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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), p1 t (ρ1) − p2 t (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 p1 t (ρ1) p2 t (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. Hence, we allow 1,2 1 2 I = (1) − (2) for the general situation of a biased preparation by Alice. int(t) p1ρS (t) p2ρS (t) 1 (9) She finally sends the prepared state to Bob, who performs and a single (strong) measurement to infer which state he had I = (1) − (2) − I received [17]. To guess the state from the measurement with ext(t) p1ρSE(t) p2ρSE(t) 1 int(t), (10) possible outcomes , Bob defines two sets of possible results † ρ(i) t = U ρ(i) ⊗ ρ U R and \R and assigns the state to be ρ1 if the measurement where SE( ) t S E t refers to the state of the system t U outcome is in R and ρ2 if a value in \R is obtained. This and environment at time subject to a unitary evolution t . (i) = (i) strategy results in an effective two-valued positive-operator- We thus have ρS (t) TrEρSE(t) and, similarly, the state of { − } (i) = (i) valued measure TR,1 TR , where TR refers to the collection the environment at time t is given by ρE (t) TrS ρSE(t). of effects corresponding to outcomes in R. The probability for As discussed in [21], Iint(t) describes the distinguishability correct state discrimination via this strategy is then given by of the open system at time t, while Iext(t) can be interpreted as the information on the total system that is not accessible by P = p {T ρ }+p { 1 − T ρ } success 1Tr R 1 2Tr ( R) 2 measurements on the open system only. Due to contractivity = p2 + Tr{TR}, (5) of the trace norm under positive and trace-preserving maps
042108-2 GENERALIZED TRACE-DISTANCE MEASURE CONNECTING . . . PHYSICAL REVIEW A 92, 042108 (2015)
[cf. Eq. (7)], we conclude that both quantities are positive Typically, however, the inverse maps exist apart from as the partial trace is (completely) positive and preserves the isolated points in time, so one may describe the dynamical trace. Moreover, the existence of the quantum dynamical map process by a time-local master equation for the open system implies that the initial state is factorized, therefore Iext(0) = 0, during intermediate intervals. Assuming a sufficiently smooth K = ˙ −1 so unitary invariance of the trace norm implies time dependence, the generator obeys t t t and has I + I = I = the general structure int(t) ext(t) int(0) const. (11) K =− + † This relation clearly expresses the idea of exchange of t ρS (t) i[HS (t),ρS(t)] γj (t)[Aj (t)ρS(t)Aj (t) information between the open system and the environment as j an increase of I (t) necessarily forces I (t) to decrease. One 1 † ext int − {A (t)A (t),ρ (t)}], (13) may also derive upper bounds for the external information that, 2 j j S however, do not allow for a clear interpretation in terms of the formation of correlations between the system and environment which is similar to the well-known Lindblad form [2,3] apart and changes in the environmental states as proven for the trace from the fact that the system Hamiltonian HS (t), the rates distance in Ref. [22]. γj (t), and the Lindblad operators Aj (t) may depend on time as the process might not represent a semigroup. The maps of the two-parameter family are then given by B. Distinguishability and divisibility t In this section we introduce the concept of divisibility of a t,s = T← exp dτ Kτ , (14) dynamical map, which has been at the basis of several other s proposals for quantum non-Markovian dynamics [5,13,23], where T← refers to chronological time ordering. We thus d and elucidate its connection to the distinguishability measure K = | = have s dt t,s t s , so we conclude that is (C) P defined before. We start by revisiting the notion of n positivity. divisible if and only if K is the generator of a (completely) H t Amap on is said