Non-Markovian Dynamics in Open Quantum Systems

Non-Markovian dynamics in open quantum systems

Heinz-Peter Breuer Physikalisches Institut, Universit¨atFreiburg, Hermann-Herder-Straße 3, D-79104 Freiburg, Germany Elsi-Mari Laine QCD Labs, COMP Centre of Excellence, Department of Applied Physics, Aalto University, P.O. Box 13500, FI-00076 AALTO, Finland Jyrki Piilo Turku Centre for Quantum Physics, Department of Physics and Astronomy, University of Turku, FI-20014 Turun yliopisto, Finland Bassano Vacchini Dipartimento di Fisica, Universit`adegli Studi di Milano, Via Celoria 16, I-20133 Milan, Italy INFN, Sezione di Milano, Via Celoria 16, I-20133 Milan, Italy

(Dated: May 7, 2015)

The dynamical behavior of open quantum systems plays a key role in many applications of quantum mechanics, examples ranging from fundamental problems, such as the environment-induced decay of quantum coherence and relaxation in many-body systems, to applications in condensed matter theory, quantum transport, quantum chemistry and quantum information. In close analogy to a classical Markovian stochastic process, the interaction of an open quantum system with a noisy environment is often modeled phenomenologically by means of a dynamical semigroup with a corresponding time-independent generator in Lindblad form, which describes a memoryless dynamics of the open system typically leading to an irreversible loss of characteristic quantum features. However, in many applications open systems exhibit pronounced memory effects and a revival of genuine quantum properties such as quantum coherence, correlations and entanglement. Here, recent theoretical results on the rich non-Markovian quantum arXiv:1505.01385v1 [quant-ph] 6 May 2015 dynamics of open systems are discussed, paying particular attention to the rigorous mathematical definition, to the physical interpretation and classification, as well as to the quantification of quantum memory effects. The general theory is illustrated by a series of physical examples. The analysis reveals that memory effects of the open system dynamics reflect characteristic features of the environment which opens a new perspective for applications, namely to exploit a small open system as a quantum probe signifying nontrivial features of the environment it is interacting with. This article further explores the various physical sources of non-Markovian quantum dynamics, such as structured environmental spectral densities, nonlocal correlations between environmental degrees of freedom and correlations in the initial system-environment state, in addition to developing schemes for their local detection. Recent experiments addressing the detection, quantification and control of non-Markovian quantum dynamics are also briefly discussed.

