Bounds on the breaking time for entanglement-breaking channels

Quantum speed limit [1, 2, 3, 4, 5, 6] is a well-known fundamental concept related to the question about time-energy uncertainty relations in quantum mechanics. However, at the same time, several diverse applications of quantum speed limit (QSL) can be found [7, 8, 9, 10, 11, 12, 13, 14, 15]. The most popular form of QSL is due to Mandelstam and Tamm [1, 2] and depends on the variance of the generator of time evolution. However, sometimes the variance would give an unreasonable assessment, and the average value of the generator is employed, leading to the so-called Margolus-Levitin QSL [4], which in fact has been derived before by Fleming [3].

One can ask a philosophical question, namely: Why does QSL provide useful pieces of information, given that it is “just” a mere consequence of an underlying time evolution? Here we consider a problem which, on top of being interesting and relevant by itself, also perfectly illustrates the way in which QSL overcomes an overall complexity of the full description of a quantum system’s dynamics.

To be more precise, we are concerned with quantum channels generated by Markovian dynamics of open quantum systems. We ask whether such channels are entanglement breaking. Any channel $\Phi$ is called entanglement breaking (EB) if the composite state $(\Phi\otimes\mathcal{I})[\rho]$ is always separable, even for entangled input states $\rho$ [16]. Since we only consider quantum channels which belong to a semigroup parametrized by $t\geqslant 0$, i.e., channels of the form $\Phi_{t}=e^{t\mathcal{L}}$, the question we pose gains a bit of structure. In particular, for $t=0$ we have $\Phi_{0}=\mathcal{I}$, so the channel $\Phi_{0}$ is not entanglement breaking. We call such channels entanglement preserving (EP). One can expect that, depending on $\mathcal{L}$, the range of time splits into two nonempty and disjoint sets,

with selfexplanatory interpretations. Even though $t=0\in T_{\mathrm{EP}}(\mathcal{L})$, for every $\mathcal{L}$, both sets in general will not be connected. It might happen that the channel is EP for $t\in[0,t_{\mathrm{EB}}[$ and starts to be EB at $t=t_{\mathrm{EB}}$. However, at a later time it is again EP (revival of the EP property).

We might ask whether both sets can be characterized in a more direct way. It is clear that, even for qubit channels, the above task seems hopeless. In this simplest case, even though one is able to explicitly write down cumbersome conditions which describe the EB property [17], it is not possible to turn such complex inequalities into an informative description of $T_{\mathrm{EB}}(\mathcal{L})$.

In this paper we establish quantum speed limit for (potentially) entanglement-breaking channels, i.e., given the fact that $\Phi_{0}$ is entanglement preserving, we bound from below the time in which this property might be lost. In other words, we are to bound the time $t_{\mathrm{EB}}$, such that a given channel $\Phi_{t}$ is certainly entanglement preserving for $t\in[0,t_{\mathrm{EB}}[$. The standard formulation of QSL tells us what is the minimum time necessary to pass from a given state to an orthogonal state. Our results tell what is the minimal time during which the channel in question is entanglement preserving. We shall stress that this methodology is not aiming at delivering the very exact moment in which the channel becomes entanglement breaking since the problem of separability is, in general [18], NP-hard (however, in certain circumstances it can almost always be solved [19]).

The paper is organized as follows. In Sec. II we present the methodology as well as a necessary formal background concerning quantum channels. We then utilize entanglement witnesses to establish in Secs. III and IV various bounds for $t_{EB}$, in particular, bounds inspired by the Mandelstam-Tamm QSL and Margolus-Levitin QSL. In Sec. IV.2 we discuss the results with an example.

In accordance with formalism of quantum mechanics, let $\mathcal{H}_{1}$ be a Hilbert space of a system under consideration and let $\mathcal{H}_{2}$ be a Hilbert space of an auxiliary system. To study open system evolution, being a $t-$parametrized completely positive and trace-preserving map on $\mathcal{B}(\mathcal{H}_{1}),$ one can rely on expectation values [20, 21]

Here $\rho$ denotes an initial state acting on a composite Hilbert space $\mathcal{H}_{1}\otimes\mathcal{H}_{2}$, while $\hat{A}$ is an arbitrary observable therein, i.e., a Hermitian operator from $\mathcal{B}(\mathcal{H}_{1}\otimes\mathcal{H}_{2})$.

Different ways to characterize the map $\Phi_{t}$ concern different choices of initial states and observables. For instance, in quantum process tomography [22, 23], the overall profile of the map $\Phi_{t}$ is recovered by collecting several $\left\langle A(t)\right\rangle$ constructed from various pairs of $\rho$ and $\hat{A}$, so that both inputs and observables exhaust some fixed bases of the matrix space $\mathcal{B}(\mathcal{H}_{1})$. In this case full knowledge about the auxiliary space $\mathcal{H}_{2}$ is not required. Another example is the extraction of a characteristic of an external system, encoded in an interaction with the probe system $\mathcal{H}_{1}$ (signal), where in this case the auxiliary system $\mathcal{H}_{2}$ can be taken as a reference (idler), a catalyst, or the source of enhancement for the manipulation [24].

