Measure for the Non-Markovianity of Quantum Processes

Transcription

1 Measure for the Non-Markovianity of Quantum Processes Elsi-Mari Laine 1, Jyrki Piilo 1, an Heinz-Peter Breuer 1 Turku Center for Quantum Physics, Department of Physics an Astronomy, University of Turku, FI-14, Turun Yliopisto, Finlan Physikalisches Institut, Universität Freiburg, Hermann-Herer-Strasse 3, D-7914 Freiburg, Germany (Date: June 16, 1) Recently, a measure for the non-markovian behavior of quantum processes in open systems has been evelope which is base on the quantification of the flow of information between the open system an its environment [Phys. Rev. Lett. 13, 141 (9)]. The information flow is connecte to the rate of change of the trace istance between quantum states which can be interprete in terms of the istinguishability of these states. Here, we elaborate the mathematical etails of this theory, present applications to specific physical moels, an iscuss further theoretical an experimental implications, as well as relations to alternative approaches propose recently. arxiv:1.583v [quant-ph] 15 Jun 1 PACS numbers: 3.65.Yz, 3.65.Ta, 4.5.Lc I. INTRODUCTION A Markov process in the evolution of an open quantum system typically gives rise to a quantum ynamical semigroup for which the most general representation can be written in the Linbla form [1, ]. There exist however complex systems for which this relatively simple escription of the open system ynamics in terms of a Markovian master equation fails to give a comprehensive picture of the ynamics [3]. Thus in many realistic physical systems the Markovian approximation of the ynamics gives an overly simplifie picture of the open system evolution an a more rigorous treatment of the ynamics is require. To give insights into the nature of non-markovian effects many analytical methos an numerical simulation techniques have been evelope in recent years (see, for example, Refs. [4 17]). Non-Markovianity manifests itself in the ifferent approaches in a variety of ways an there exists no general recipe for comparing the egree of non-markovianity in ifferent physical moels. In orer to give a general quantity etermining the egree of non- Markovian behavior in the open system ynamics, one has to rigorously efine what makes a ynamical map non-markovian. Here, we iscuss a recently propose measure for the egree of non-markovian behavior which is base on the trace istance between quantum states [18]. The trace istance escribes the probability of successfully istinguishing two quantum states an the change in the trace istance of two open system states can be interprete as a flow of information between the system an the environment. When the trace istance ecreases information flows from the system into the environment, while an increase of the trace istance signifies a backflow of information from the environment to the system. Markovian Electronic aress: emelai@utu.fi Electronic aress: jyrki.piilo@utu.fi Electronic aress: breuer@physik.uni-freiburg.e processes ten to continuously ecrease the istinguishability between any two states of the open system, i.e., information flows continuously from the system to the environment. The conition which efines a non-markovian ynamical map is that the map allows an information flow from the environment to the system an, therefore, allows the system to gain information about its former state. This conition for a non-markovian map leas to a rigorous an general efinition of a measure for the egree of non-markovianity in open quantum systems. In Sec. II we construct the measure for non- Markovianity an iscuss its properties for some general classes of quantum processes in open systems. It is shown that the non-ivisibility of the ynamical map is necessary for the process to be non-markovian. Hence, the measure vanishes for quantum ynamical semigroups an for time-epenent Markov processes. We also emonstrate that the appearance of negative rates in the quantum master equation is a necessary conition for non- Markovianity. In Sec. III we illustrate the etermination of the measure for a two-level system an for a Λ-type atom in a cavity. Section IV contains a etaile iscussion of several alternative ways for efining a measure for non-markovianity. Moreover, we present possible experimental strategies for the etection of non-markovian effects. The conclusions are rawn in Sec. V. II. THE MEASURE FOR NON-MARKOVIANITY A. Construction of the measure To construct the measure for non-markovianity we nee a measure for the istance between any pair of quantum states represente by ensity matrices ρ 1 an ρ. Such a measure is given by the trace istance, which is efine as D(ρ 1, ρ ) = 1 Tr ρ 1 ρ, (1)