to be n positive if and only if the positive semigroup for all fixed t 0. The famous Gorini- ⊗ ⊗ linear tensor extension of the map given by ( 1n)(A Kossakowski-Sudarshan-Lindblad theorem [2,3] and a result = ⊗ H n B) (A) B for linear operators A on and B on C is on generators of positive semigroups by Kossakowski [19] ⊗ positive, that is, ( 1n)C 0 for all positive linear operators then allow for a characterization of CP and P divisibility in H ⊗ n C on C . terms of the generator [21]. Clearly, 1 positivity is equivalent to the map being Theorem 1. The dynamics generated by Kt (13) (i) is CP positive and the concept of n positivity is hierarchical, which divisible if and only if γj (t) 0 holds for all j and t 0 and means that n positivity of implies that the map is also k (ii) is P divisible if and only if for all n = m positive for all 1 k n [24]. The converse, however, is 2 not true in general [25,26]. Finally, one speaks of complete γj (t)| m|Aj (t)|n | 0 (15) ∈ positivity if a map is n positive for all n N.Fora j finite-dimensional system with dimH = N this property is equivalent to N positivity [27]. holds for any orthonormal basis {|n } of H and all t 0. Proof. The first statement is precisely the Gorini- If the maps t comprising a dynamical process are −1 Kossakowski-Sudarshan-Lindblad theorem [2,3] and the sec- invertible for all t 0 with inverse t , then one can define a two-parameter family of maps given by ond can be derived from Kossakowski’s result on generators of positive semigroups [19], which states that the dynamics = −1 t,s t s (12) generated by L is P divisible if and only if for any set of projections ={m}m∈I defining a resolution of the identity, for all t s 0 such that t,0 = t and t = t,s s .We i.e., = 1H, the relations remark that although the existence of a left-inverse requiring m m −1 injectivity of the maps t would be sufficient, we assume t amm() 0,m∈ I (16) to be the left and right inverse. The notion of divisibility deals a () 0,m = n ∈ I (17) with the properties concerning n positivity of these maps, i.e., mn Definition 2. A dynamical process is called P divisible amn() = 0,n∈ I (18) (CP divisible) if t,s is a positive (completely positive) map m∈I for all t s 0. for a () ≡ Tr{ (L )} are satisfied. Even though t is completely positive for all times by mn m n It is easily seen that the condition (18) refers to the definition, this does not imply (complete) positivity of t,s in general as the inverse of a CP map need not be positive, preservation of trace, which is always met for the generator K showing that the above definition is nontrivial. However, t (13) by its very structure. Rearranging terms of Eq. (18), one then finds the concept of divisibility relies on the existence of the two-parameter family that is limited to processes for which a () =− a () (19) −1 exists for all times. This property though cannot be taken nn mn t m =n for granted as, e.g., the damped Jaynes-Cummings model on resonance [1,10,21] or examples on quantum semi-Markov for all n ∈ I. The constraints thus reduce to the single processes [28] show, thus making the concept of divisibility relation (17), which can additionally be restricted to sets sometimes ill defined. of rank-1 projections by virtue of linearity. Evaluating
042108-3 WIßMANN, BREUER, AND VACCHINI PHYSICAL REVIEW A 92, 042108 (2015) amn()form = n and rank-1 projections associated with an from which positivity of t,s follows according to Eq. (7)by orthonormal basis {|n } of H, one finds employing Lemma 1 and the fact that t is bijective for all t. Hence, is P divisible. a () = Tr{|m m|K (|n n|)} mn t 2 = γj (t)| m|Aj (t)|n | , (20) C. Connection to classical Markovian stochastic processes j The definition of Markovianity stated above is thus equiv- which is condition (15). alent to P divisibility of a dynamical process if this We point out that condition (15) only guarantees that the notion is well defined at all. However, the measure for dynamics generated by Kt is positive. To warrant complete non-Markovianity (37) may be evaluated for any dynamical positivity of the dynamics as required for a dynamical map, process, showing its great benefit. Similarly, one could obtain one has to impose further constraints on the generator, which the equivalence of CP divisibility with Markovian behavior are unfortunately not completely known. Positivity of the rates if the dilated process ⊗ 1H is considered as shown in γj (t) for all j and t 0 represents, for example, a sufficient Ref. [13]. However, P divisible quantum processes offer (but not necessary) condition for this property. the relevant feature of a distinct connection to classical Obviously, the conditions for P and CP divisibility coincide Markovian stochastic processes. To show this, we employ the for a master equation with a single decay channel. We may characterization of P divisible processes given in Theorem 1. alternatively characterize P divisibility by employing the Let ρ(t) be the solution of the master equation contraction property (7) and link it to the quantum Markovian d behavior defined above (see Definition 1). To this end, we first ρ(t) = Kt ρ(t), (24) prove the following lemma. dt Lemma 1. For any Hermitian operator X = 0 there exists a with initial state ρ(0) where Kt is given by Eq. (13). The time- real number λ>0 and a Helstrom matrix such that X = λ. evolved state admits an instantaneous spectral decomposition † Proof. Let X = X = 0 be given. If X 0, then ρ1 = − = | | (TrX) 1X defines a state so that for λ = TrX the Helstrom ρ(t) pm(t) φm(t) φm(t) , (25) m matrix characterized by p1 = 1 and p2 = 0 with arbitrary ρ2 proves the claim, and similarly for X 0. so {|φm(t) } defines an orthonormal basis on H and {pm(t)} Hence, suppose that X is indefinite. Employing the Jordan- represents a classical probability distribution for all t 0. Hahn decomposition, we thus find nonzero operators Y1,2 0, By virtue of the orthonormality, the eigenvalues pm(t) = ⊥ = − | |= where Y1 Y2 such that X Y1 Y2 and therefore Tr X φm(t)|ρ(t)|φm(t) obey the following closed differential −1 TrY1 + TrY2 > 0. Clearly, the operators ρi = (TrYi ) Yi de- equation: fine states and we have d = − pm(t) = [Wmn(t)pn(t) − Wnm(t)pm(t)], (26) X λ(p1ρ1 p2ρ2) (21) dt n for λ = Tr|X| and p = λ−1TrY . These quantities are indeed i i where positive and sum to one, thus representing a probability 2 distribution, which concludes the proof. Wmn(t) = γj (t)| φm(t)|Aj (t)|φn(t) | . (27) Since dynamical maps t as well as the trace norm j are homogeneous (with respect to positive numbers) it thus = suffices to apply the characterization of positivity (7)to Obviously, the term in Eq. (26) with m n drops out. Given Helstrom matrices, which yields the following theorem. any solution of a quantum master equation, one thus obtains Theorem 2. If the dynamical maps defining a process are a classical jump process. We emphasize that the rates Wmn(t) {| } bijective, then is Markovian if and only if it is P divisible. depend in general on the initial eigenbasis φm(0) as well as the initial probability distribution {pm(0)}. Proof. We first note that p1 t (ρ1) − p2 t (ρ2) = t () According to the theory of classical stochastic pro- holds for any probability distribution {pi } and pair of states ρi . Hence, it suffices to consider the time evolution of Helstrom cesses, any hierarchy of n-point probability distributions matrices when studying quantum Markovianity. Suppose is Pn(yn,tn; ...; y1,t1) with discrete sample space satisfying P divisible. It follows that the consistency relations = P (y ,t ; ...; y ,t ) 0, (28) t () 1 t,s( s ()) 1 s () 1 (22) n n n 1 1 for all t s 0 and Helstrom matrices due to positivity of P1(y1,t1) = 1, (29) ∈ . Hence, () is a monotonically