.84544pt{\rm Im}\big{[}\bra{\phi_{n},\chi_{n}}H\ket{\phi_{m},\chi_{m}}\big{]}.$$

(30)

Now, let us consider one such example where adding ancillas allows one to increase capacity of entanglement more efficiently. Let us consider the case in which the ancillas are also qubits. Letting $\lambda_{1}=p$ and $\lambda_{2}=\lambda_{3}=\lambda_{4}=(1-p)/3$, Eq. (29) simplifies to

$$\tilde{\Gamma}=\tilde{f}(p)\tilde{h}(H,\phi_{n},\chi_{n}),$$

(31)

where

$$\tilde{f}(p)=2\sqrt{p(1-p)/3}\bigg{[}(1-2p)\log^{2}{\frac{3p}{1-p}}+2\log{\frac{3p}{1-p}}\bigg{]},$$

(32)

$$\tilde{h}(H,\phi_{n},\chi_{n})=\sum_{n=2}^{4}{\rm Im}[\bra{\phi_{n},\chi_{n}}H\ket{\phi_{m},\chi_{m}}].$$

(33)

We have a freedom to choose the phase of states $\ket{\phi_{n}}$ such that all terms add with the same sign thus allowing us to replace the imaginary parts of the above terms by their absolute values, i.e., $\tilde{f}(p)$ by $\left|\tilde{f}(p)\right|$. We find that $\tilde{p}_{0}\simeq 0.6036$ corresponding to capacity of entanglement $C_{E}(\tilde{p}_{0})\simeq 0.5523$ maximizing $\tilde{f}(p)$ to $\left|\tilde{f}(p_{0})\right|\simeq 1.4459$. Further, proceeding to maximize $\tilde{h}$, we obtain that the maximum value is $\tilde{h}_{max}=\mu_{1}+\mu_{2}+\mu_{3}$, which occurs when $\ket{\phi_{n}}$ and $\ket{\chi_{n}}$ are both orthogonal maximally entangled states between the qubit and the ancilla.

Upon comparing the cases in which ancillas are used to those in which they are not used, we can either have $\left|\tilde{f}(\tilde{p}_{0})\right|\geq\left|f(p_{0})\right|$ or $\tilde{h}_{max}\geq h_{max}$. For the case when $\mu_{3}\neq 0$, we can use ancillas to increase the maximum rate of capacity of entanglement $\Gamma_{max}$ as well as $\Gamma$ for a given capacity of entanglement of the state $\ket{\Psi}$.

In this section, we will show that the capacity of entanglement plays an important role in providing an upper bound for the entanglement rate for the non-local Hamiltonian. Specifically, we will show that the entanglement rate is upper bounded by the speed of transportation of the bipartite state and the time average of square root of the capacity of entanglement. Also, this sets a quantum speed limit on the entanglement production and degradation for pure bipartite states. Thus, the capacity of entanglement has a physical meaning in deciding how much time a bipartite states takes to produce a certain amount of entanglement.

Let us consider a bipartite system initially in a pure state. Let $\ket{\Psi(0)}_{AB}$ denote the initial state of the system. We consider the dynamics generated by a non-local Hamiltonial $H_{AB}$. The time evolved state at later time $t$ is given by $\ket{\Psi(t)}_{AB}=U_{AB}(t)|\Psi(0)\rangle_{AB}$, where $U_{AB}(t)=e^{-iH_{AB}t}$ with $\hbar=1$.

Now, we apply the Heisenberg-Robertson uncertainty relation Robertson (1929) for two non-commuting operators $K_{A}$ and $H_{AB}$. This leads to

$$\frac{1}{2}\left|\langle\Psi(t)|[K_{A}\otimes I_{B},H_{AB}]|\Psi(t)\rangle\right|\leq\Delta K_{A}\Delta H_{AB}\,.$$

(34)

Recall that the evolution of average of any self adjoint operator $O$ is given by

$$i\hbar\frac{{{\rm d}\langle O\rangle}}{{\rm d}t}=\langle[O,H]\rangle\,.$$

(35)

Using Eq. (35) (for $O=K_{A}$) in Eq. (34), we then obtain

$$\frac{\hbar}{2}\left|\frac{{\rm d}\langle K_{A}\rangle}{{\rm d}t}\right|\leq\Delta K_{A}\Delta H_{AB}\,.$$