2 where the moulus of an operator A is efine by A = A A. The trace istance D yiels a natural metric on the state space an satisfies D 1. It has many nice properties that make it a useful measure for the istance between quantum states [19]. First, the trace istance is preserve uner unitary transformations U, D(Uρ 1 U, Uρ U ) = D(ρ 1, ρ ). () Secon, all completely positive an trace preserving (CPT) maps Φ (trace preserving quantum operations) are contractions for this metric, D(Φρ 1, Φρ ) D(ρ 1, ρ ). (3) Thir, the trace istance has a physical interpretation as a measure of state istinguishability. Suppose Alice prepares a quantum system in the state ρ 1 with probability 1/, an in the state ρ with probability 1/. She gives the system to Bob, who performs a measurement to istinguish the two states. The maximal probability that Bob can ientify the state given to him is [] p max = 1 [1 + D(ρ 1, ρ )]. (4) Hence, the trace istance represents the maximal bias in favor of the correct state ientification which Bob can achieve through an optimal strategy. For example, if ρ 1 an ρ have orthogonal supports the trace istance becomes D(ρ 1, ρ ) = 1 an thus p max = 1, which means that Bob is able to istinguish the states with certainty. The change in the istinguishability of states of an open system can be interprete as a flow of information between the system an the environment. We consier here quantum processes given by a ynamical CPT map Φ(t, ) which transforms the initial states ρ() at time zero to the states ρ(t) at time t, ρ() ρ(t) = Φ(t, )ρ(). (5) When such a quantum process reuces the istinguishability of states, information is flowing from the system to the environment. Likewise, the increase of the istinguishability signifies that information flows from the environment to the system. The invariance uner unitary transformations () inicates that information is preserve uner the ynamics of close systems. The contraction property of Eq. (3) guarantees that the maximal amount of information the system can recover from the environment is the amount of information earlier flowe out the system. The basic iea unerlying our construction for the measure of non-markovianity in a quantum process is that for Markovian processes information flows continuously from the system to the environment. In orer to give rise to non-markovian effects there must be, for some interval of time, an information flow from the environment back to the system. The information flowing from the environment back to the system allows the earlier states of the system to have an effect on the later ynamics of the system, i.e., it allows the emergence of memory effects. We efine the rate of change of the trace istance of a pair of states by means of σ(t, ρ 1, ()) = t D(ρ 1(t), ρ (t)), (6) where ρ 1, (t) = Φ(t, )ρ 1, (). For a non-markovian process escribe by a ynamical map Φ(t, ), information must flow from the environment to the system for some interval of time an thus we must have σ > for this time interval. A measure of non-markovianity shoul measure the total increase of istinguishability over the whole time evolution, i.e., the total amount of information flowing from the environment back to the system. This suggests efining the measure N (Φ) for the non- Markovianity of the quantum process Φ(t, ) through N (Φ) = max tσ(t, ρ 1, ()). (7) ρ 1,() σ> The time integration is extene over all time intervals (a i, b i ) in which σ is positive an the maximum is taken over all pairs of initial states. Due to Eq. (6) the measure can be written as N (Φ) = max ρ 1,() [D(ρ 1 (b i ), ρ (b i )) D(ρ 1 (a i ), ρ (a i ))]. i (8) To calculate this quantity one first etermines for any pair of initial states the total growth of the trace istance over each time interval (a i, b i ) an sums up the contribution of all intervals. N (Φ) is then obtaine by etermining the maximum over all pairs of initial states. While it may be ifficult to erive an analytical expression for the measure efine in Eq. (8), the numerical evaluation of the measure is relatively easy provie the ynamical map is known explicitly. We will iscuss in Sec. III the etermination of N (Φ) for some specific examples. B. Classification of quantum processes Having efine our measure for non-markovianity we iscuss in this section the properties of this measure for some general classes of quantum processes. Specific physical systems will be investigate in Sec. III. 1. Divisible maps The ynamical map Φ(t, ) is efine to be ivisible if for all t, τ the CPT map Φ(t+τ, ) can be written as composition of the two CPT maps Φ(t + τ, t) an Φ(t, ), Φ(t + τ, ) = Φ(t + τ, t)φ(t, ). (9) We note that this efinition iffers slightly from the usual efinition of ivisibility accoring to which a CPT map