decreasing function y1 t,s t 1 of time for any showing that the process is Markovian. − P (y ,t ; ...; y ,t ; ...; y ,t ) = P − (y ,t ; ...; y ,t ) For the converse, we first note that the inverse map 1 n n n m m 1 1 n 1 n n 1 1 t ∈ exists for all t as the dynamical maps are bijective on the set ym of Hermitian operators by assumption. Thus, the maps t,s (30) exist for all t s 0. Now let the process be Markovian, for all 1 m n and 0 t 042108-4 GENERALIZED TRACE-DISTANCE MEASURE CONNECTING . . . PHYSICAL REVIEW A 92, 042108 (2015) For classical Markov processes obeying the relation equation has the property that the eigenbases {|φn(t) } of ρ(t) run over all orthonormal bases when varying the initial P | − (y ,t |y − ,t − ; ...; y ,t ) 1 n 1 n n n 1 n 1 1 1 state ρ(0). This is satisfied if the maximally mixed state is = P1|1(yn,tn|yn−1,tn−1) (31) in the image of the dynamical map t , which holds true for sufficiently small times due to continuity of the process for any conditional probability distribution P | − , the full hi- 1 n 1 and as we have = 1. For two-level systems it is shown erarchy is completely determined by the one-point probability 0 in Appendix A that this constraint is not only sufficient but distribution P and the so-called transition probability P | . 1 1 1 also necessary. If any orthonormal basis is encountered, all However, these two non-negative functions cannot be chosen classical processes derived from the quantum master equation arbitrarily but must satisfy are classically Markovian if and only if the quantum dynamics P1(y2,t2) = P1|1(y2,t2|y1,t1)P1(y1,t1), (32) is P divisible. y ∈ 1 III. GENERALIZATION OF THE P | (y ,t |y ,t ) = P | (y ,t |y ,t )P | (y ,t |y ,t ), 1 1 3 3 1 1 1 1 3 3 2 2 1 1 2 2 1 1 TRACE-DISTANCE-BASED NON-MARKOVIANITY y ∈ 2 MEASURE (33) Having stated the generalized definition for quantum non- where Eq. (33) is called the Chapman-Kolmogorov equation. Markovianity, which allows for a connection to classical These equations thus characterize uniquely a Markov process. Markov processes, we now consider the expression of the One may equivalently write this relation for the transition corresponding measure and address its mathematical and probability in its differential form physical features. According to the definition of a quantum d Markovian process given in Sec. II A, it is natural to con- P | (y,t|x,s) dt 1 1 sider the following measure for quantum non-Markovianity, quantifying the degree of memory effects with respect to this = W | − W | [ yz(t)P1|1(z,t x,s) zy (t)P1|1(y,t x,s)], (34) generalized definition [21]: z∈ where the transition probability per unit time W (t) is non- N ( ) ≡ max dt σ(t,pi ,ρi ) (37) yz { } negative and represents the probability for a transition to y pi ,ρi σ>0 given the classical state was z at time t. It is clear that the with (differential) Chapman-Kolmogorov equation characterizes d σ (t,p ,ρ ) ≡ p (ρ ) − p (ρ ) , (38) the transition probability P1|1 but a similar equation, the so- i i dt 1 t 1 2 t 2 1 called Pauli master equation, can be derived for the one-point probability distribution, that is, where the integration runs over all intervals where the dis- tinguishability p (ρ ) − p (ρ ) increases. A process d 1 t 1 2 t 2 1 P (y,t) = [W (t)P (z,t) − W (t)P (y,t)]. (35) is said to be non-Markovian if N ( ) > 0. As discussed dt 1 yz 1 zy 1 z∈ in Sec. II A, this measure still admits the interpretation as a One thus concludes that Eq. (26) can be interpreted as a Pauli quantifier for the information flow from the environment back master equation for the one-point probability distribution of to the open system. In addition, as discussed in Sec. II B, a classical Markov process with ={1,...,dimH} if and Markovianity will now be equivalent to P divisibility of a only if quantum process. Note that this feature was only sufficient in the original definition (see Sec. III B). 