(36)

Let $\Gamma(t)$ denote the rate of entanglement. Recall that the average of the modular Hamiltonian is the entanglement entropy $S_{EE}$. In terms of the entanglement rate $\Gamma(t)$, the above equation can be written as

$$\left|\Gamma(t)\right|\leq\frac{2}{\hbar}\Delta K_{A}\Delta H_{AB}\,.$$

(37)

Since the square of the standard deviation of modular Hamiltonian is the capacity of entanglement, so in terms of the capacity of entanglement, we can write above bound as

$$\left|\Gamma(t)\right|\leq\frac{2}{\hbar}\sqrt{C_{E}(t)}\Delta H_{AB}\,.$$

(38)

To interpret the above equation, first note that $\frac{2}{\hbar}\Delta H_{AB}$ is nothing but the speed of transportation of the bipartite pure entangled state on the projective Hilbert Space of the composite system. If we use the Fubini-Study metric for two nearby states Anandan and Aharonov (1990); Pati (1991, 1995), then the infinitesimal distance between two nearby states is defined as

$${\rm d}S^{2}=4\left(1-\left|\langle{\Psi(t)}\ket{\Psi(t+{\rm dt})}\right|^{2}\right)=\frac{4}{\hbar^{2}}\Delta H_{AB}^{2}{\rm d}t^{2}.$$

(39)

Therefore, the speed of transportation as measured by the Fubini-Study metric is given by $V=\frac{{\rm d}S}{{\rm d}t}=\frac{2}{\hbar}\Delta H_{AB}$. Thus, the entanglement rate is upper bounded by the speed of quantum evolution Deffner and Campbell (2017) and the square root of the capacity of entanglement, i.e., $|\Gamma(t)|\leq\sqrt{C_{E}(t)}V$.

It was shown in Ref Bravyi (2007) that for ancilla unassisted case, the entanglement rate is upper bounded by $c\|H\|\log d$, where $d={\rm min(dim}{\cal{H}}_{A},{\rm dim}{\cal{H}}_{B})$, $c$ is constant between 0 and 1, and $\|H\|$ is operator norm of Hamiltonian which corresponds to $p=\infty$ of the Schatten p-norm of $H$ which is defined as $\|H\|_{p}=[\operatorname{tr}\left(\sqrt{H^{\dagger}H}\right)^{p}]^{\frac{1}{p}}$. Now, using the fact that the maximum value of capacity of entanglement is proportional to $S_{max}(\rho_{A})^{2}$ de Boer et al. (2019), where $S_{max}(\rho_{A})$ is maximum value of von Neuamnn entropy of subsystem which is upper bounded by $\log{d}_{A}$, where $d_{A}$ is the dimension of Hilbert space of subsystem $A$, and $\Delta H\leq\|H\|$, a similar bound on the entanglement rate can be obtained from Eq. (38). Thus, the bound on the entanglement rate given in Eq. (38) is stronger than the previously known bounds.

The bound on the entanglement rate can be used to provide a quantum speed limit for the creation or degradation of entanglement. The notion of quantum speed limit (QSL) decides how fast a quantum state evolves in time from an initial state to a final state Pfeifer (1993). Even though it was discovered by Mandelstam and Tamm Mandelstam and Tamm (1945), over last one decade, there have been active explorations on generalising the notion of quantum speed limit for mixed states xiong Wu and shui Yu (2018); Mondal and Pati (2016) and on resources that a quantum system might posses Campaioli et al. (2022). Recently, the notion of generalised quantum speed limit has been defined in Ref. Thakuria et al. (2022). In addition, the quantum speed limit for observables has been defined and it was shown that the QSL for state evolution is a special case of the QSL for observable Mohan and Pati (2021). For a quantum system evolving under a given dynamics, there exists fundamental limitations on the speed for entropy $S(\rho)$, maximal information $I(\rho)$, and quantum coherence $C(\rho)$ Mohan et al. (2022) as well as on other quantum correlations like entanglement, quantum mutual information and Bell-CHSH correlation Pandey et al. (2022). Below, we provide a speed limit bound for the entanglement entropy which can be applied for scenario where entanglement can be generated or degraded, based on the capacity of entanglement. Our bound highlights the non-trivial role played by the capacity of entanglement in deciding the QSL.