3 3 Λ (quantum channel) is sai to be ivisible if there exist CPT maps Λ 1 an Λ such that Λ = Λ 1 Λ, where it is assume that neither Λ 1 nor Λ is a unitary transformation [1]. In Eq. (9) the left-han sie as well as the secon factor on the right-han sie are fixe by the given ynamical map. Hence, Eq. (9) requires the existence of a certain linear transformation Φ(t + τ, t) which maps the states at time t to the states at time t + τ an represents a CPT map (that may be a unitary transformation) for all t an all τ. There are many quantum processes which are not ivisible. For instance, if Φ(t, ) is not invertible, a linear map Φ(t + τ, t) which fulfills Eq. (9) may not exist. Moreover, even if a linear map Φ(t + τ, t) satisfying Eq. (9) oes exist, this map nees not be completely positive, an not even positive. We claim that all ivisible ynamical maps are Markovian. To prove this statement suppose that Φ(t, ) is ivisible. For any pair of initial states ρ 1, () we then have ρ 1, (t + τ) = Φ(t + τ, t)ρ 1, (t). (1) Since Φ(t + τ, t) is a CPT map we can apply the contraction property (3) to obtain: D(ρ 1 (t + τ), ρ (t + τ)) D(ρ 1 (t), ρ (t)). (11) This shows that for all ivisible ynamical maps the trace istance ecreases monotonically, i.e., σ(t, ρ 1, ()) an, therefore, N (Φ) =. Thus, we conclue that all ivisible processes are Markovian an that non-markovian processes must necessarily be escribe by a nonivisible ynamical map.. Quantum ynamical semigroups The prototype of a Markovian ynamics is provie by a Markovian master equation for the ensity matrix, ρ(t) = Lρ(t), (1) t with a generator in Linbla form [1, ] Lρ = i[h, ρ] + [ γ i A i ρa i 1 { A i A i, ρ} ], (13) i involving a time-inepenent Hamiltonian H as well as time-inepenent Linbla operators A i an positive ecay rates γ i. Such a master equation leas to a ynamical semigroup of CPT maps, Φ(t, ) = exp(lt). With Φ(t + τ, t) = exp(lτ) the ivisibility conition (9) is trivially satisfie. Hence, we have N (Φ) = for all ynamical semigroups, i.e., for all processes escribe by a master equation in the Linbla form. 3. Time-epenent Markov processes The ivisibility property hols for a much larger class of quantum processes than those escribe by a master equation of the form (1). Suppose we have a time-local master equation of the form ρ(t) = K(t)ρ(t) (14) t with a time-epenent generator K(t). It can be shown that in orer to preserve the Hermiticity an trace of the ensity matrix this generator must be of the form [1, 8] K(t)ρ = i [H(t), ρ] (15) + [ γ i (t) A i (t)ρa i (t) 1 { A i (t)a i(t), ρ} ]. i By contrast to the assumptions in Eq. (13) the Hamiltonian H(t), the Linbla operators A i (t) an the ecay rates γ i (t) may now epen on time. If the ecay rates are positive functions, γ i (t), the generator (15) is in Linbla form (13) for each fixe t. Such a process with γ i (t) may be calle time-epenent Markovian although the corresponing ynamical map [ ] Φ(t, ) = T exp t K(t ) (16) oes not yiel a ynamical semigroup (T enotes the chronological time-orering operator). However, one can easily see that the ivisibility conition (9) still hols because the map [+τ ] Φ(t + τ, t) = T exp t K(t ) t (17) is CPT for γ i (t). Thus we can conclue that for all time-epenent Markovian processes we again have N (Φ) =. We have just seen that a quantum process given by the time-local master equation (14) with positive rates leas to a ivisible ynamical map. Uner certain conitions the converse of this statement is also true. More precisely, if the ynamical map Φ(t, ) is ivisible with a unique map Φ(t+τ, t) epening smoothly on τ, then the corresponing ensity matrix ρ(t) obeys a master equation of the form (14) with positive rates in the generator (15). In fact, using ρ(t + τ) = Φ(t + τ, t)ρ(t) we fin t ρ(t) = τ Φ(t + τ, t)ρ(t), (18) τ= an, hence, we obtain the master equation (14), where the generator is given by K(t) = τ Φ(t + τ, t). (19) τ= Since Φ(t + τ, t) is CPT an satisfies Φ(t, t) = I, this generator must be in Linbla form for each fixe t, i.e., it must have the form (15) with γ i (t).