2 Wmn(t) = γj (t)| φm(t)|Aj (t)|φn(t) | 0 (36) To begin with, we concentrate on the maximization pro- j cedure contained in the quantifier for non-Markovianity (37), = which, contrary to the original definition [9], now even requires for all t 0 and m n. Hence, P divisibility of the one to sample also over binary probability distributions. Fortu- quantum process is a sufficient condition to warrant nately, a similar characterization of pairs of states maximizing positivity of the rates Wmn(t) [see Eq. (15)]. This shows Eq. (37) can be proven [15] and yet also a local representation that quantum non-Markovianity defined with respect to the is admitted [14], simplifying the sampling significantly. We generalized trace-distance-based measure not only provides finally illustrate the present definition of non-Markovianity by an interpretation in terms of an information backflow but means of examples that show the difference from the original also allows for a connection to classical Markovian stochastic characterization and other approaches to non-Markovianity. processes. Namely, to each P divisible quantum process, given as the solution of a master equation of the form (13) with Lindblad operators and rates satisfying Eq. (15) for an A. Expressions of the generalized non-Markovianity measure initial state ρ(0), one associates a classical Markovian process Aset{pi ,ρ1,2}, where ρ1,2 ∈ S(H) and {pi } defines a binary obtained as the solution of the classical master equation (26) probability distribution, is said to be optimal if the maximum with transition rates given by Eq. (27) and initial condition in Eq. (37) is attained for it. Note that the quantum states of specified by the eigenvalues of ρ(0). an optimal set are necessarily nonequal as the dynamics of the Finally, we note that P divisibility of the quantum process norm of a nonindefinite Helstrom matrix under any dynamical is indeed equivalent to the positivity of the rates Wmn(t)of map is trivial due to trace preservation and positivity of the the classical Pauli master equation if the quantum master map. 042108-5 WIßMANN, BREUER, AND VACCHINI PHYSICAL REVIEW A 92, 042108 (2015) Theorem 3. The states of an optimal set must be orthogonal, a real number λ>0 with λ|TrX| < 1 such that i.e., 1 [p+ρ − sgn(TrX)λX] ∈ ∂U(ρ), (43) p− N ( ) = max dtσ(t,pi ,ρi ). (39) { ⊥ } pi ,ρ1 ρ2 σ>0 1 ± ≡ ± | | where p 2 (1 λ TrX ) and Proof. Suppose {pi ,ρ1,2} is optimal with ρ1 ⊥ ρ2. Along −1forx 0 the lines of the proof of Lemma 1 we obtain a probability = { } sgn(x) (44) distribution qi and two orthogonal states 1,2 such that +1 otherwise. − = − p1ρ1 p2ρ2 λ(q1 1 q2 2), (40) Proof. Let ∂U(ρ) be an enclosing surface of ρ. Consider a nonzero, Hermitian, indefinite operator X. The operator where λ = Tr|p1ρ1 − p2ρ2|.Asρ1 ⊥ ρ2,wehave0<λ<1 according to Eq. (3). Here λ>0 holds as states of an optimal Y = sgn(TrX)[(TrX)ρ − X] (45) set are nonequal by definition. By means of linearity of the defines a nonzero, Hermitian, and traceless operator. By dynamical maps t and homogeneity of the trace norm one finally obtains definition, there exists a real number μ>0 such that 1 = ρ + 2μY ∈ ∂U(ρ). (46) q1 1(t) − q2 2(t)1 = p1ρ1(t) − p2ρ2(t)1 (41) λ We then define a real number λ by means of − for all t 0 where λ 1 > 1. This shows that any in- λ − μ = (47) crease of p1ρ1(t) p2ρ2(t) 1 is exceeded by the increase 1 − λ|TrX| of q1 1(t) − q2 2(t)1. Hence, {qi , 1,2} yields a non- { } so that λ = μ/(1 + μ|TrX|) > 0 holds and hence λ|TrX| < 1. Markovianity strictly larger than the set pi ,ρ1,2 contradicting 1 ± = ± | | its optimality. For p 2 (1 λ TrX ) one finally obtains We define S˚(H) to be the interior of the state space, i.e., 2λ = ρ + {sgn(TrX)[(TrX)ρ − X]} the set of all quantum states ρ for which there is an >0 1 − λ|TrX| such that all Hermitian operators X with unit trace satisfying || − || S H λ p+ − p− ρ X 1 belong to ( ). Hence, all states in the interior = ρ − sgn(TrX)X + ρ have full rank. Note that S˚(H) =∅ if dimH =∞, which p− p− implies that the local representation of the measure (37) 1 = [p+ρ − sgn(TrX)λX], (48) introduced below is not available in infinite dimensions. p− However, these quantum systems can frequently be accurately described by finite-dimensional Hilbert spaces. which is Eq. (43). Based on the orthogonality of states of an optimal set, we As proven in Appendix B, Lemma 2 can actually be can also establish a local representation for the generalized augmented to show that the characterization (43)ofan measure as provided for the original definition in Ref. [14]. To enclosing surface is equivalent to Eq. (42). this end, one first has to prove a characterization of enclosing Theorem 4. The generalized measure of quantum non- Markovianity admits a local representation, i.e., surfaces, which are defined as follows (see Fig. 1): A set ∂U(ρ) ⊂ S(H) not containing ρ ∈ S˚(H) is called an enclosing N ( ) = max dtσ˜ (t,pi,ρi ) (49) { } ∈ surface of ρ if and only if for any nonzero, Hermitian, and pi ,ρ2 ∂U(ρ1) σ>˜ 0 traceless operator Y there exists a real number μ>0 such that with ρ + 2μY ∈ ∂U(ρ). (42) 1 d σ t,p ,ρ ≡ p ρ − p ρ , ˜ ( i i ) − 1 t ( 1) 2 t ( 2) 1 Lemma 2. Let ∂U(ρ) be an enclosing surface and X a p1ρ1 p2ρ2 1 dt nonzero, Hermitian, and indefinite operator. Then there exists (50) where ρ1 ∈ S˚(H) is any fixed inner point of the state space and ∂U(ρ1) refers to an arbitrary enclosing surface of ρ1. We emphasize thatσ ˜ (t,pi ,ρi ) is well defined as 0 < p1ρ1 − p2ρ21 for any state ρ2 ∈ ∂U(ρ1) and probability distribution {pi} as by definition of an enclosing surface we have ρ2 = ρ1 [cf. Eq. (3)]. Proof. Let ∂U(ρ1) be an enclosing surface of ρ1. To prove that the corresponding local representation yields a value smaller than or equal to the original definition (39) one follows the lines of the proof of Theorem 3. Let ρ2 ∈ ∂U(ρ1) and a probability distribution {pi } be given. Now, following the proof of Lemma 1 there exist two states 1 ⊥ 2 and a probability distribution {qi } such that FIG. 1. (Color online) Illustration of an enclosing surface ∂U(ρ) of an inner point ρ of the state space S(H). p1ρ1 − p2ρ2 = λ(q1 1 − q2 2), (51) 042108-6 GENERALIZED TRACE-DISTANCE MEASURE CONNECTING . . . PHYSICAL REVIEW A 92, 042108 (2015) where λ =p1ρ1 − p2ρ21 > 0. We then obtain master equation (24) where the generator Kt obeys 3 σ˜ (t,pi,ρi ) = σ(t,qi , i ) (52) γ (t) Kt ρ(t) = [σj ρ(t)σj − ρ(t)] (55) = 4 for all t 0 due to linearity of t and homogeneity of the j 1 trace norm and the derivative. We conclude that the right-hand for 0 t t with t 042108-7 WIßMANN, BREUER, AND VACCHINI PHYSICAL REVIEW A 92, 042108 (2015) according to the described dynamical map, is given by IV. CONCLUSION AND OUTLOOK = 1 {| − +| || + | − −| ||} We have introduced a generalization of the criterion of (t) 1 2 p1 p2 w(t) p1 p2 w(t) , (60) quantum non-Markovianity based on the flow of information. where w(t) = p1v1(t) − p2v2(t). Clearly, if p1 = p2 = 1/2, This characterization relies on the trace norm of Helstrom = 1 | − | matrices, which can also be interpreted as a measure for the then (t) 1 2 v1(t) v2(t) , showing that the unbiased case is unable to detect the non-Markovianity of the process distinguishability. By virtue of this property, the generalized resulting from the uniform translation. measure still admits an interpretation as quantifier of an Without restriction we may assume p1 p2 so that information backflow from the environment to the open quantum system. p − p for p − p > |w(t)| It was shown that the generalized criterion is equivalent to = 1 2 1 2 (t) 1 (61) P divisibility of the dynamical process, which has an explicit |w(t)| for p1 − p2 |w(t)| connection to classical Markovian stochastic processes. That