PACS numbers: 03.65.Yz, 03.65.Ta, 03.67.-a, 42.50.-p, 42.50.Lc 2

CONTENTS stationary nonequilibrium state and the decay of quantum coherences and correlations (Alicki and Lendi, 1987; I. Introduction 2 Breuer and Petruccione, 2002; Davies, 1976). II. Definitions and measures for quantum non-Markovianity 3 There is a well-established treatment of the dynam- A. Basic concepts 3 ics of open quantum systems in which the open system’s 1. Open quantum systems and dynamical maps 3 time evolution is represented by a dynamical semigroup 2. Divisibility and time-local master equations 4 and a corresponding master equation in Lindblad form B. Classical versus quantum non-Markovianity 5 et al. 1. Classical stochastic processes and the Markov (Gorini , 1976; Lindblad, 1976). If one adopts a mi- condition 5 croscopic system-environment approach to the dynamics 2. Non-Markovianity in the quantum regime 6 of open systems, such a master equation may be derived, C. Quantum non-Markovianity and information flow 6 for example, with the help of the Born-Markov approx- 1. Trace distance and distinguishability of quantum states 7 imation by assuming a weak coupling between the sys- 2. Definition and quantification of memory effects 7 tem and its environment. However, it turns out that for 3. Generalizing the trace distance measure 9 many processes in open quantum systems the approxi- 4. Connection between quantum and classical mations underlying this approach are not satisfied and non-Markovianity 10 that a description of the dynamics by means of a dy- D. Alternative approaches 10 1. Quantum non-Markovianity and CP-divisibility 11 namical semigroup fails. Typically, this is due to the fact 2. Monotonicity of correlations and entropic that the relevant environmental correlation times are not quantities 12 small compared to the system’s relaxation or decoherence time, rendering impossible the standard Markov approx- III. Models and applications of non-Markovianity 12 A. Prototypical model systems 12 imation. The violation of this separation of time scales 1. Pure decoherence model 13 can occur, for example, in the cases of strong system- 2. Two-level system in a dissipative environment 14 environment couplings, structured or finite reservoirs, 3. Single qubit with multiple decoherence channels 15 low temperatures, or large initial system-environment 4. Spin-boson model 16 B. Applications of quantum non-Markovianity 16 correlations. 1. Analysis and control of non-Markovian dynamics 17 If the dynamics of an open quantum system substan- 2. Open systems as non-Markovian quantum probes 18 tially deviates from that of a dynamical semigroup one often speaks of a non-Markovian process. This term IV. Impact of correlations and experimental realizations 18 A. Initial system-environment correlations 19 refers to a well-known concept of the theory of classical 1. Initial correlations and dynamical maps 19 stochastic processes and is used to loosely indicate the 2. Local detection of initial correlations 19 presence of memory effects in the time evolution of the B. Nonlocal memory effects 20 open system. However, the classical notions of Marko- C. Experiments on non-Markovianity and correlations 21 1. Control and quantification of memory effects in vianity and non-Markovianity cannot be transferred to photonic systems 21 the quantum regime in a natural way since they are based 2. Experiments on the local detection of correlations 22 on a Kolmogorov hierarchy of joint probability distribu- 3. Non-Markovian quantum probes detecting nonlocal tions which does not exist in quantum theory (Accardi correlations in composite environments 22 et al., 1982; Lindblad, 1979; Vacchini et al., 2011). There- V. Summary and outlook 23 fore, the concept of a quantum non-Markovian process requires a precise definition which cannot be based on Acknowledgments 24 classical notions only. Many important questions need References 24 to be discussed in this context: How can one rigorously define non-Markovian dynamics in the quantum case, how do quantum memory effects manifest themselves in I. INTRODUCTION the behavior of open systems, and can such effects be uniquely identified experimentally through monitoring of The observation and experimental control of charac- the open system? The definition of non-Markovianity teristic quantum properties of physical systems is of- should provide a general mathematical characterization ten strongly hindered by the coupling of the system which does not rely on any specific representation or ap- to a noisy environment. The unavoidable interaction proximation of the dynamics, e.g. in terms of a pertur- of the quantum system with its surroundings generates bative master equation. Moreover, definitions of non- system-environment correlations leading to an irretriev- Markovianity should lead to quantitative measures for able loss of quantum coherence. Realistic quantum me- the degree of non-Markovianity, enabling the compari- chanical systems are thus open systems governed by a son of the amount of memory effects in different physical non-unitary time development which describes all fea- systems. tures of irreversible dynamics such as the dissipation Recently, a series of different proposals to answer these of energy, the relaxation to a thermal equilibrium or a questions has been published, rigorously defining the bor- 3 der between the regions of Markovian and non-Markovian state and leads to schemes for the local detection of quantum dynamics and developing quantitative measures such correlations through monitoring of the open system for the degree of memory effects (see, e.g., (Breuer et al., (Gessner and Breuer, 2011; Laine et al., 2010b). 2009; Chru´sciński et al., 2011; Chru´scińskiand Manis- In recent years, several of the above features of non- calco, 2014; Luo et al., 2012; Rivas et al., 2010; Wolf et al., Markovian quantum dynamics have been observed exper- 2008), the tutorial paper (Breuer, 2012) and the recent imentally in photonic and trapped ion systems (Cialdi review by Rivas et al. (2014)). Here, we describe and et al., 2014; Gessner et al., 2014b; Li et al., 2011; Liu discuss several of these ideas, paying particular attention et al., 2011; Smirne et al., 2011; Tang et al., 2014). We to those concepts which are based on the exchange of in- briefly discuss the results of these experiments which formation between the open system and its environment, demonstrate the transition from Markovian to non- and on the divisibility of the dynamical map describing Markovian quantum dynamics, the non-Markovian be- the open system’s time evolution. We will also explain havior induced by nonlocal environmental correlations, the relations between the classical and the quantum no- and the local scheme for the detection of nonclassical ini- tions of non-Markovianity and develop a general classi- tial system-environment correlations. fication of quantum dynamical maps which is based on these concepts. The general theory will be illustrated by a series of II. DEFINITIONS AND MEASURES FOR QUANTUM NON-MARKOVIANITY examples. We start with simple prototypical models describing pure decoherence dynamics, dissipative pro- A. Basic concepts cesses, relaxation through multiple decay channels and the spin-boson problem. We then continue with the 1. Open quantum systems and dynamical maps study of the dynamics of open systems which are coupled to interacting many-body environments. The examples An open quantum system S (Alicki and Lendi, 1987; include an Ising and a Heisenberg chain in transverse Breuer and Petruccione, 2002; Davies, 1976) can be re- magnetic fields, as well as an impurity atom in a Bose- garded as subsystem of some larger system composed of Einstein condensate (Apollaro et al., 2011; Haikka et al., S and another subsystem E, its environment, see Fig. 1. 2012, 2011). The discussion will demonstrate, in partic- The Hilbert space of the total system S + E is given by ular, that memory effects of the open system dynamics the tensor product space reflect characteristic properties of the environment. This = , (1) fact opens a new perspective, namely to exploit a small HSE HS ⊗ HE open system as a quantum probe signifying nontrivial fea- where S and E denote the Hilbert spaces of S and tures of a complex environment, for example the critical E, respectively.H H Physical states of the total system are point of a phase transition (Gessner et al., 2014a; Smirne represented by positive trace class operators ρSE on SE et al. H , 2013). Another example to be discussed here is with unit trace, satisfying ρSE > 0 and trρSE = 1. Given the use of non-Markovian dynamics to determine nonlo- a state of the total system, the corresponding states of cal correlations within a composite environment, carrying subsystems S and E are obtained by partial traces over out only measurements on the open system functioning and , respectively, i.e., we have ρ = tr ρ and HE HS S E SE as quantum probe (Laine et al., 2012; Liu et al., 2013a; ρE = trSρSE. We denote the convex set of physical states Wißmann and Breuer, 2014). belonging to some Hilbert space by ( ). A large variety of further applications of quantum H S H memory effects is described in the literature. Interested readers may find examples dealing with, e.g., phenomenological master equations (Mazzola et al., 2010), chaotic environment! open system interaction H I Hilbert space et al. Hilbert space E S systems (Znidaric , 2011), energy transfer processes H Hamiltonian H Hamiltonian HE S system state ⇢ in photosynthetic complexes (Rebentrost and Aspuru- environment state ⇢E S Guzik, 2011), continous variable quantum key distribution (Vasile et al., 2011b), metrology (Chin et al., 2012), steady state entanglement (Huelga et al., 2012), Coulomb crystals (Borrelli et al., 2013), symmetry breaking (Chan- FIG. 1 (Color online) Sketch of an open quantum system cellor et al., 2013), and time-invariant quantum discord described by the Hilbert space HS and the Hamiltonian HS , (Haikka et al., 2013). which is coupled to an environment with Hilbert space HE and Hamiltonian H through an interaction Hamiltonian H . The standard description of the open system dynamics E I in terms of a dynamical map is based on the assumption We suppose that the total system S + E is closed and of an initially factorizing system-environment state. The governed by a Hamiltonian of the general form approach developed here also allows to investigate the impact of correlations in the initial system-environment H = H I + I H + H , (2) S ⊗ E S ⊗ E I 4 where HS and HE are the free Hamiltonians of system map Φ In operating on the combined system by linear and environment, respectively, and H is an interaction extension⊗ of the relation (Φ I )(A B) = (ΦA) B. I ⊗ n ⊗ ⊗ Hamiltonian. The corresponding unitary time evolution This map Φ In thus describes a quantum operation of operator is thus given by the composite⊗ system which acts nontrivially only on the first factor representing subsystem S. One then defines U(t) = exp( iHt)(~ = 1). (3) the map Φ to be n-positive if Φ I is a positive map, − ⊗ n and completely positive if Φ In is a positive map for all The dynamics of the total system states is obtained from n. We note that the existence⊗ of maps which are positive the von Neumann equation, but not completely positive is closely connected to the d existence of entangled states. We note further that posi- ρSE(t) = i[H, ρSE(t)], (4) tivity is equivalent to 1-positivity, and that for a Hilbert dt − space with finite dimension N = dim complete pos- S HS which yields the formal solution itivity is equivalent to NS-positivity. If the time parameter t now varies over some time in- ρ (t) = ρ (0) U(t)ρ (0)U †(t). (5) SE Ut SE ≡ SE terval from 0 to T , where T may be finite or infinite, we obtain a one-parameter family of dynamical maps, An important concept in the theory of open quantum systems is that of a dynamical map. To explain this con- Φ = Φ 0 t T, Φ = I , (10) cept we assume that the initial state of the total system { t | 6 6 0 } is an uncorrelated tensor product state where I denotes the unit map, and the initial environmental state ρE(0) used to construct Φt is still kept fixed. ρSE(0) = ρS(0) ρE(0), (6) ⊗ This family contains the complete information on the dy- which leads to the following expression for the reduced namics of all initial open system states over the time in- open system state at any time t > 0, terval [0,T ] we are interested in. ρ (t) = tr U(t)ρ (0) ρ (0)U †(t) . (7) S E S ⊗ E 2. Divisibility and time-local master equations Considering a fixed initial environmental state ρE(0) and any fixed time t > 0, Eq. (7) defines a linear map Let us suppose that the inverse of Φt exists for all times t 0. We can then define a two-parameter family of Φt : S( S) S( S) (8) > H −→ H maps by means of on the open system’s state space S( S) which maps any H Φ = Φ Φ−1, t s 0, (11) initial open system state ρS(0) to the corresponding open t,s t s > > system state ρS(t) at time t: such that we have Φt,0 = Φt and ρ (0) ρ (t) = Φ ρ (0). (9) S 7→ S t S Φt,0 = Φt,sΦs,0. (12) Φt is called quantum dynamical map. It is easy to verify that it preserves the Hermiticity and the trace of oper- The existence of the inverse for all positive times thus al- ators, and that it is a positive map, i.e., that it maps lows us to introduce the very notion of divisibility. While positive operators to positive operators. Thus, Φt maps Φt,0 and Φs,0 are completely positive by construction, the physical states to physical states. map Φt,s need not be completely positive and not even −1 A further important property of the dynamical map positive since the inverse Φs of a completely positive Φt is that it is not only positive but also completely pos- map Φs need not be positive. The family of dynamical itive. Maps with this property are also known as trace maps is said to be P-divisible if Φt,s is positive, and CP- preserving quantum operations or quantum channels in divisible if Φt,s is completely positive for all t > s > 0. quantum information theory. Let us recall that a linear A simple example for a CP-divisible process is provided Lt map Φ is completely positive if and only if it admits a by a semigroup Φt = e with a generator in Lindblad LL(t−s) Kraus representation (Kraus, 1983), which means that form. In this case we obviously have Φt,s = e which there are operators Ωi on the underlying Hilbert space is trivially completely positive. As we will discuss later P † S such that ΦA = i ΩiAΩi , and that the condition on there are many physically relevant models which give H P † of trace preservation takes the form i Ωi Ωi = IS. An rise to dynamical maps which are neither CP-divisible equivalent definition of complete positivity is the follow- nor P-divisible. ing. We consider for any number n = 1, 2,... the tensor An interesting property of the class of processes for n −1 product space S C which represents the Hilbert space which Φt exists is given by the fact that they always of S combinedH with⊗ an n-level system. We can define a lead to a time-local quantum master equation for the 5 open system states with the general structure ing values in a discrete set x ∈ is characterized { i}i N d by a hierarchy of joint probability distributions Pn = ρ = ρ (13) S t S Pn(xn, tn; xn−1, tn−1; ... ; x1, t1) for all n N and times dt K ∈ t t − ... t 0. The distribution P yields the = i [HS(t), ρS] n > n 1 > > 1 > n − probability that the process takes on the value x1 at time X h † 1 † i + γi(t) Ai(t)ρSA (t) A (t)Ai(t), ρS . t , the value x at time t , . . . , and the value x at time i − 2 i 1 2 2 n i tn. In order for such a hierarchy to represent a stochastic The form of this master equation is very similar to that process the Kolmogorov consistency conditions must be of a Lindblad master equation, where however the Hamil- satisfied which, apart from conditions of positivity and tonian contribution HS(t), the Lindblad operators Ai(t) normalization, require in particular the relation (which can be supposed to be linearly independent), and X Pn(xn, tn; ... ; xm, tm; ... ; x1, t1) the decay rates γi(t) may depend on time since the process does not represent a semigroup, in general. Note xm that Eq. (13) involves a time-dependent generator t, = Pn−1(xn, tn; ... ; x1, t1), (15) but no convolution of the open system states with a mem-K connecting the n-point probability distribution Pn to the ory kernel as in the Nakajima-Zwanzig equation (Naka- (n 1)-point probability distribution Pn−1. jima, 1958; Zwanzig, 1960). Master equations of the form A− stochastic process X(t) is said to be Markovian if (13) can be derived employing the time-convolutionless the conditional probabilities defined by projection operator technique (Chaturvedi and Shibata, 1979; Shibata et al., 1977). P (x , t x , t ; ... ; x , t ) 1|n n+1 n+1| n n 1 1 It is an important open problem to formulate general Pn+1(xn+1, tn+1; ... ; x1, t1) necessary and sufficient conditions under which the mas- = (16) Pn(xn, tn; ... ; x1, t1) ter equation (13) leads to a completely positive dynamics. If the process represents a semigroup, the Hamiltonian satisfy the relation HS, the operators Ai and the rates γi must be time- P | (xn+1, tn+1 xn, tn; ... ; x1, t1) independent and a necessary and sufficient condition for 1 n | = P (x , t x , t ). (17) complete positivity of the dynamics is simply that all de- 1|1 n+1 n+1| n n cay rates are positive, γi > 0. This is the famous Gorini- This is the classical Markov condition which means that Kossakowski-Sudarshan-Lindblad theorem (Gorini et al., the probability for the stochastic process to take the value 1976; Lindblad, 1976). However, we will see later by sev- xn+1 at time tn+1, under the condition that it assumed eral examples that in the time-dependent case the rates values xi at previous times ti, only depends on the last γi(t) can indeed become temporarily negative without previous value xn at time tn. In this sense the pro- violating the complete positivity of the dynamics. cess is said to have no memory, since the past history On the other hand, for divisible quantum processes it prior to tn is irrelevant to determine the future once we is indeed possible to formulate necessary and sufficient know the value xn the process assumed at time tn. Note conditions. In fact, the master equation (13) leads to a that Eq. (17) imposes an infinite number of conditions CP-divisible dynamics if and only if all rates are positive for all n-point probability distributions which cannot be for all times, γi(t) > 0, which follows from a straight- checked if only a few low-order distributions are known. forward extension of the Gorini-Kossakowski-Sudarshan- The Markov condition (17) substantially simplifies the Lindblad theorem. Moreover, the master equation (13) mathematical description of stochastic processes. In fact, leads to a P-divisible dynamics if and only if the weaker one can show that under this condition the whole hi- conditions erarchy of joint probability distributions can be recon- X γ (t) n A (t) m 2 0 (14) structed from the initial 1-point distribution P1(x0, 0) i |h | i | i| > i and the conditional transition probability hold for all orthonormal bases n of the open system T (x, t y, s) P1|1(x, t y, s) (18) and all n = m. This statement can{| i} be obtained by apply- | ≡ | 6 ing a characterization of the generators of positive semi- by means of the relations groups due to Kossakowski (1972a,b). Pn(xn, tn; xn−1, tn−1; ... ; x1, t1) n−1 Y B. Classical versus quantum non-Markovianity = T (x , t x , t )P (x , t ) (19) i+1 i+1| i i 1 1 1 i=1 1. Classical stochastic processes and the Markov condition and X In classical probability theory (Gardiner, 1985; van P (x , t ) = T (x , t x , 0)P (x , 0). (20) 1 1 1 1 1| 0 1 0 Kampen, 1992) a stochastic process X(t), t > 0, tak- x0 6