4 4 4. Non-Markovian processes The measure for quantum non-markovianity oes not epen on any specific mathematical representation of the ynamics. There are many ifferent such representations, e.g., through generalize master equations involving a certain memory kernel. However, quantum master equations with the time-local structure given by Eqs. (14) an (15) are also very useful for the escription of non- Markovian processes. It follows from the preceing results that in orer for such a master equation to yiel a nonzero measure, N (Φ) >, at least one of the rates γ i (t) must take on negative values for some interval of time. We emphasize that temporarily negative rates in the master equation o in general not lea to a violation of the complete positivity of the ynamical map. Many examples for time-local master equations with negative rates are known in the literature. Further examples will be iscusse in the next section. III. EXAMPLES A. Two-level system We stuy the ynamics of a two-level atom with excite state + an groun state which is couple to a reservoir of fiel moes initially in the vacuum state. In Ref. [18] we have escribe the etune Jaynes- Cummings moel, while here we treat the resonant case. We will show that the pair of states maximizing the measure for non-markovinity is ifferent in the two cases. This emonstrates that the change in both the populations an the coherences plays a crucial role in the flow of information between the system an the environment. The two-level atom moel can easily be solve exactly [3] an leas to a ynamical map Φ(t, ) which can be represente in terms of the elements ρ ±± (t) = ± ρ(t) ± of the ensity matrix ρ(t) as follows, ρ ++ (t) = G(t) ρ ++ (), ρ (t) = ρ () + (1 G(t) )ρ ++ (), ρ + (t) = G(t)ρ + (), ρ + (t) = G (t)ρ + (). () Here, the function G(t) is efine as the solution of the integroifferential equation t G(t) = t 1 f(t t 1 )G(t 1 ) (1) corresponing to the initial conition G() = 1, where f(t t 1 ) enotes the two-point reservoir correlation function (Fourier transform of the spectral ensity). The map () is completely positive if an only if G(t) 1. One can easily check that Φ(t, ) can be ecompose as in Eq. (9), where the map Φ(t + τ, t) is given by ρ ++ (t + τ) = G(t + τ) G(t) ρ ++ (t), ( ρ (t + τ) = ρ (t) + 1 G(t + τ) ) ρ ++ (t), G(t) ρ + (t + τ) = G(t + τ) ρ + (t), G(t) ρ + (t + τ) = G (t + τ) G ρ + (t). () (t) It follows from these equations that a necessary an sufficient conition for the complete positivity of Φ(t + τ, t) is given by G(t + τ) G(t). Thus we see that the ynamical map of the moel is ivisible if an only if G(t) is a monotonically ecreasing function of time. Note that this statement hols true also for the case that G(t) vanishes at some finite time. With the help of the above results one can easily erive an analytical formula for the time erivative of the trace istance, σ(t, ρ 1, ()) = G(t) a + b G(t), (3) G(t) a + b t where a = ρ ++ 1 () ρ ++ () enotes the ifference of the populations an b = ρ + 1 () ρ + () the ifference of the coherences of the initial states. This relation shows that the trace istance increases at some point if an only if G(t) increases at this point. We conclue that the measure for non-markovianity is positive, N (Φ) >, if an only if the ynamical map is nonivisible. A positive measure for non-markovianity is not only linke to a breakown of the ivisibility of the ynamical map, but also to the emergence of a negative rate in the corresponing master equation (14). In fact, as long as G(t) one can write an exact master equation of this form with the generator K(t)ρ = i S(t)[σ +σ, ρ] (4) [ +γ(t) σ ρσ + 1 ] {σ +σ, ρ}, where we use the efinitions ) ) (Ġ(t) (Ġ(t) γ(t) = R, S(t) = I. (5) G(t) G(t) Writing the rate γ(t) as γ(t) = G(t) (6) G(t) t we see that an increase of G(t) an, hence, a breakown of the ivisibility leas to a negative rate in the generator of the master equation. Thus we fin that for the present

