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Generalized Trace-Distance Measure Connecting Quantum and Classical Non-Markovianity

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Generalized Trace-Distance Measure Connecting Quantum and Classical Non-Markovianity View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by AIR Universita degli studi di Milano PHYSICAL REVIEW A 92, 042108 (2015) Generalized trace-distance measure connecting quantum and classical non-Markovianity Steffen Wißmann and Heinz-Peter Breuer Physikalisches Institut, Universitat¨ Freiburg, Hermann-Herder-Straße 3, D-79104 Freiburg, Germany Bassano Vacchini Dipartimento di Fisica, Universita` degli Studi di Milano, Via Celoria 16, I-20133 Milan, Italy and INFN, Sezione di Milano, Via Celoria 16, I-20133 Milan, Italy (Received 31 July 2015; published 12 October 2015) We establish a direct connection of quantum Markovianity of an open system to its classical counterpart by generalizing the criterion based on the information flow. Here the flow is characterized by the time evolution of Helstrom matrices, given by the weighted difference of statistical operators, under the action of the quantum dynamical map. It turns out that the introduced criterion is equivalent to P divisibility of a quantum process, namely, divisibility in terms of positive maps, which provides a direct connection to classical Markovian stochastic processes. Moreover, it is shown that mathematical representations similar to those found for the original trace-distance-based measure hold true for the associated generalized measure for quantum non-Markovianity. That is, we prove orthogonality of optimal states showing a maximal information backflow and establish a local and universal representation of the measure. We illustrate some properties of the generalized criterion by means of examples. DOI: 10.1103/PhysRevA.92.042108 PACS number(s): 03.65.Yz, 03.65.Ta, 03.67.−a, 02.50.−r I. INTRODUCTION to the characterization proposed in this article, which turns out to be equivalent to divisibility in terms of positive maps Even though experimental techniques are developing (P divisibility) and has in addition a clear-cut connection rapidly, perfect isolation of fragile quantum systems from to classical Markov stochastic processes. The generalized noisy environments cannot be warranted in general, which ne- criterion introduced in Sec. II thus combines a physical cessitates effective descriptions for nonunitary dynamics [1]. interpretation of non-Markovianity in terms of an information A well-known treatment of the dynamics of such systems is flow and a relation to the well-known classical definition. provided by a quantum dynamical semigroup represented by Moreover, the associated measure has mathematical fea- a generator of the Gorini-Kossakowski-Sudarshan-Lindblad tures and representations similar to those found for the original form [2,3]. definition [14,15], which simplify its analytical, numerical, However, this description uses several rather drastic ap- and experimental determination drastically. Indeed, we prove proximations that do not apply to open quantum systems in in Sec. III A that, first, optimal initial states for non-Markovian general. The mismatch is typically associated with the neglect dynamics, experiencing a maximal backflow of information, of memory effects. Classically, there exists a well-established must be orthogonal and, second, the measure admits a local mathematical theory dealing with stochastic dynamics fea- representation showing locality and universality of quantum turing memory effects, based on the theory of stochastic memory effects. processes, yet the definition of classical Markovian stochastic An illustration of the essential features of the generalized processes cannot be straightforwardly carried over to the characterization of non-Markovianity in comparison with the quantum regime. This led to an intense debate along with original definition and approaches based on CP divisibility different proposals for the characterization and quantification is provided in Sec. III B. The examples show the sensitivity of memory effects in the dynamics of open quantum systems. of the present approach to memory effects of dynamics to Among others, approaches to non-Markovianity were based which the original definition was unsusceptible and illustrate on deviation from semigroup dynamics [4], on divisibility the existing difference between different types of divisibility in terms of completely positive maps [5], on dynamics of of a dynamical process. Finally, we summarize our results in entanglement [5] and correlations [6], and on the Fisher Sec. IV. information [7](see[8]forareview). In this work we focus on a straightforward extension of the measure of non-Markovianity introduced in Refs. [9,10] and reviewed in [11], which characterizes memory effects II. STATE DISCRIMINATION AND P DIVISIBILITY by an exchange of information between the open quantum system and the environment. This interpretation applies also Henceforth, H refers to the Hilbert space of the open to the generalized measure that uses Helstrom matrices known quantum system and the corresponding set of physical states, from quantum estimation theory [12]. A similar approach was i.e., the set of positive trace class operators with unit trace, is already introduced in Ref. [13], focusing, at variance with the denoted by S(H). Moreover, we assume that the dynamics of present proposal, on divisibility in terms of completely positive the open quantum system is determined by a one-parameter maps (CP divisibility). However, CP divisibility yields only family ={t |0 t T } of completely positive, trace- a sufficient condition for quantum Markovianity with respect preserving linear maps t , where 0 refers to the identity [1]. 