and, due to Theorem 3, we may restrict ourselves to orthogonal is, any rate equation obtained from a quantum master equation states corresponding to antipodal unit vectors. That is, v1 = of a P divisible process can be interpreted as the Pauli master −v2 with |v1|=1, which implies w(t) =v1(t). equation of a classical Markov process. However, the presented If the probability distribution is such that p1 − p2 > |v1(t)| approach is more general since it can also be applied even holds for all t1 042108-8 GENERALIZED TRACE-DISTANCE MEASURE CONNECTING . . . PHYSICAL REVIEW A 92, 042108 (2015) the set of time-evolved states Im t ={ t (ρ)|ρ ∈ S(H)} for 2 two-level systems, i.e., H = C . Note that Im t defines a nonempty, convex, and compact set for any finite-dimensional Hilbert space due to linearity of t along with compactness and convexity of the state space. Lemma 3. Any orthonormal basis {|i } of H represents the eigenbasis of a quantum state ρ(t) ∈ Im t if and only if 1 ∈ 2 12 Im t . The “if” statement is clear from the fact that any orthonor- 1 ∈ mal basis defines a resolution of identity. Hence, if 2 12 Im t then any basis {|i } defines at least the eigenbasis of the maximally mixed state, proving the claim. It is clear that the same reasoning actually applies to any finite-dimensional FIG. 3. (Color online) Two-dimensional cut of a hyperplane in Hilbert space. the Bloch representation separating the two disjoint nonempty, 1 To show the reverse we employ the Hahn-Banach separation convex, and compact sets { 12} and Im t . As the plane intersects the 1 ∈ 2 theorem for normed vector spaces. Suppose that 2 12 / maximally mixed state its intersection with the state space comprises Im t . Then there exists a real-valued, linear, and continuous an infinite set of pure and orthogonal states defining orthonormal 2 functional ϕA separating the disjoint nonempty, convex, and bases of C . 1 compact sets Im t and { 12}.Thatis,wehave 2 1 { | ∈ } ϕA 2 12 < inf ϕA(ρ(t)) ρ(t) Im t , (A1) Proof. That any enclosing surface obeys a characterization = { } where ϕA(X) Tr AX for some Hermitian operator A due to in terms of nonzero, Hermitian, and indefinite operators has the Riesz representation theorem. By means of the transforma- already been proven in Lemma 2. Conversely, suppose the → − tion A A TrA one may also assume that the operator A states in ∂U(ρ) are characterized by Eq. (B1). Hence, for a { | = † { }= = } is traceless. The set X X X ,Tr X 1,ϕA(X) 0 thus nonzero, indefinite, and Hermitian operator X there exists a describes a hyperplane that intersects the maximally mixed real number λ>0 with λ|TrX| < 1 such that state and separates it from Im t . Employing the Bloch representation, this hyperplane de- 1 3 = [p+ρ − sgn(TrX)λX] ∈ ∂U(ρ), (B3) fines an ordinary plane in R that contains the origin and p− therefore intersects the surface of the Bloch sphere in a 2 1 unit circle. As pure, orthogonal states on C correspond to ± = ± | | where p 2 (1 λ TrX ). Now consider the map antipodal points on the surface of the Bloch sphere, we thus conclude that the (hyper)plane contains an infinite set of ρ (X) ≡ sgn(TrX)[(TrX)ρ − X](B4) orthonormal bases that do not represent eigenbases of quantum defined on the set of nonzero, Hermitian, and indefinite states ρ(t) ∈ Im t (cf. Fig. 3). operators. The operator Y = ρ (X) represents a traceless Hermitian operator and Y = 0 if and only if (TrX)ρ = X, APPENDIX B: CHARACTERIZATIONS OF ENCLOSING contradicting that X is indefinite. Thus, we have Y = 0 and SURFACES one finds In this section we show that the relation for enclosing 1 surfaces in terms of nonzero, Hermitian, and indefinite = {p+ρ − λ[|TrX|ρ − Y ]} operators derived in Lemma 2 is equivalent to the original p− definition, i.e., the following lemma holds. p+ − λ|TrX| λ = ρ + Y Lemma 4. 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