5 Γ Λ FIG. 1: The non-markovianity N (Φ) for the ampe Jaynes- Cummings moel as a function of the coupling strength γ. Gray ots: 1 ranomly rawn pairs of pure an mixe initial states. Black circles: The initial pair given by Eq. (9) which leas to the maximum in Eq. (7). moel a nonzero measure for non-markovianity is equivalent to the non-ivisibility of the ynamical map an to the occurrence of a negative rate in the master equation. As an example we consier the case of a Lorentzian reservoir spectral ensity which is on resonance with the atomic transition frequency an leas to an exponential two-point correlation function f(τ) = 1 γ λe λ τ, (7) where γ escribes the coupling strength an λ the spectral with (ampe Jaynes-Cummings moel). Solving Eq. (1) with this correlation function we fin G(t) = e λt/ [cosh ( ) t + λ ( )] t sinh, (8) where = λ γ λ. We see that for small couplings, γ < λ/, the function G(t) ecreases monotonically. The ynamical map is thus ivisible in the weak coupling regime, the rate γ(t) is positive, an the measure for non- Markovianity vanishes. However, in the strong coupling regime, γ > λ/, the function G(t) starts to oscillate, showing a non-monotonic behavior. Consequently, the ynamical map is then no longer ivisible an N (Φ) >. We note that in the strong coupling regime the rate γ(t) iverges at the zeros of G(t). However, the master equation can still be use to escribe the evolution between successive zeros an, therefore, the connection between a positive measure an negative rates in the master equation remains vali. There is thus a threshol γ = λ/ for the systemreservoir coupling below which N (Φ) =. We fin that the measure increases monotonically with increasing coupling for γ > λ/. This is illustrate in Fig. 1. The maximization over the pair of initial states ρ 1, () in expression (7) has been performe here by a Monte Carlo sampling of pairs of initial states. Our simulations provie strong evience that the maximum is attaine for the initial states ρ 1 () =, ρ () = 1 ( + + )( + + ). (9) In Ref. [18] we calculate the measure for the etune Jaynes-Cummings moel in the weak coupling limit. In this example the maximum of the measure was obtaine for the initial states ρ 1 () = an ρ () = + +, i.e. for the invariant groun state an the excite state. The ifference in the maximization for the resonant an the off-resonant case arises from the fact that the rate at which the populations an the coherences initially ecay is much larger for the resonant case. Consequently, the growth of the trace istance occurs after the excite state population an the coherences have reache the value zero. After this point, the increase of the coherences yiels the ominant contribution to the increase of the trace istance. Therefore, the maximal growth of the trace istance for the resonant case is reache for the invariant state an the state with maximal initial coherence. B. Λ-moel The Λ-moel escribes a three-level atom with excite state a an two groun states b an c interacting off-resonantly with a cavity fiel. This example allows us to emonstrate how the measure for non-markovianity operates in a multi-channel case an how there can exist simultaneously positive an negative ecay rates for ifferent channels. The spectral ensity we use is J(ω) = γ λ π (ω cav ω) + λ, (3) where ω cav is the resonance frequency of the cavity. Further etails an the master equation escribing the ynamics of the Λ-type atom are presente in the Appenix. The generator of the master equation is of the form of Eq. (15) with two Linbla operators b a an c a, an two time-epenent ecay rates γ 1 (t) an γ (t). The etunings of the transition frequencies of the Λ- atom from the cavity resonance frequency are enote by i = ω i ω cav. When the etuning parameters 1 an are both sufficiently large, the ecay rates γ 1 (t) an γ (t) get temporarily negative values an this gives rise to an information flow from the environment to the system. On one han, the two ecay rates γ 1 (t) an γ (t) have simultaneous negative regions when 1 =. On the other han, when 1, the ecay rates can have opposite signs. In this case, the co-operative action of the other channel reuces the amount of information flowing from the environment to the system. The maximum of the measure over the initial states is reache when the states are chosen to be a a an b b, or a a an c c, epening on which of the channels has more information flow from the environment to the system. When