For a Markov process the transition probability has to which yields the probability of observing a certain se- obey the Chapman-Kolmogorov equation quence xn, xn−1, . . . , x1 of measurement outcomes at the respective times t , t , . . . , t . Thus, it is of course pos- X n n−1 1 T (x, t y, s) = T (x, t z, τ)T (z, τ y, s) (21) sible in quantum mechanics to define a joint probabil- | | | z ity distribution for any sequence of measurement results. However, as is well known these distribution do in gen- for t τ s. Thus, a classical Markov process is > > eral not satisfy condition (15) since measurements carried uniquely characterized by a probability distribution for out at intermediate times in general destroy quantum in- the initial states of the process and a conditional tran- terferences. The fact that the joint probability distribu- sition probability satisfying the Chapman-Kolmogorov tions (24) violate in general the Kolmogorov condition equation (21). Indeed, the latter provides the necessary (15) is even true for closed quantum system. Thus, the condition in order to ensure that the joint probabilities quantum joint probability distributions given by (24) do defined by (19) satisfy the condition (15), so that they not represent a classical hierarchy of joint probabilities actually define a classical Markov process. Provided the satisfying the Kolmogorov consistency conditions. More conditional transition probability is differentiable with generally, one can consider other joint probability distri- respect to time (which will always be assumed here) one butions corresponding to different quantum operations obtains an equivalent differential equation, namely the describing non-projective, generalized measurements. Chapman-Kolmogorov equation: For an open system coupled to some environment mea- d surements performed on the open system not only influ- T (x, t y, s) (22) dt | ence quantum interferences but also system-environment X h i = W (t)T (z, t y, s) W (t)T (x, t y, s) , correlations. For example, if the system-environment xz | − zx | z state prior to a projective measurement at time ti is given by ρSE(ti), the state after the measurement conditioned where Wzx(t) > 0, representing the rate (probability per on the outcome x is given by unit of time) for a transition to the state z given that the state is x at time t. An equation of the same struc- 0 xρSE(ti) x ρSE(ti) = M = ϕx ϕx ρE(ti), (25) ture holds for the 1-point probability distribution of the tr xρSE(ti) | ih | ⊗ process: M x where ρE is an environmental state which may depend d X h i on the measurement result x. Hence, projective mea- P (x, t) = W (t)P (z, t) W (t)P (x, t) , (23) dt 1 xz 1 − zx 1 surements completely destroy system-environment corre- z lations, leading to an uncorrelated tensor product state which is known as Pauli master equation for a classical of the total system, and, therefore, strongly influence the Markov process. The conditional transition probability subsequent dynamics. of the process can be obtained solving Eq. (23) with the We conclude that an intrinsic characterization and initial condition P1(x, s) = δxy. quantification of memory effects in the dynamics of open quantum systems, which is independent of any prescribed measurement scheme influencing the time evolution, has 2. Non-Markovianity in the quantum regime to be based solely on the properties of the dynamics of the open system’s density matrix ρS(t). The above definition of a classical Markov process cannot be transferred immediately to the quantum regime (Vacchini et al., 2011). In order to illustrate the arising C. Quantum non-Markovianity and information flow difficulties let us consider an open quantum system as described in Sec. II.A. Suppose we carry out projective The first approach to quantum non-Markovianity to measurements of an open system observable Xˆ at times be discussed here is based on the idea that memory ef- tn > tn−1 > ... > t1 > 0. For simplicity we assume that fects in the dynamics of open systems are linked to the this observable is non-degenerate with spectral decom- exchange of information between the open system and ˆ P position X = x x ϕx ϕx . As in Eq. (5) we write the its environment: While in a Markovian process the open | ih | † unitary evolution superoperator as tρSE = UtρSEUt system continuously looses information to the environ- and the quantum operation correspondingU to the mea- ment, a non-Markovian process is characterized by a flow surement outcome x as xρSE = ϕx ϕx ρSE ϕx ϕx . of information from the environment back into the open Applying the Born rule andM the projection| ih | postulate| ih one| system (Breuer et al., 2009; Laine et al., 2010a). In such a can then write a joint probability distribution way quantum non-Markovianity is associated to a notion of quantum memory, namely information which has been Pn(xn, tn; xn−1, tn−1; ... ; x1, t1) transferred to the environment, in the form of system- = tr − ... ρ (0) (24) environment correlations or changes in the environmen- Mxn Utn tn−1 Mx1 Ut1 SE 7 tal states, and is later retrieved by the system. To make t this idea more precise we employ an appropriate distance 1 1 ⇢ ⇢S(t) measure for quantum states. S D(⇢1 , ⇢2 ) 1 2 Alice S S D(⇢S(t), ⇢S(t)) Bob 1. Trace distance and distinguishability of quantum states 2 2 ⇢S ⇢S(t) t The trace norm of a trace class operator A is defined by A = tr A , where the modulus of the operator is given || || | | by A = √A†A. If A is selfadjoint the trace norm can be FIG. 2 (Color online) Illustration of the loss of distinguisha- | | bility quantified by the trace distance D between density ma- expressed as the sum of the moduli of the eigenvalues ai P trices as a consequence of the action of a quantum dynamical of A counting multiplicities, A = ai . This norm || || i | | map Φt. Alice prepares two distinct states, but a dynamical leads to a natural measure for the distance between two map acts as a noisy channel, thus generally reducing the in- 1 2 quantum states ρ and ρ known as trace distance: formation available to Bob in order to distinguish among the states by performing measurements on the system only. 1 D(ρ1, ρ2) = ρ1 ρ2 . (26) 2|| − || preserving map Λ is a contraction for the trace distance, The trace distance represents a metric on the state space i.e. we have ( ) of the underlying Hilbert space . We have the S H 1 2 H1 2 1 2 1 2 bounds 0 6 D(ρ , ρ ) 6 1, where D(ρ , ρ ) = 0 if and D(Λρ , Λρ ) 6 D(ρ , ρ ) (28) only if ρ1 = ρ2, and D(ρ1, ρ2) = 1 if and only if ρ1 and ρ2 are orthogonal. Recall that two density matrices are for all states ρ1,2. In view of the interpretation of the said to be orthogonal if their supports, i.e. the subspaces trace distance we thus conclude that a trace preserving spanned by their eigenstates with nonzero eigenvalue, are quantum operation can never increase the distinguisha- orthogonal. The trace distance has a series of interest- bility of any two quantum states. We remark that the ing mathematical and physical features which make it a equality sign in (28) holds if Λ is a unitary transforma- useful distance measure in quantum information theory tion, and that (28) is also valid for trace preserving maps (Nielsen and Chuang, 2000). There are two properties which are positive but not completely positive. which are the most relevant for our purposes. The first property is that the trace distance between two quantum states admits a clear physical interpreta- 2. Definition and quantification of memory effects tion in terms of the distinguishability of these states. To explain this interpretation consider two parties, Alice and In the preceding subsection we have interpreted the Bob, and suppose that Alice prepares a quantum system trace distance of two quantum states as the distinguisha- 1 2 1 bility of these states, where it has been assumed that in one of two states, ρ or ρ , with a probability of 2 each, and sends the system to Bob. The task of Bob the quantum state Bob receives is identical to the state prepared by Alice. Suppose now that Alice prepares is to find out by means of a single quantum measure- 1,2 ment whether the system is in the state ρ1 or ρ2. It can her states ρS (0) as initial states of an open quantum be shown that the maximal success probability Bob can system S coupled to some environment E. Bob will then receive at time t the system in one of the states achieve through an optimal strategy is directly connected 1,2 1,2 to the trace distance: ρS (t) = ΦtρS (0), where Φt denotes the corresponding quantum dynamical map. This construction is equiva- 1 1 2 lent to Alice sending her states through a noisy quantum Pmax = 1 + D(ρ , ρ ) . (27) 2 channel described by the completely positive and trace preserving map Φ . According to (28) the dynamics gen- Thus, we see that the trace distance represents the bias t erally diminishes the trace distance and, therefore, the in favor of a correct state discrimination by Bob. For this distinguishability of the states, reason the trace distance D(ρ1, ρ2) can be interpreted as the distinguishability of the quantum states ρ1 and ρ2. 1 2 1 2 D(ρS(t), ρS(t)) 6 D(ρS(0), ρS(0)), (29) For example, suppose the states prepared by Alice are orthogonal such that we have D(ρ1, ρ2) = 1. In this such that it will in general be harder for Bob to discrimi- case we get Pmax = 1, which is a well known fact since nate the states prepared by Alice. This fact is illustrated orthogonal states can be distinguished with certainty by schematically in Fig. 2, in which the distinguishability of a single measurement, an optimal strategy of Bob being states prepared by Alice decreases due to the action of a to measure the projection onto the support of ρ1 or ρ2. dynamical map, so that Bob is less able to discriminate The second important property of the trace distance is among the two. Thus, we can interpret any decrease of 1 2 given by the fact that any completely positive and trace the trace distance D(ρS(t), ρS(t)) as a loss of information 8 from the open system into the environment. Conversely, the total system states minus the distinguishability of the 1 2 if the trace distance D(ρS(t), ρS(t)) increases, we say that open system states at time t. In other words, ext(t) can information flows back from the environment into the be viewed as the gain in the state discriminationI Bob open system. could achieve if he were able to carry out measurements This interpretation naturally leads to the following def- on the total system instead of measurements on the open inition: A quantum process given by a family of quantum system only. We can therefore interpret (t) as the Iint dynamical maps Φt is said to be Markovian if the trace information inside the open system and ext(t) as the 1 2 I distance D(ρS(t), ρS(t)) corresponding to all pair of ini- amount of information which lies outside the open sys- 1 2 tial states ρS(0) and ρS(0) decreases monotonically for all tem, i.e., as the information which is not accessible when times t > 0. Quantum Markovian behavior thus means only measurements on the open system can be performed. a continuous loss of information from the open system Obviously, we have int(t) > 0 and ext(t) > 0. Since the to the environment. Conversely, a quantum process is total system dynamicsI is unitary theI distinguishability of 1 non-Markovian if there is an initial pair of states ρS(0) the total system states is constant in time. Moreover we 2 1 2 1 2 1 2 and ρS(0) such that the trace distance D(ρS(t), ρS(t)) have D(ρSE(0), ρSE(0)) = D(ρS(0), ρS(0)) because, by is non-monotonic, i.e. starts to increase for some time assumption, the initial total states are uncorrelated with t > 0. If this happens, information flows from the envi- the same reduced environmental state. Hence, we obtain ronment back to the open system, which clearly expresses the presence of memory effects: Information contained int(t) + ext(t) = int(0) = const. (32) in the open system is temporarily stored in the environ- I I I ment and comes back at a later time to influence the Thus, initially there is no information outside the open system. A crucial feature of this definition of quantum system, ext(0) = 0. If int(t) decreases ext(t) must Markovian process is the fact that non-Markovianity can increaseI and vice versa, whichI clearly expressesI the idea be directly experimentally assessed, provided one is able of the exchange of information between the open system to perform tomographic measurement of different initial and the environment illustrated in Fig. 3. Employing states at different times during the evolution. No prior the properties of the trace distance one can derive the information on the dynamical map Φt is required, apart following general inequality (Laine et al., 2010b) which from its very existence. holds for all t > 0:

(t) D(ρ1 (t), ρ1 (t) ρ1 (t)) (33) Iext 6 SE S ⊗ E environment! environment! +D(ρ2 (t), ρ2 (t) ρ2 (t)) + D(ρ1 (t), ρ2 (t)). SE S ⊗ E E E The right-hand side of this inequality consists of three terms: The first two terms provide a measure for the cor- system system 1,2 d d relations in the total system states ρSE(t), given by the int < 0 > 0 dtI dtIint trace distance between these states and the product of their marginals, while the third term is the trace distance between the corresponding environmental states. Thus, when (t) increases over the initial value (0) = 0 Iext Iext FIG. 3 (Color online) Illustration of the information flow system-environment correlations are built up or the envi- between an open system and its environment according to ronmental states become different, implying an increase Eq. (32). Left: The open system looses information to the en- of the distinguishability of the environmental states. This vironment, corresponding to a decrease of Iint(t) and Marko- clearly demonstrates that the corresponding decrease of vian dynamics. Right: Non-Markovian dynamics is charac- the distinguishability (t) of the open system states terized by a backflow of information from the environment to int always has an impactI on degrees of freedom which are the system and a corresponding increase of Iint(t). inaccessible by measurements on the open system. Con- versely, if (t) starts to increase at some point of time t, In order to explain in more detail this interpretation int the correspondingI decrease of (t) implies that system- in terms of an information flow between system and en- ext environment correlations mustI be present already at time vironment let us define the quantities t, or that the environmental states are different at this (t) = D(ρ1 (t), ρ2 (t)) (30) point of time. Iint S S On the ground of the above definition for non- and Markovian dynamics one is naturally led to introduce the following measure for the degree of memory effects (t) = D(ρ1 (t), ρ2 (t)) D(ρ1 (t), ρ2 (t)). (31) Iext SE SE − S S Z Here, (t) is the distinguishability of the open system Iint (Φ) = max dt σ(t), (34) states at time t, while (t) is the distinguishability of N ρ1,2 Iext S σ>0 9 where Furthermore, the representation reveals the universality of memory effects, namely that for a non-Markovian dy- d 1 2 σ(t) D ΦtρS, ΦtρS (35) namics memory effects can be observed everywhere in ≡ dt state space. denotes the time derivative of the trace distance of the evolved pair of states. In Eq. (34) the time integral is extended over all intervals in which σ(t) > 0, i.e., in 3. Generalizing the trace distance measure which the trace distance increases with time, and the 1,2 There is an interesting generalization of the above defi- maximum is taken over all pairs of initial states ρS of the open system’s state space ( S). Thus, (Φ) is a nition and quantification of non-Markovian quantum dy- positive functional of the familyS H of dynamicalN maps Φ namics first suggested by Chru´sciński et al. (2011). Re- which represents a measure for the maximal total flow turning to the interpretation of the trace distance de- of information from the environment back to the open scribed in Sec. II.C.1, we may suppose that Alice pre- 1 2 system. By construction we have (Φ) = 0 if and only pares her quantum system in the states ρS or ρS with N if the process is Markovian. corresponding probabilities p1 and p2, where p1 +p2 = 1. The maximization over all pairs of quantum states in Thus, we assume that Alice gives certain weights to her Eq. (34) can be simplified substantially employing sev- states which need not be equal. In this case one can again eral important properties of the functional (Φ) and derive an expression for the maximal probability for Bob the convex structure of the set of quantumN states. A to identify the state prepared by Alice: certain pair of states ρ1,2 is said to be an optimal state S 1 pair if the maximum in Eq. (34) is attained for this pair P = 1 + p ρ1 p ρ2 . (38) max 2 1 S 2 S of states. Thus, optimal state pairs lead to the maxi- k − k mal possible backflow of information during their time As follows from (38) the quantity p ρ1 p ρ2 gives 1,2 k 1 S − 2 Sk evolution. One can show that optimal states pairs ρS the bias in favor of the correct identification of the state lie on the boundary of the state space, and are in par- 1 prepared by Alice. We note that the operator ∆ = p1ρS ticular always orthogonal (Wißmann et al., 2012). This 2 − p2ρS is known as Helstrom matrix (Helstrom, 1967), and is a quite natural result in view of the interpretation in that Eq. (38) reduces to Eq. (27) in the unbiased case 1 terms of an information flow since orthogonality implies p1 = p2 = . Note that the interpretation in terms 1 2 2 that D(ρS, ρS) = 1, which shows that optimal state pairs of an information flow between system and environment have maximal initial distinguishability, corresponding to still holds for the biased trace distance, as can be seen a maximal amount of initial information. An even more replacing in (30) and (31) the trace distance by the norm drastic simplification is obtained by employing the fol- of the Helstrom matrix. lowing equivalent representation (Liu et al., 2014): Following this approach we can define a process to be 1 2 Z Markovian if the function Φt(p1ρ p2ρ ) decreases k S − S k (Φ) = max dt σ¯(t), (36) monotonically with time for all p1,2 and all pairs of ini- N ρ∈∂U(ρ0) σ>¯ 0 1,2 tial open system states ρS . Consequently, the extended where measure for non-Markovianity is then given by Z d D (Φ ρ, Φ ρ ) dt t t 0 (Φ) = max dt σ(t), (39) σ¯(t) (37) i ≡ D (ρ, ρ0) N pi,ρS σ>0 is the time derivative of the trace distance at time t di- where vided by the initial trace distance. In Eq. (36) ρ0 is a d 1 2 fixed point of the interior of the state space and ∂U(ρ0) σ(t) Φt(p1ρS p2ρS) . (40) an arbitrary surface in the state space enclosing, but not ≡ dtk − k containing ρ0. The maximization is then taken over all This generalized definition of quantum non-Markovianity points of such an enclosing surface. This representation leads to several important conclusions. First, we note often significantly simplifies the analytical, numerical or that also for the measure (39) optimal state pairs are or- experimental determination of the measure since it only 1 2 thogonal, satisfying p1ρS p2ρS = 1 and, therefore, involves a maximization over a single input state ρ. It corresponding to maximalk initial− distinguishability.k The is particularly advantageous if the open system dynam- maximum in (39) can therefore be taken over all Hermi- ics has an invariant state and if this state is taken to be tian (Helstrom) matrices ∆ with unit trace norm. This ρ0, such that only one state of the pair evolves in time. leads to a local representation analogous to Eq. (36): Equation (36) may be called local representation for it shows that optimal state pairs can be found in any lo- Z (Φ) = max dt σ¯(t), (41) cal neighborhood of any interior point of the state space. N ∆∈∂U(0) σ>¯ 0 10 whereσ ¯(t) = d Φ ∆ / ∆ , and ∂U(0) is a closed sur- 4. Connection between quantum and classical non-Markovianity dt k t k k k face which encloses the point ∆ = 0 in the R-linear vector space of Hermitian matrices. The definition of quantum non-Markovianity of Sec. II.C.3 allows to establish an immediate general connection to the classical definition. Suppose as in Sec. II.A.2 that the inverse of the dynamical maps Φt exists such that we have the time-local quantum mas- Markovian non-Markovian ter equation (13), in which for the sake of simplicity we ()=0 () > 0 consider only the rates to be time dependent. Suppose N N P-divisible not P-divisible further that Φt maps operators which are diagonal in a fixed basis n to operators diagonal in the same ba- {| i} sis. If ρS(0) is an initial state diagonal in this basis, then CP-divisible ρS(t) can be expressed at any time in the form: X ρ (t) = P (t) n n , (42) S n | ih | n