The speed limit for entanglement entropy can be calculated from Eq. (38) by taking the absolute value on both the sides and integrating over time. Thus, we have

$$\int_{0}^{T}\left|\frac{{\rm d}S_{EE}(t)}{{\rm d}t}\right|{{\rm d}t}\leq\int_{0}^{T}\frac{2}{\hbar}\sqrt{C_{E}(t)}\Delta H{\rm d}t.\\$$

(40)

For the time independent Hamiltonian, we obtain the following bound for the quantum speed limit for entanglement

$\displaystyle T\geq T_{\rm QSL}^{E}:=\frac{\left|S_{EE}(T)-S_{EE}(0)\right|}{\frac{2}{\hbar}\Delta H\frac{1}{T}\int_{0}^{T}\sqrt{C_{E}(t)}{\rm d}t}.$

(41)

In the case of time dependent Hamiltonian $H(t)$, we can apply the Cauchy-Schwarz inequality in Eq. (40) and obtain the following inequality

$$\int_{0}^{T}\left|\frac{{\rm d}S_{EE}(t)}{{\rm d}t}\right|{{\rm d}t}\leq\sqrt{\int_{0}^{T}\frac{2}{\hbar}\sqrt{C_{E}(t)}{\rm dt}}\sqrt{\int_{0}^{T}\frac{2}{\hbar}\Delta H_{t}{\rm dt}}.\\$$

(42)

From the above inequality, we get the bound for the speed limit for entanglement entropy change as given by

$\displaystyle T\geq T_{\rm QSL}^{E}:=\frac{\left|S_{EE}(T)-S_{EE}(0)\right|}{\frac{2}{\hbar}\Delta\bar{H}\sqrt{\frac{1}{T}\int_{0}^{T}\sqrt{C_{E}(t)}\rm dt}}\,,$

(43)

where $\Delta\bar{H}=\frac{1}{T}\int_{0}^{T}\sqrt{\langle H(t)^{2}\rangle-\langle H(t)\rangle^{2}}\,\rm dt$, is the time averaged fluctuation in the Hamiltonian. In both these bounds (time dependent and time independent Hamiltonian) it is clear that evolution speed for entanglement generation (or degradation) is a function of capacity of entanglement $C_{E}$. Thus, we can say that $C_{E}$ controls how much time a system may take to produce certain amount of entanglement.

Now, one may ask how tight is the QSL bound for the entanglement generation of degradation? Here, we illustrate with a specific example that the quantum speed limit for the creation of entanglement is actually tight. Consider the initial state at $t=0$ as given in Eq.(12). The time evolution of the state is given by Eq.(13). Estimating the speed limit bound on the entanglement entropy in Eq. (41), for the considered state, would need following quantities:

$$C_{E}(t)=\frac{\left(1-\eta(t)^{2}\right)\tanh^{-1}\left(\eta(t)\right)^{2}}{\ln^{2}(2)},$$

(44)

where $\eta(t)=(1-2p)\cos(2\theta t)$.

	$\displaystyle\Delta H=$	$\displaystyle\theta(1-2p),$		(45)
	$\displaystyle S_{EE}=$	$\displaystyle-\frac{\log_{2}\left(\left(p-\frac{1}{2}\right)\cos(2\theta t)+\frac{1}{2}\right)}{2}$
		$\displaystyle+\frac{\log_{2}\left((\frac{1}{2}-p)\cos(2\theta t)+\frac{1}{2}\right)}{2}$
		$\displaystyle+\frac{(1-2p)\cos(2\theta t)\tanh^{-1}((1-2p)\cos(2\theta t))}{\ln(2)}.$		(46)

The plot in Fig. 2 for $T_{QSL}^{E}$ vs $T\in[0,0.45]$ is shown under unitary dynamics through a general two qubit non-local Hamiltonian $H^{+}_{AB}$, beginning with an initial state of the system $\ket{\Psi(0)}=|0\rangle|0\rangle$ (taking $p=1$ in Eq. (13)). Our example shows that for the case of $\theta=\mu_{1}-\mu_{2}=0.5$ and $1.0$, the QSL for the entanglement creation is indeed tight and attainable.