1050-2947/2015/92(4)/042108(10) 042108-1 ©2015 American Physical Society WIßMANN, BREUER, AND VACCHINI PHYSICAL REVIEW A 92, 042108 (2015) A. Ensemble discrimination which is maximal if TR is the projection {0} on the The previously introduced definition of quantum non- subspace spanned by eigenvectors corresponding to positive | |= { − Markovianity [9,10], which was based on the concept of an eigenvalues of . Employing that Tr X Tr X({X0} } information flow between system and environment, relied on {X<0}) for any Hermitian operator X, one shows that the maximal success probability for correct discrimination the trace distance between two quantum states ρ1 and ρ2 defined by [16] obeys [17] max 1 1 P = max P = (1 + ). (6) D(ρ ,ρ ) ≡ ρ − ρ . (1) success success 2 1 1 2 2 1 2 1 0TR 1 = | | = − Here A 1√ Tr A , where the modulus of an operator is given Hence, the trace norm of the Helstrom matrix p1ρ1 † by |A|= A A, refers to the so-called trace norm on the p2ρ2 is the bias in favor of the correct state identification set of trace class operators [17]. A natural and interesting of the state prepared by Alice, so it may be interpreted as a generalization of this quantity is obtained when one also allows measure for the distinguishability of the two states ρ1 and ρ2 for biased probability distributions. That is, one considers with associated weights p1 and p2. Due to a result by Kossakowski [19,20], the trace norm can − = | − | p1ρ1 p2ρ2 1 Tr p1ρ1 p2ρ2 , (2) also be used to witness positivity of a trace-preserving map . That is, a trace-preserving map is positive if and only where {pi } refers to an arbitrary binary probability distribution, i.e., p 0 and p + p = 1. Clearly, choosing an unbiased if it defines a contraction for any Hermitian operator X with 1,2 1 2 respect to the trace norm, i.e., distribution, i.e., p1,2 = 1/2, one obtains the original expres- † sion (1). This generalization was first proposed in this context X1 X1 for all X = X . (7) by Chrusci´ nski´ et al. [13]. We thus obtain for two states evolving according to a The Hermitian operator = p1ρ1 − p2ρ2 is also known as the Helstrom matrix [12,18] and we find, applying the triangle dynamical map t , inequality to Eq. (2), p1t (ρ1) − p2t (ρ2)1 p1ρ1 − p2ρ2 (8) | − | + = p1 p2 1 p1 p2 1. (3) for all t where we also used linearity of the map. Adopting the previous characterization for quantum non- Clearly, the lower bound is obtained for ρ1 = ρ2, whereas the Markovianity [9,10], one then directly derives a generalized upper bound is attained if and only if ρ1 ⊥ ρ2, meaning that the two states have orthogonal support (which is defined as the criterion that still relies on the concept of an information flow. subspace spanned by the eigenvectors with nonzero eigenval- Definition 1. A quantum process is defined to be − ues). This is easily shown by means of the characterization of Markovian if p1t (ρ1) p2t (ρ2) 1 is a monotonically { } the trace norm decreasing function of t 0 for all sets pi ,ρi with pi 0, p1 + p2 = 1, and ρi ∈ S(H). 1 = 2maxTr{}+p2 − p1, (4) We stress that the authors of Ref. [13] introduced a different definition for Markovianity, which used the Helstrom matrices where the maximum is taken over all projection operators of a dilated system making it equivalent to CP divisibility of . Here Eq. (4) is derived by employing the Jordan-Hahn the dynamical process. However, as we shall see in Sec. II B, decomposition of the Hermitian Helstrom matrix in terms Markovianity as defined here can be related to the concept of of two positive and orthogonal operators, i.e., = S − Q, P divisibility and provides a clear-cut connection to classical where S,Q 0 and S ⊥ Q. Markov stochastic processes. The interpretation of the trace distance of two states We conclude this section by showing that the previously ρ1,2 ∈ S(H) as a measure of their distinguishability directly developed explanation to substantiate the interpretation of carries over to the trace norm of a Helstrom matrix . Consider quantum memory as an information backflow from the a one-shot two-state discrimination problem where Alice environment to the system [21] also applies here. To this end, prepares one out of two quantum states ρ1,2 with corresponding we define the quantities probability p , i.e., we have p + p = 1.
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