t invertible where Pn(t) denote the time-dependent eigenvalues. The quantum master equation then leads to the following equation of motion for the probabilities P (t): t not invertible n d X h i P (t) = W (t)P (t) W (t)P (t) , (43) dt n nm m − mn n FIG. 4 (Color online) Schematic picture of the relations be- m tween the concepts of quantum Markovianity and divisibility of the family of quantum dynamical maps Φt. The blue area where to the left of the thick black line represents the set of Marko- X W (t) = γ (t) n A m 2. (44) vian quantum processes for which the inverse of the dynam- nm i |h | i| i| ical map exists, which is identical to the set of P-divisible i processes. This set contains as subset the set of CP-divisible We observe that Eq. (43) has exactly the structure of a processes, depicted in green. The blue area to the right repre- classical Pauli master equation (23). Therefore, Eq. (43) sents the non-Markovian processes which are therefore neither CP nor P-divisible. The zig-zag line dividing the two regions can be interpreted as differential Chapman-Kolmogorov points to the fact that neither of them is convex. The red equation (22) for the conditional transition probability of areas mark the Markovian or non-Markovian processes for a classical Markov process if and only if the rates (44) are which the inverse of Φt does not exist. positive. As we have seen a quantum process is Marko- vian if and only if the dynamics is P-divisible. Accord- ing to condition (14) P-divisibility implies that the rates (44) are indeed positive. Thus, we conclude that any quantum Markovian process which preserves the diago- Another important conclusion is obtained if we as- −1 nal structure of quantum states in a fixed basis leads to a sume, as was done in Sec. II.A.2, that Φt exists. It can classical Markovian jump process describing transitions be shown that under this condition the quantum process between the eigenstates of the density matrix. is Markovian if and only if Φt is P-divisible, which follows immediately from a theorem by Kossakowski (1972a,b). The relationships between Markovianity and divisibility D. Alternative approaches of the dynamical map are illustrated in Fig. 4, where also the situation in which the inverse of Φt as a linear map In Sec. II.C we have considered an approach to quan- does not exist for all times has been considered. The tum non-Markovianity based on the study of the dynam- generalized characterization of non-Markovianity is sen- ical behavior of the distinguishability of states prepared sitive to features of the open system dynamics neglected by two distinct parties, Alice and Bob. This viewpoint by the trace distance measure (Liu et al., 2013b). It is directly addresses the issue of the experimental detection worth stressing that in keeping with the spirit of the ap- of quantum non Markovian processes, which can be ob- proach, this generalized measure of non-Markovianity, in tained by suitable quantum tomographic measurements. addition to its clear connection with the mathematical It further highlights a strong connection between quan- property of P-divisibility of the dynamical map, can still tum non-Markovianity and quantum information theory, be directly evaluated by means of experiments. More or more specifically quantum information processing. specifically, both the quantification of non-Markovianity Alternative approaches to the study of quantum non- according to the measure (34), as well as its generaliza- Markovian dynamics of open systems have also been re- tion (39) can be obtained from the very same data. cently proposed. We shall try to briefly discuss those 11

1 2 which are more closely related to the previous study, P1 (t) and P1 (t). Therefore these two equivalent prop- without any claim to properly represent the vast liter- erties capture the feature of a classical Markovian pro- ature on the subject. For a review on recent results cess at the level of the one-point probability distribution, on quantum non-Markovianity see Rivas et al. (2014). which is all we are looking for in order to obtain an in- The different approaches can to some extent be grouped trinsic definition of quantum non-Markovianity. according to whether they propose different divisibility Relying on the fact that in the theory of open quantum properties of the quantum dynamical map (Chru´sciński systems complete positivity suggests itself as a natural and Maniscalco, 2014; Hall et al., 2014; Hou et al., 2011; counterpart of positivity, as an alternative approach to Rivas et al., 2010; Wolf et al., 2008), different quantifiers quantum non-Markovianity it has been proposed to iden- of the distinguishability between states (Chru´scińskiand tify quantum Markovian dynamics with those dynamics Kossakowski, 2012; Dajka et al., 2011; Lu et al., 2010; which are actually CP-divisible. This criterion was first Vasile et al., 2011a; Wißmann et al., 2013) or the study of suggested by Rivas et al. (2010). In order to check CP- other quantities, be they related to quantum information divisibility of the time evolution one actually needs to concepts or not, which might exhibit both a monotonic know the mathematical expression of the transition maps and an oscillating behavior in time (Bylicka et al., 2014; Φ or of the infinitesimal generator . Indeed given the t,s Kt Fanchini et al., 2014; Haseli et al., 2014; Lu et al., 2010; maps Φt,s one can determine their complete positivity Luo et al., 2012). studying the associated Choi matrices, while as discussed in Sec. II.A CP-divisibility corresponds to positivity at all times of the rates γi(t) appearing in (13). In order to 1. Quantum non-Markovianity and CP-divisibility quantify non-Markovianity as described by this criterion, different related measures have been introduced, relying Let us first come back to the notion of divisible maps on an estimate of the violation of positivity of the Choi introduced in Sec. II.A.2, which we express in the form matrix (Rivas et al., 2010) or on various quantifiers of the negativity of the rates (Hall et al., 2014). In particu- Φt,s = Φt,τ Φτ,s, t > τ > s > 0, (45) lar, Rivas et al. (2010) suggest to consider the following measure of non-Markovianity based on CP-divisibility where the maps have been defined according to (11). In the classical case considering the matrices whose el- Z ∞ ements are given by the conditional transition prob- RHP(Φ) = dt g(t) (48) N 0 abilities (18) according to (Λt,s)xy = T (x, t y, s) the Chapman-Kolmogorov equation takes the form| with