For the sake of studying the dynamics of entanglement, concerning the action of the map $\Phi_{t}$, an entanglement witness $\hat{W}$ [25, 26, 27, 28] will be an appropriate choice for the observable $\hat{A}$. With this choice

and

since by definition $\operatorname{tr}(\sigma\hat{W})\geqslant 0$ if $\sigma$ is a separable state, and we require $\rho$ to be entangled.

In a more formal fashion we now recall that $\Phi_{t}$ is called entanglement breaking [17, 16] if $(\Phi_{t}\otimes\mathcal{I})[\sigma]$ is separable for all density matrices $\sigma$ acting on $\mathcal{H}_{1}\otimes\mathcal{H}_{2}$, where $\mathcal{H}_{2}$ is a finite dimensional Hilbert space. By setting $\rho$ to be entangled, as exposed in Eq. (4), and if $\Phi_{t}$ becomes entanglement breaking at a time moment $t_{\mathrm{EB}}$, this property will be reflected in the sign of the function $w_{\rho}(t_{\mathrm{EB}})$. This sign shall change before $t_{\mathrm{EB}}$, or, in an optimal case, $w_{\rho}(t_{\mathrm{EB}})=0$.

Interestingly, when both subsystems are finite dimensional and symmetric, i.e., $\mathcal{H}_{1}=\mathcal{H}_{2}\simeq\mathbb{C}^{d}$, the initial state $\rho$ can be chosen to be a maximally entangled, pure state $\rho=\ket{\Psi_{+}\vphantom{\Psi_{+}}}\!\bra{\Psi_{+}\vphantom{\Psi_{+}}}$ [23], where

and $\{\ket{k}\}$ is a basis of $\mathbb{C}^{d}$. In such a case, the final state $\rho_{\Phi_{t}}=(\Phi_{t}\otimes\mathcal{I})[\rho]$ is simply a Choi-Jamiołkowski isomorphic representation of the map $\Phi_{t}$ [29, 30]. In this sense one can say that entanglement of $\rho_{\Phi_{t}}$ certifies that the corresponding map $\Phi_{t}$ is entanglement preserving.

For nonsymmetric finite dimensional cases, and also for infinite dimensional case, even though the notion of isomorphism may not be applicable, the basic concepts can be illustrated in the similar way, i.e., an entanglement witness for the outcome state can also be an EP witness for the dynamical map. However, here we shall restrict our attention only to the finite dimension and symmetric situation (both subsystems have the same dimension).

From the perspective of our main question, we are particularly interested in a configuration-breaking time $t_{CB}{(\rho,\hat{W})}$ defined as

In other words, this is minimal time $t$ in which, for a given prepare-measure configuration $(\rho,\hat{W})$, the state $\rho_{\Phi_{t}}$ is no longer classified as entangled. Clearly

because on the one hand, for a given $\rho$, we shall find an optimal entanglement witness which works for the longest possible time, while in the second step we need to find the optimal entangled state $\rho$.

The aim of this paper is to find lower bounds for $t_{CB}(\rho,\hat{W})$, and consequently, bounds for $t_{\mathrm{EB}}$. From now on, we shall omit the arguments of $t_{CB}$. We note in passing that the time $t_{\mathrm{EB}}$ is a notion which relies on a possibility of entanglement detection (through a witness or an entanglement measure in general). Therefore, this time cannot by itself play the role of a witness.

In general, the problem of finding $t_{CB}(\rho,\hat{W})$ defined in Eq. (6), with explicit form of the input function taken from Eq. (14), is very complicated. This is due to its non-linear character and plenty of involved parameters. In fact, since $w_{\rho}(0)<0$, this problem boils down to solving highly nonlinear equation $w_{\rho}(t)=0$, and further seeking for the minimum root. Obviously, if all parameters are known, one can employ numerical methods to find the exact breaking time. Therefore, our goal is to get explicit results without specifying the parameters. To this end we shall follow three routes in order to bound $t_{EB}$.

In this section we consider a specific choice for the entanglement witness operator

with the maximally entangled state $\left|\Psi_{+}\right\rangle$ already defined in (5). One can observe that the average value of this witness is $0$ for all separable states, a fact which suggests a certain optimality of this choice of the witness.

Moreover, in order to strengthen the configuration we also select the state to be maximally entangled, i.e. $\hat{\rho}=\left|\Psi_{+}\right\rangle\left\langle\Psi_{+}\right|$. We shall call this whole choice a symmetric configuration.

As a result, the time-dependent average value of the witness becomes

where

In the vectorized notation, the state

is represented by a vector

where $\left|e_{jj^{\prime}}\right)$ are members of a canonical basis of the matrix space $\mathcal{B}(\mathcal{H}_{1})$. In other words, $\left|e_{jj^{\prime}}\right)$ is a vectorized form of $\ket{j\vphantom{j^{\prime}}}\!\bra{j^{\prime}\vphantom{j}}$ (we remember that $\mathcal{H}_{1}=\mathcal{H}_{2}\simeq\mathbb{C}^{d}$).