6 6 1 an are such that the channel corresponing to the ecay rate γ i (t) (i = 1 or ) causes more information flow from the environment to the system we get the expression σ(t) = γ i (t)ρ aa (t). (31) The function ρ aa (t) is specifie in the Appenix. Eq. (31) shows that the Λ-system is non-markovian if one of the ecay rates γ 1 (t) or γ (t) takes on negative values. The maximization over the the pair of initial states is emonstrate in Fig., where the measure was again calculate numerically from a large sample of initial states FIG. : The non-markovianity N (Φ) for the Λ-moel as a function of the etuning 1 for /λ = 5 an γ /λ =.1. Gray ots: 1 ranomly rawn pairs of initial states. Circles: The initial pair ρ 1() = a a an ρ () = b b. Pluses: The initial pair ρ 1() = a a an ρ () = c c. At 1 = the pair which yiels the maximum in Eq. (7) changes from the latter to the former pair of initial states. IV. 1 Λ DISCUSSION A. Alternative istance measures We have base our efinition of the measure of non- Markovianity on the trace istance (1). An alternative measure is obtaine if one replaces the trace istance by the relative entropy S(ρ 1 ρ ) = Tr [ρ 1 (log ρ 1 log ρ )]. (3) Using this quantity as a measure for the istance between quantum states one is le to a similar interpretation as before because the relative entropy also ecreases uner CPT maps []. There are however some technical problems an limitations in the usefulness of the relative entropy which arise from the fact that for many pairs ρ 1 an ρ the relative entropy becomes infinite [3] an thus leas to singularities in the efinition of the measure. This situation can occur even in the simple case of a two-state system, emonstrating the problems of the relative entropy concept in efining a general measure for non-markovianity. No such problems occur for the trace istance which is well-efine an finite for all physical states represente by positive trace class operators. Another common measure for the istance between two states is the Hilbert-Schmit istance D HS (ρ 1, ρ ) = Tr [(ρ 1 ρ ) ]. (33) For two-imensional Hilbert spaces the Hilbert-Schmit istance an the trace istance coincie an correspon to the Eucliean istance between the Bloch vectors representing the states (up to numerical factors). However, the Hilbert-Schmit istance is not suitable for a efinition of non-markovianity since CPT maps are in general not contractions for this metric [4]. Thus, the Hilbert- Schmit istance oes not provie a natural way to efine the information flow between system an environment. B. Experimental issues The exact etermination of the measure generally requires solving the complete reuce ynamics which can be a ifficult task for more complex systems. However, any observe growth of the trace istance is a clear signature for non-markovian behavior an leas to a lower boun for N (Φ). The measure for non-markovianity introuce here coul therefore be useful also for the experimental etection of non-markovianity. In an experiment one has to perform a state tomography on ifferent ensembles at ifferent times in orer to ecie whether or not the trace istance has increase. Such an experiment also allows the valiation of theoretical moels or approximation schemes. Consier a theoretical moel preicting σ(t, ρ 1, ()) > for some interval t (t 1, t ) an for some pair of initial states ρ 1, (). In the experiment one shoul then etect the increase of the trace istance between the states ρ 1 (t) an ρ (t) in this time interval. This type of experiment coul be base, e. g., on the recent proposal to use a trappe ion to stuy quantum Brownian motion in the non-markovian regime [5]. The explicit experimental implementation of this system can be one, e. g., by using reservoir engineering techniques [6] or by using the trappe ion as a quantum simulator for non-markovian ynamics [7]. One of the possibilities here to etect non-markovianity is to prepare the ion in various Fock states, an to stuy the trace istance ynamics as escribe above. A great avantage of the present approach is that it also allows to plan experiments for testing non- Markovianity without knowing the properties of the environment or the system-environment interaction. The interactions an environmental properties can be quite ifficult to moel in an experimental setup. By performing a state tomography for two states of the open system uner stuy at many ifferent times, one can etermine whether there has been any increase in the trace istance an, hence, non-markovian behavior in the ynamics. From this information one can conclue whether or not non-markovian effects are crucial in the ynamics an