(Φt+,t I)( Ψ Ψ ) 1 Λt,s = Λt,τ Λτ,s, t > τ > s > 0, (46) g(t) = lim || ⊗ | ih | || − , (49) →0+ where each Λ is a stochastic matrix, which obviously t,s where Ψ is a maximally entangled state between open bears a strict relationship with (45), once one consid- system| andi ancilla. ers that positive maps are natural quantum counter- A further feature which can be considered in this con- part of classical stochastic matrices. Given the fact that text, and which has been proposed in Lorenzo et al. in the classical case the Markov condition (17) entails (2013) as another signature of non-Markovianity is the the Chapman-Kolmogorov equation, and building on the dynamical behavior of the volume of the accessible states, conclusion drawn at the end of Sec. II.B.2, i.e. that according to their parametrization in the Bloch repre- an intrinsic characterization of memory effects has to be sentation, identifying non-Markovianity with the growth based on the system’s density operator only, whose dy- of this volume, which also allows for a geometrical de- namics is fully described in terms of the time evolution scription. This study can be performed e.g. upon knowl- maps, one is led to associate quantum Markovianity with edge of the rates appearing in the time-local master equa- P-divisibility. This observation reinforces the result ob- tion, and appear to be a strictly weaker requirement for tained in Sec. II.C.3 relying on the study of the informa- Markovianity with respect to P-divisibility (Chru´sciński tion flow as quantified by the generalized trace distance and Wudarski, 2013). measure. Actually it can be shown that the P-divisibility To explain the relation between P-divisibility and CP- condition (46) is equivalent to the monotonicity property divisibility suppose that Alice prepares statesρ ˜1 orρ ˜2 on of the function S S an extended space S S and sends the states through X ˜ H ⊗H ˜ K(P 1(t),P 2(t)) = p P 1(x, t) p P 2(x, t) (47) the channel Φt = Φt I to Bob. The dynamical map Φt is 1 1 | 1 1 − 2 1 | ⊗ x then Markovian, in the sense of P-divisibility, if and only if Φt is CP-divisible. Note however that this construction which provides a generalization of the Kolmogorov dis- requires the use of a physically different system, described tance between two classical probability distributions by the tensor product space , which carries the HS ⊗ HS 12 information transferred from Alice to Bob. More specifi- quantum dynamical maps as a collection of time depen- cally, the maximization over the initial states must allow dent channels. In this respect the natural question is 1,2 for entangled statesρ ˜S which clearly explains the differ- how well Bob can recover information on the state pre- ence between the definitions of non-Markovianity for Φt pared by Alice performing a measurement after a given and Φ˜ t. This implies in particular that an experimental time t. A proposal related to this viewpoint has been assessment of non-Markovianity according to the crite- considered in (Bylicka et al., 2014), where two types rion of CP-divisibility would in general imply the capabil- of capacity of a quantum channel, namely the so-called ity to prepare entangled states and perform tomographic entanglement-assisted classical capacity and the quan- measurements on the extended space. Alternatively one tum capacity, have been considered. While the quantum should be able to perform process tomography instead of data processing inequality (Nielsen and Chuang, 2000) state tomography, with an analogous scaling with respect warrants monotonicity of these quantities provided the to the dimensionality of the system. time evolution is CP-divisible, lack of this monotonicity has been taken as definition of non-Markovianity of the dynamics. The connection between non-Markovianity 2. Monotonicity of correlations and entropic quantities and the quantum data processing inequality has also been recently studied by Buscemi and Datta (2014). Another line of thought about non-Markovianity of a quantum dynamics is based on the study of the behavior in time of quantifiers of correlations between the open III. MODELS AND APPLICATIONS OF system of interest and an ancilla system. In this respect NON-MARKOVIANITY both entanglement (Rivas et al., 2014) and quantum mutual information have been considered (Luo et al., 2012). In this section we first address the description of simple In this setting one considers an ancilla system and stud- prototypical systems, for which one can exactly describe ies the behavior in time of the entanglement or the mu- the non-Markovian features of the dynamics according tual information of an initially correlated joint system- to the concepts and measures introduced in Sec. II. In ancilla state. Given the fact that both quantities are particular we will stress how a non-Markovian dynamics non-increasing under the local action of a completely pos- leads to a recovery of quantum features according to the itive trace preserving transformation, CP-divisibility of notion of information backflow. In the second part we the time evolution would lead to a monotonic decrease present applications of the introduced non-Markovianity of these quantities. Their non monotonic behavior can measures to a number of more involved models illus- therefore be taken as a signature or a definition of quan- trating how memory effects allow to analyze their dy- tum non-Markovianity. Actually, Rivas et al. (2014) have namics and can be related to characteristic properties of considered CP-divisibility as the distinguishing feature complex environments, thus suggesting to use the non- of non-Markovianity, so that a revival of entanglement Markovianity of the open system dynamics as a quantum in the course of time is just interpreted as a witness of probe for features of the environment. non-Markovianity. In (Luo et al., 2012) instead failure of monotonicity in the loss of quantum mutual information is proposed as a new definition of non-Markovianity. A. Prototypical model systems In both cases one considers as initial state a maximally entangled state between system and ancilla. The quan- We consider first a two-level system undergoing pure tification of the effect is obtained by summing up the in- decoherence. This situation can be described by a time- creases over time of the considered quantity. Still other convolutionless master equation of the form (13) with approaches have connected non-Markovianity with non a single decay channel and allows to demonstrate in a monotonicity of the quantum Fisher information, consid- simple manner how the properties of the environment ered as a quantifier of the information flowing between and the system-environment coupling strength influence system and environment. In this case however the infor- the transition from a Markovian to a non-Markovian dy- mation flow is not directly traced back to a distance on namics for the open quantum system. A similar anal- the space of states (Lu et al., 2010). ysis is then performed for a dynamics which also takes In the trace distance approach one studies the behav- into account dissipative effects, further pointing to the ior of the statistical distinguishability between states, phenomenon of the failure of divisibility of the quantum and relying on the introduction of the Helstrom matrix dynamical map. We further discuss a dynamics driven (Chru´sciński et al., 2011) this notion can be directly re- by different decoherence channels, which allows to bet- lated to a divisibility property of the quantum dynamical ter discriminate between different approaches to the de- map. The generalization of the trace distance approach scription of non-Markovian behavior. Finally, we present also allows for a stronger connection to a quantum infor- some results on the non-Markovian dynamics of the spin- mation theory viewpoint. Indeed one can consider the boson model. 13

1. Pure decoherence model Note that for a generic microscopic or phenomenological decoherence model the decoherence function is in general Let us start considering a microscopic model for a pure a complex quantity even in the interaction picture, and decoherence dynamics which is amenable to an exact so- in this case the modulus of the function should be consid- lution and describes a single qubit interacting with a ered in formula (55). The map Φt is completely positive bosonic reservoir. The total Hamiltonian can be writ- since G(t) 6 1 which is equivalent to the positivity of the R t 0 0 ten as in Eq. (2) with the system Hamiltonian HS and time integral of the decay rate Γ(t) = 0 dt γ(t ). In or- the environmental Hamiltonian HE given by der to discuss the non-Markovianity of the obtained time evolution as discussed in Sec. II.C and Sec. II.D.1 we 1 X H = ω σ ,H = ω a† a , (50) consider the divisibility property of the time evolution. S 2 0 z E k k k k The master equation (54) has a single channel with decay rate γ(t) and thanks to the strict positivity of the deco- where ω0 is the energy diﬀerence between ground state −1 herence function Φt always exists. However, according 0 and excited state 1 of the system, σz denotes a Pauli to the criterion given by Eq. (14), P-divisibility fails at | i † | i matrix, ak and ak are annihilation and creation opera- times when the decoherence rate becomes negative, which tors for the bosonic reservoir mode labelled by k with is exactly when CP-divisibility is lost. This is generally frequency ωk, obeying the canonical commutation rela- true when one has a master equation in the time-local † tions [ak, ak0 ] = δkk0 . The interaction term is taken to form (13) with a single Lindblad operator, since in this be case the condition for CP-divisibility, given by the re-

X ∗ † quirement of positivity of the decay rate γ(t), coincides HI = σz gkak + gkak (51) with the condition (14) for P-divisibility. k The trace distance between time evolved states corresponding to the initial conditions ρ1,2 is given according with coupling constants gk. Considering a factorized ini- S tial condition with the environment in a thermal state at to (26) by inverse temperature β the model can be exactly solved 1 2 p 2 2 2 D ΦtρS, ΦtρS = a + G (t) b , (56) leading to a quantum dynamical map Φt which leaves | | the populations invariant and modifies the off-diagonal 1 2 1 2 where a = ρ11 ρ11 and b = ρ10 ρ10 denote the difference matrix elements according to of the populations− and of the− coherences of the initial states respectively. Its time derivative corresponding to ρ (t) = ρ (0), ρ (t) = G(t)ρ (0), (52) 11 11 10 10 the quantity (35) in terms of which the non-Markovianity where ρ (t) = i ρ (t) j denote the elements of the in- measure can be constructed then reads ij h | S | i teraction picture density matrix ρS(t). The function G(t) G(t) b 2 d is often called decoherence function and in the present σ(t) = p | | G(t). (57) a2 + G2(t) b 2 dt case it is real since we are working in the interaction pic- | | ture. It can be expressed in the form The non-monotonicity in the behavior of the trace dis- Z ∞ tance which is used as a criterion for non-Markovianity βω 1 cos (ωt) in Sec. II.C.2 is therefore determined by the non mono- G(t) = exp dω J(ω) coth − 2 , (53) − 0 2 ω tonic behavior of the decoherence function. In order to evaluate the measure Eq. (34) one has to maximize over where we have introduced the spectral density J(ω) = all pairs of initial states. As shown by Wißmann et al. P g 2δ(ω ω ) which keeps track of the features of k k k (2012) and discussed in Sec. II.C.2 the latter can be re- the environment| | − relevant for the reduced system descrip- stricted to orthogonal state pairs. For the case at hand it tion, containing informations both on the environmental follows from Eq. (57) that the maximum is obtained for density of the modes and on how strongly the system antipodal points on the equator of the Bloch sphere, so couples to each mode. It can be shown that the inter- that a = 0 and b = 1. The non-Markovianity measure action picture operator ρ (t) obeys a master equation of S (34) is then given| | by the form (13) with a single Lindblad operator, X h f i i d (Φ) = G(tk ) G tk , (58) ρ (t) = γ(t)[σ ρ (t)σ ρ (t)] , (54) N − dt S z S z − S k i f where the time dependent decay rate γ(t) is related to where tk and tk denote initial and final time point of the decoherence function G(t) by the k-th interval in which G(t) increases. Since the increase of G(t) coincides with the negativity of the de- 1 d coherence rate according to Eq. (55), the growth of the γ(t) = G(t). (55) −G(t) dt trace distance is in this case equivalent to breaking both 14