We are in position to use the above vectorization to prove two technical results concerning the choice (38):

To show this result, one shall perform the calculation as follows (we omit summation ranges for brevity):

Passing from the second to the third line we used orthogonality of the vectors $\left|e_{jj^{\prime}}\right)$.

First of all, since the witness under discussion is vectorized to the form $\big{|}\hat{W}\big{)}=\big{|}0\big{)}^{\otimes 2}-d\left|\Psi_{+}\right)$, for the symmetric configuration we know that

Therefore, given Lemma 1 we only need to explicitly calculate the first term. To this end, we observe that $\left|0\right)$, which corresponds to the identity operator $\mathbbm{1}=\sum_{j=1}^{d}\ket{j\vphantom{j}}\!\bra{j\vphantom{j}}$, is represented as $\left|0\right)=\sum_{j=1}^{d}\left|e_{jj}\right)$. We can then explicitly calculate

This finalizes the proof. Given both lemmas above, we reach the following conclusion

Lemma 2 says that $r_{0}=1-\frac{1}{d}\geqslant 0$ and consequently $\phi_{0}=0$. On the other hand, for $\alpha\neq 0$ (i.e., for members of classes II and III) we can see that $r_{\alpha}e^{i\phi_{\alpha}}$ is real and negative. Therefore, $\phi_{\alpha}=\pi$ in all these cases. These are exactly the conditions defining the good configuration.

As we can see, one can find a good configuration for any channel $\Phi_{t}$, simply by means of the symmetric configuration discussed in this section. Moreover, since the entanglement breaking time $t_{EB}$ is lower bounded by all $t_{CB}$, we conclude that all three bounds derived in the previous section do apply to $t_{EB}$.

In the following, we simplify these bounds given the symmetric configuration. Before doing so, we note in passing that the symmetric configuration has an additional interesting feature, namely, it can be related to the geometric measure of entanglement [32]. The latter was recently shown to be equal to a minimal time required for a unitary (global) transformation to transform a given pure entangled state to a closed separable state [33]. An analog of $t_{EB}$ in this problem reads $\Omega^{-1}\arccos\left(\mathchoice{{\hbox{$\displaystyle\sqrt{d\,}$}\lower 0.4pt\hbox{\vrule height=6.94444pt,depth=-5.55559pt}}}{{\hbox{$\textstyle\sqrt{d\,}$}\lower 0.4pt\hbox{\vrule height=6.94444pt,depth=-5.55559pt}}}{{\hbox{$\scriptstyle\sqrt{d\,}$}\lower 0.4pt\hbox{\vrule height=4.8611pt,depth=-3.8889pt}}}{{\hbox{$\scriptscriptstyle\sqrt{d\,}$}\lower 0.4pt\hbox{\vrule height=3.47221pt,depth=-2.77779pt}}}\right)$ with $\Omega$ defined as an energy scale of a Hamiltonian rendering the global time evolution. Recently, a similar problem has been studied in Refs. [34, 35], where the dynamical behavior of several entanglement quantifiers has been considered for a generic form of dynamics, revealing a relation between quantum speed limit and the change in entanglement.

Since time in quantum mechanics is not an observable — it is “just” a parameter — the rich formalism of uncertainty relations cannot be utilized to describe the dynamics. However, quantum speed limit, even though based on slightly different foundations, often seems to offer a sufficient quantification of various dynamical features of quantum systems. Here we propose yet another aspect in which the methodology behind the QSL can successfully be applied. We study Markovian open-system dynamics which, when described in the language of quantum channels, always corresponds to an identity channel in the starting moment of the evolution. Consequently, such a dynamics always starts with an entanglement-preserving channel. Therefore, depending on the details of the Markovian dynamics, such a channel sooner (but not instantly) or later (perhaps never) becomes entanglement breaking. While the task to describe this transition exactly is computationally laborious as it would pretend to solving an NP-hard problem [18], using the QSL we managed to derive three lower bounds for the time moment in which it happens (see Sec. IV.1 for a summary). These bounds do only depend on the parameters describing the associated master equation, so that they can be expressed in terms of decay rates and oscillation frequencies. Despite fundamental aspects pertaining to a better understanding of quantum open-system dynamics, the presented results in a way complement other efforts aiming at description of generation and degradation of quantum resources [38].

Future, more technically oriented studies can already start from the case of qutrits ($d=3$). Even for a specific map similar to (52), namely,

where all parameters are defined in the same spirit as $\gamma_{\parallel}$, $\gamma_{\perp}$, and $\omega$ in the qubit case, one can see that there are four pairs of real parameters to be considered. This setting immediately leads to more complex relations among the obtained bounds. We shall leave this example, as well as more elaborate cases, as an open question for further development.

We thank Kavan Modi for a stimulating discussion and Sevag Gharibian for fruitful correspondence. We acknowledge support by the Foundation for Polish Science (IRAP project, ICTQT, Contract No. 2018/MAB/5, co-financed by EU within the Smart Growth Operational Programme).