7 7 in this way also gain some knowlege on the nature of the environment an the interactions. An example uner active investigation, where nevertheless a complete characterization of the environment is still missing an where non-markovianity coul play a role, is given by the energy transfer in photosynthetic systems [8]. C. Other approaches to non-markovianity Recently, other interesting approaches to the characterization an quantification of non-markovianity have been propose. The measure suggeste in Ref. [5] quantifies non-markovianity in terms of the minimal amount of noise require to make a given quantum channel Markovian. The most important ifference to our approach is that this measure is base on the properties of the ynamical map at a given time, i.e., on the properties of the quantum channel represente by a snapshot of the time evolution. Hence, this approach assesses to what exten the ynamical map at each fixe time t eviates from an element of a Markovian process. The funamental ifference between the notion of non-markovianity use in Ref. [5] an ours can be seen from the following simple example. We consier the ynamical map Φ(t, ) of a two state system unergoing a pure e- an re-phasing ynamics which is given by (using the notation of Sec. III A) ρ ++ (t) = ρ ++ (), ρ (t) = ρ (), ρ + (t) = g(t)ρ + (), ρ + (t) = g(t)ρ + (), (34) where the function g(t) = 1 [1 + cos ωt] escribes a perioic oscillation of the coherences. The trace istance for this moel is given by D(ρ 1 (t), ρ (t)) = a + g (t) b (35) where a = ρ ++ 1 () ρ ++ () an b = ρ + 1 () ρ + (). For b the trace istance thus oscillates perioically an, hence, N (Φ) = + accoring to the efinition (7) of our measure. On the other han, the non-markovianity in the sense of Ref. [5] is zero because for any fixe t the ynamical map (34) can be written as an element of a Markovian semigroup: Φ(t, ) = exp(l) with the Linbla generator Lρ = Γ (σ 3ρσ 3 ρ), where Γ = ln g(t ). A further interesting measure propose recently [6] is closely connecte to the measure iscusse here. In fact, the measure of Ref. [6] quantifies eviations from the ivisibility of the ynamical map. As we have seen, the non-ivisibility of the ynamical map is a necessary conition for N (Φ) to be nonzero. However, we conjecture that our notion of non-markovianity an the one use in [6] are not strictly equivalent, i.e., that there are nonivisible maps with N (Φ) =. Further consierations concerning this point will be publishe elsewhere. V. CONCLUSIONS We have constructe a measure N (Φ) for the non- Markovianity of quantum processes in open systems in terms of the information flowing from the environment to the system uring the time evolution. The flow of information is characterize by the change of the istinguishability between a pair of quantum states which, in turn, is linke to the change of the trace istance between these states. We have also argue why the trace istance represents the most suitable istance measure for quantum states to be use in this context. Furthermore, since we have evelope a genuine quantitative measure, the results presente here also allow to compare the egree of non-markovianity of ifferent types of physical systems. It has been emonstrate that a nonzero measure for non-markovianity requires the ynamical map to be nonivisible, a property which is thus necessary for the presence of memory effects in the open system ynamics. It has also been shown that Markovian semigroups an time-epenent Markov processes are ivisible an, hence, lea to N (Φ) =. The examples iscusse here illustrate how the measure can be calculate for a given open system ynamics an that a nonzero measure for non-markovianity is linke to the emergence of negative ecay rates in the corresponing master equation. Our measure for non-markovianity has a clear operational meaning base on the interpretation of the trace istance in terms of the istinguishability of states, an suggests various ways to experimentally ecie whether a system uner stuy is non-markovian. The