P-divisibility and CP-divisibility. Moreover, the gener- where the decoherence function is a complex function de- alized trace distance measure considered in Sec. II.C.3 termined by the spectral density J(ω) of the model. In leads to the same expression (58) for non-Markovianity. particular, denoting by f(t t ) the two-point correlation − 1 The measure for non-Markovianity of Eq. (48) based function of the environment corresponding to the Fourier on CP-divisibility takes in this case the simple expression transform of the spectral density, the function G(t) is determined as the solution of the integral equation Z RHP(Φ) = 2 dt γ(t), (59) d Z N − γ<0 G(t) = dt f(t t )G(t ) (62) dt − 1 − 1 1 which is the integral of the decoherence rate in the time with initial condition G(0) = 1. Also in this case it is intervals in which it becomes negative. An example of possible to write down the exact master equation obeyed an experimental realization of a pure decoherence model by the density matrix ρ (t) of the system in the form of is considered in Sec. IV.C.1, where the decoherence func- S Eq. (13), which in the interaction picture reads (Breuer tion is determined by the interaction between the polar- et al., 1999) ization and the frequency degrees of freedom of a photon. In this case the decoherence function is, in general, com- d i ρ (t) = S(t)[σ , ρ ] (63) plex valued, so that the previous formulae hold with G(t) dt S − 4 z S replaced by G(t) . The geometric measure discussed in 1 | | + γ(t) σ−ρ (t)σ σ σ−, ρ (t) , Sec. II.D.1, which connects non-Markovian behavior with S + − 2 { + S } the growth in time of the volume of accessible states, also leads to the same signature for non-Markovianity: where the time dependent Lamb shift is given by S(t) = ˙ The volume can again be identified with the decoherence 2 (G(t)/G(t)), and the decay rate can be written as −γ(t=) = 2 (G˙ (t)/G(t)), or equivalently function G(t) and therefore as soon as G(t) increases the − < volume grows. 2 d γ(t) = G(t) . (64) As we have discussed, in a simple decoherence model − G(t) dt| | non-Markovianity can be identified with the revival of | | coherences of the open system, corresponding to a back- The analysis of non-Markovianity in the present model flow of information from the environment to the system. closely follows the discussion in III.A.1 due to the cru- In this situation the time-local master equation (54) de- cial fact that the master equation (63) still has a single scribing the dynamics exhibits just a single channel, so Lindblad operator. The expression of the trace distance that all discussed criteria of non-Markovianity do coin- is determined by the decoherence function and, with the cide. same notation as in (56), reads p D Φ ρ1 , Φ ρ2 = G(t) G(t) 2a2 + b 2, (65) t S t S | | | | | | 2. Two-level system in a dissipative environment so that one has the time derivative We now consider an example of dissipative dynamics 2 G(t) 2a2 + b 2 d σ(t) = p| | | | G(t) . (66) in which the system-environment interaction influences G(t) 2a2 + b 2 dt| | both coherences and populations of a two-state system. | | | | The environment is still taken to be a bosonic bath, so Complete positivity of the map is ensured by positivity of that the free contributions to the Hamiltonian are again the integrated decay rate Γ(t), while the trace distance given by Eq. (50), but one considers an interaction term shows a non monotonic behavior when the modulus of in rotating wave approximation given by the decoherence function grows with time. A typical expression for the spectral density is given X ∗ † by a Lorentzian HI = gkσ+ak + gkσ−ak , (60) k J(ω) = γ λ2/2π (ω + ∆ ω)2 + λ2 (67) 0 0 − where σ = 0 1 and σ = 1 0 are the lowering and − + of width λ and centered at a frequency detuned from the raising operators| ih | of the system.| ih Such| an interaction de- atomic frequency ω by an amount ∆, while the rate γ scribes, e.g., a two-level atom in a lossy cavity. For a 0 0 quantifies the strength of the system-environment cou- factorized initial state with the environment in the vac- pling. Let us first discuss the case when the qubit is in uum state, this model is again exactly solvable, thanks resonance with the central frequency of the spectral den- to the conservation of the number of excitations. The sity, so that one has a vanishing detuning, ∆ = 0. The quantum dynamical map Φ transforms populations and t decoherence function then takes the form coherences according to −λt/2 dt λ dt 2 G(t) = e cosh + sinh , (68) ρ11(t) = G(t) ρ11(0), ρ10(t) = G(t)ρ10(0), (61) 2 d 2 | | 15

p where d = λ2 2γ λ. For weak couplings, correspond- expression for the decoherence function. For this case − 0 ing to γ0 < λ/2, the decoherence function G(t) is a real, the derivative of the trace distance (66) takes the simple monotonically decreasing function so that (66) is always R t 0 0 form σ(t) = γ(t) exp[ Γ(t)] with Γ(t) = 0 γ(t ) dt , negative and the dynamics is Markovian. In this case, as which puts into− evidence− the direct connection between discussed in Sec. II.A.2, the map is invertible and since the direction of information flow and the sign of the decay the time-convolutionless form of the master equation has rate. just a single channel, it is at the same time P-divisible and CP-divisible. Note that in the limit γ λ/2 the rate 0 γ(t) becomes time-independent, γ = γ0, such that the 3. Single qubit with multiple decoherence channels master equation (63) is of Lindblad form and leads to a Markovian semigroup. However, for larger values of the In the preceding examples we have shown that for a coupling between system and environment, namely for master equation in the time-convolutionless form (13) γ0 > λ/2, the function G(t) displays an oscillatory be- with a single Lindblad operator A(t), all the different havior leading to a non-monotonic| | time evolution of the considered criteria for non-Markovianity coincide. It is trace distance and, hence, to non-Markovian dynamics. therefore of interest to discuss an example in which one Thus, the trace distance measure of Eq. (34) is zero for has multiple decoherence channels with generally differ- γ0 < λ/2 and starts at the threshold γ0 = λ/2 to increase ent rates. As we shall shortly see, such model allows continuously to positive values (Laine et al., 2010a). It is to discriminate among the different definitions of non- interesting to note that the transition point γ0 = λ/2 ex- Markovian dynamics. To this end, we take a phenomeno- actly coincides with the point where the perturbation ex- logical approach and following Chru´scińskiand Manis- pansion of the time-convolutionless master equation (63) calco (2014); Chru´scińskiand Wudarski (2013); and Vac- breaks down. We also remark that for this model opti- chini (2012) we consider for a two-level system the master mal state pairs still correspond to antipodal points of the equation equator of the Bloch sphere (Xu et al., 2010), and that 3 the criterion for non-Markovianity based on the Helstrom d 1 X ρ (t) = γ (t)[σ ρ (t)σ ρ (t)] , (69) matrix considered in Sec. II.C.3 leads to the same results. dt S 2 i i S i − S i=1 In the parameter regime γ0 > λ/2 the map Φt defined by (61) is no longer invertible for all times due to the where σi with i = x, y, z denote the three Pauli opera- existence of zeros of the decoherence function. Thus, the tors. Considering, e.g., the decoherence dynamics of a 1 model provides an example for a family of maps lying in spin- 2 particle in a complex environment the rates γi(t) the lower (red) region depicted in Fig. 4. Physically, the would correspond to generally time dependent transver- two-level system reaches the ground state before memory sal and longitudinal decoherence rates. The dynamical effects revive the coherences and the population of the map corresponding to (69) can be exactly worked out and upper state. Indeed, also in this case non-Markovianity is given by the random unitary dynamics can be traced back to the revival of populations and 3 coherences, corresponding to a backflow of information X Φ (ρ ) = p (t)σ ρ σ , (70) from the environment to the system. Since the inverse t S i i S i i=0 of Φt does not exist for all times, the criterion based on CP-divisibility cannot strictly speaking be applied. This where σ0 denotes the identity operator and the coeffi- is reflected by the fact that the corresponding measure, cients pi(t) summing up to one are determined from the taking in this case again the expression (59), jumps from decoherence rates. Introducing the quantities Aij(t) = zero to infinity. exp[ (Γ (t)+ Γ (t))], where as usual we have denoted by − i j In the non-resonant case, i.e., for nonzero detuning, Γk(t) the time integral of the k-th decay rate, the coeffi- one can observe a transition from Markovian to non- cients pi(t) take the explicit form p0,1 = (1/4)[1 A12(t) Markovian dynamics for a fixed value of the coupling A (t) + A (t)] and p = (1/4)[1 A (t) ±A (t) ± 13 23 2,3 ∓ 12 ± 13 − in the weak coupling regime by increasing the detun- A23(t)]. These equations show that the dynamical map ing ∆. In this case the map is invertible for all times, actually depends on the sum of the integrals of the de- since the decay rate γ(t) is finite for all times, so that cay rates. According to its explicit expression the map also the measure (48) remains well defined. Again, the Φt is completely positive provided the coefficients pi(t) regions of non-Markovianity coincide for all approaches remain positive, in which case they can be interpreted as since the constraint in order to have CP-divisibility or a probability distribution and thus characterize the ran- P-divisibility is the same. The pair of states maximiz- dom unitary dynamics (70). Note that this can be the ing the expression of the measure (34) are however now case even if a decoherence rate stays negative at all times given by the projections onto the excited and the ground as in Hall et al. (2014) and Vacchini et al. (2011). state. This shows how the optimal pair, which is always As explained in Sec. II.A.2, the map is CP-divisible given by orthogonal states, does depend on the actual when all of the decoherence rates remain positive for all 16 t 0, so that as soon as at least one of the rates becomes (Φ) > 10 N > 1 negative the measure (48) becomes positive indicating a 9 non-Markovian behavior. However, the condition (14) 1 10− for having P-divisibility is weaker. Indeed, in order to 8 7 2 have P-divisible dynamics only the sum of all pairs of 10− 6

distinct decoherence rates has to remain positive, i.e., 0 3 γi(t) + γj(t) 0 for all j = i. The same constraint war- 5 10− > 6 T/ω rants monotonicity in time of the behavior of the trace 4 4 10− distance, so that also in this case the measure (34) of non- 3