measurement scheme iscusse here has the great avantage that it oes not presuppose any knowlege about the structure of the environment or about the system-environment interaction an, therefore, also gives valuable information on the theoretical moelling of the open system ynamics. If, for example, a substantial increase of the trace istance is observe experimentally, a mathematical escription of the ynamics through any equation escribing a Markovian or time-epenent Markovian process is exclue. This shows that our measure is a useful tool for the characterization of non-markovianity, both in experiments on open systems an in their theoretical analysis an moelling. We have argue that the characteristics of the information exchange between the system an its environment etermine the egree of non-markovian behavior in an open system. This exchange of quantum information has been efine here in very general terms through the change of the istinguishability of quantum states, an oes not presuppose anything about the specific physical carriers of the information, e.g., energy or particles. Moreover, the measure oes not epen on any specific representation of the open system s ynamics. It therefore opens the possibility to compare an assess ifferent mathematical formulations of ynamical processes in their ability to escribe memory effects, in orer to unerstan better the mathematical escription of non-

8 8 Markovian quantum ynamics. where Acknowlegments The authors thank the Acaemy of Finlan (projects 13368, 11598, 11568), Magnus Ehrnrooth Founation, an the Finnish National Grauate School of Moern Optics an Photonics for financial support. Appenix Here we present some etails of the Λ-moel stuie in Sec. III B. The weak-coupling master equation for this moel is given by t ρ(t) = iλ 1(t) [ a a, ρ(t)] iλ (t) [ a a, ρ(t)] +γ 1 (t) [ b a ρ(t) a b 1 ] {ρ(t), a a } +γ (t) [ c a ρ(t) a c 1 ] {ρ(t), a a }, where λ i (t) = γ i (t) = s s Introucing the efinitions ωj(ω) sin [(ω ω i )s], ωj(ω) cos [(ω ω i )s]. f(t) = e [D1(t)+D(t)]/ e i[l1(t)+l(t)], g i (t) = sγ i (s)e [D1(s)+D(s)], D i (t) = sγ i (s), L i (t) = sλ i (s), the solution of the master equation can be represente as follows, ρ aa (t) = f(t) ρ aa (), ρ bb (t) = g 1 (t)ρ aa () + ρ bb (), ρ cc (t) = g (t)ρ aa () + ρ cc (), ρ ab (t) = f(t)ρ ab (), ρ ac (t) = f(t)ρ ac (), ρ bc (t) = ρ bc (). These equations efine the ynamical map Φ(t, ) of the Λ-moel. Employing the results of Choi [9] one can check that a necessary an sufficient conition for the complete positivity of this map is given by g 1 (t), g (t). These conitions are satisfie for the parameters use in the simulations of Sec. III B. [1] V. Gorini, A. Kossakowski an E. C. G. Suarshan, J. Math. Phys. 17, 81 (1976). [] G. Linbla, Comm. Math. Phys. 48, 119 (1976). [3] H. P. Breuer an F. Petruccione, The Theory of Open Quantum Systems (Oxfor University Press, Oxfor, ). [4] J. Piilo, S. Maniscalco, K. Härkönen an K.-A. Suominen, Phys. Rev. Lett. 1, 184 (8); J. Piilo, K. Härkönen, S. Maniscalco an K.-A. Suominen, Phys. Rev. A 79, 611 (9); H. P. Breuer an J. Piilo, Europhys. Lett. 85, 54 (9). [5] M. M. Wolf, J. Eisert, T. S. Cubitt an J. I. Cirac, Phys.Rev. Lett. 11, 154 (8). [6] A. Rivas, S. F. Huelga an M. B. Plenio (9), eprint arxiv:911.47v1. [7] H. P. Breuer an B. Vacchini, Phys. Rev. Lett. 11, 144 (8); Phys. Rev. E 79, (9). [8] H. P. Breuer, Phys. Rev. A 7, 116 (4). [9] D. Chruściński, A. Kossakowski an S. Pascazio, Phys. Rev. A 81, 311 (1). [1] H. Krovi, O. Oreshkov, M. Ryazanov an D. A. Liar, Phys. Rev. A 76, 5117 (7). [11] A. J. van Woneren an K. Leni, Europhys. Lett. 71, 737 (6). [1] A. A. Buini, Phys. Rev. A 74, (6). [13] S. M. Barnett an S. Stenholm, Phys. Rev. A 64,3388 (1). [14] J. Wilkie an Y. M. Wong, J. Phys. A 4, 156 (9). [15] S. Daffer, K. Wókiewicz, J. D. Cresser an J. K. McIver, Phys. Rev. A 7, 134(R) (4). [16] A. Kossakowski an R. Rebolleo, Open Syst. Inf. Dyn. 15, 135 (8); Open Syst. Inf. Dyn. 16, 59 (9). [17] A. Shabani an D. A. Liar, Phys. Rev. Lett. 1, 14 (9). [18] H. P. Breuer, E.-M. Laine, J. Piilo, Phys. Rev. Lett. 13, 141 (9). [19] M. A. Nielsen an I. L. Chuang, Quantum Computation an Quantum Information (Cambrige University Press,

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