Markovianity and its generalized version (39) based on 2 5 10− the Helstrom matrix and corresponding to P-divisibility 1 6 lead to the same result. In the characterization of non- 0.2 < 10− Markovianity based on the backflow of information from 0.2 1 2 3 4 5 6 7 8 9 10 Ω/ω the environment to the system memory effects therefore 0 only appear whenever the sum of at least a pair of deco- FIG. 5 (Color online) The non-Markovianity measure N (Φ) herence rates becomes negative. To better understand, defined by Eq. (34) for the spin-boson model as a function of how this fact can be related to the dynamics of the sys- temperature T and cutoff frequency Ω in units of the transi- tem, one can notice (Chru´scińskiand Maniscalco, 2014) tion frequency ω0 (Clos and Breuer, 2012). that the Bloch vector components σi(t) , according to Eq. (69), obey the equation h i

d 1 Figure 5 shows the non-Markovianity measure (Φ) for σi(t) = σi(t) , (71) N dth i −Ti(t)h i this model obtained from numerical simulations of the corresponding second-order time-convolutionless master where the relaxation times are given by Ti(t) = [γj(t) + equation, using an Ohmic spectral density of Lorentz- −1 γk(t)] (with all three indices taken to be distinct) and Drude shape with cutoff frequency Ω, reservoir tem- correspond to experimentally measurable quantities. Ap- perature T and a fixed small coupling strength of size pearance of non-Markovianity and therefore failure of P- γ = 0.1ω0 (Clos and Breuer, 2012). As can be seen from divisibility is thus connected to negative relaxation rates, the figure the dynamics is strongly non-Markovian both corresponding to a rebuild of quantum coherences. For for small cutoff frequencies and for small temperatures the present model one can also easily express the con- (note the logarithmic scale of the color bar). The emer- dition leading to a growth of the volume of accessible gence of a Markovian region within the non-Markovian states within the Bloch sphere. In fact, this volume of regime for small cutoffs and temperatures can be under- P3 accessible states is proportional to exp[ Γi(t)]. As stood in terms of a resonance between the transition fre- − i=1 a consequence, the dynamics according to the geometric quency and the maximum of an effective, temperature- criterion by Lorenzo et al. (2013) is Markovian if and only dependent spectral density of the environmental modes. P3 if i=1 γi(t) > 0 for t > 0, which is a strictly weaker criterion with respect to either P or CP-divisibility. While in the examples considered above the non- Markovianity measure (34) based on the trace distance and its generalization (39) based on the Helstrom ma- B. Applications of quantum non-Markovianity trix agree in the indication of non-Markovian dynamics, a situation in which these criteria are actually different In Sec. III.A we have illustrated quantum non- has been considered by Chru´sciński et al. (2011). Markovian behavior by means of simple model systems. The goal of the present section is to demonstrate that 4. Spin-boson model memory effects and their control also open new perspec- tives for applications. In fact, understanding various as- Finally, we briefly discuss a further paradigmatic pects of non-Markovianity makes it possible to develop model for dissipative quantum dynamics with many ap- more refined tools for reservoir engineering and to use plications, namely the spin-boson model (Leggett et al., memory effects as an indicator for the presence of typi- 1987). The system and the environmental Hamiltonian cal quantum features. Moreover, it becomes possible to are again given by Eq. (50), while the interaction Hamil- develop schemes where a small open system is used as tonian has the non-rotating-wave structure a quantum probe detecting characteristic properties of the complex environment it is interacting with. Here, we X ∗ † present a few examples from the recent literature illus- HI = σx gkak + gkak . (72) k trating these points. 17

1. Analysis and control of non-Markovian dynamics a steady state which is independent of the initial state. On the other hand, for δh/J > 1/2 the trace distance 1 We first consider the model of a spin- 2 particle with does not decay to zero asymptotically in time. This fea- 1 spin operator s0 coupled to a chain of N spin- 2 parti- ture can be interpreted as information trapping since it cles with spin operators sn (n = 1, 2,...,N), which has implies that the distinguishability of states does not van- been investigated by Apollaro et al. (2011). The system ish asymptotically. We also remark that the origin of Hamiltonian is given by H = 2h sz, where h denotes memory effects and the differences in the dynamical be- S − 0 0 0 a local field acting on the system spin s0. The environ- havior can be understood in more detail by studying the mental Hamiltonian represents an XX-Heisenberg spin spectrum of the total Hamiltonian (Apollaro et al., 2011). chain with nearest neighbor interactions of strength J in As our second example we discuss applications to a a transverse magnetic field h: physical system which has been studied extensively during the last twenty years, namely Bose-Einstein conden- N−1 N X X H = 2J sxsx + sy sy 2h sz . (73) sates (BECs) which can be manipulated with high preci- E − n n+1 n n+1 − n sion by current technologies and thus provide means for n=1 n=1 environment engineering (Dalfovo et al., 1999; Pethick The interaction Hamiltonian takes the form and Smith, 2008; Pitaevskii and Stringari, 2003). Fol- lowing Haikka et al. (2011) we consider an impurity-BEC H = 2J (sxsx + sysy) , (74) I − 0 0 1 0 1 system, where the Markovian to non-Markovian transition can be controlled by tuning the properties of the describing an energy-exchange interaction between the BEC. An impurity atom is trapped within a double-well system spin s and the first spin s of the chain with 0 1 potential, where the left ( L ) and right ( R ) states rep- coupling strength J . This model leads to a rich dynami- 0 resent the two qubit states.| i The BEC, which| i constitutes cal behavior of the system spin and allows to demonstrate the environment of the qubit, is trapped in a harmonic not only the impact of the system-environment interac- potential. The Hamiltonian for the total impurity-BEC tion, but also how the open system dynamics changes system reads when manipulating the interactions between the constituents of the environment and their local properties. X † X † ∗ X † ∗ It can be shown analytically that optimal state pairs H = Ekckck + (ξkck +ξkck)+ σz(gkck +gkck). for the trace distance measure correspond to antipodal k k k (76) points on the equator of the surface of the Bloch sphere. Here, E denotes the energy of the Bogoliubov mode c Moreover, when the local fields of the system and envi- k k of the condensate, σ = R R L L , while g and ξ ronmental sites are equal, h = h , one can derive an an- z k k 0 describe the impurity-BEC| ih and| − the | intra-BECih | couplings, alytic expression for the rate of change σ(t) of the trace respectively. distance [see Eq. (35)] for optimal state pairs: For a background BEC at zero temperature the open

σ(t) = (2/t) sgn[ 1(2t)] 2(2t), (75) qubit dynamics is described by a dephasing master equa- − J J tion of the form of Eq. (54) and it turns out that the where sgn is the sign function, and 1 and 2 denote decoherence rate γ(t) can be tuned by changing the the Bessel functions of order 1 and 2, respectively.J J In the inter-well distance, and by controlling the dimensional- physically more interesting case h = h0 the trace distance ity and the interaction strength (scattering length) of measure (Φ) [see Eq. (34)] has to6 be determined numer- the BEC. The detailed studies demonstrate that for a ically. LetN us first consider the case, where the system 3D gas, an increase of the well-separation and of the local field is zero, h0 = 0, the couplings between the spins intra-environment interaction leads to an increase of the are equal, J = J0, and we tune the ratio h/J of the local non-Markovianity measure for the impurity dynamics. field with respect to the spin-spin interaction strength. It Moreover, the dimensionality of the environment influ- turns out that the open system is then Markovian only ences the emergence of memory effects. Studying the at the point h/J = 1/2, i.e., when the strength of the Markovian to non-Markovian transition as a function of local field is half of the coupling between the spins. For the intra-environment interaction strength shows that a all other values of h/J the open system dynamics dis- 3D environment is the most sensitive. The Markovian plays memory effects with (Φ) > 0. Thus, the point of – non-Markovian crossover occurs in 3D for weaker val- Markovianity separates twoN regions where memory effects ues of the scattering length than in 1D and 2D environ- are present. This behavior persists for h0 = 0: Denoting ments, and a 1D environment requires the largest scat- the detuning between the local fields by δ6 = h h and tering length for the crossover. This behavior can be − 0 still keeping J = J0, the point of Markovianity occurs at ultimately traced back to the question of how the di- δh/J = 1/2. On both sides of this point the system again mensionality of the environment influences the spectral exhibits memory effects, but interestingly their proper- density which governs the open system dynamics. In gen- ties are different. For δh/J < 1/2 the system approaches eral, for small frequencies the spectral density shows a 18 power law behavior, J(ω) ωs. The spectral density is () called sub-Ohmic, Ohmic,∝ and super-Ohmic for s < 1, N s = 1, and s > 1, respectively. For a 3D gas, even without interactions within the gas, the spectral density has super-Ohmic character. Introducing interactions within the environment then makes this character stronger and thereby the 3D gas is most sensitive to the Markovian to non-Markovian transition. In contrast, for the 1D gas with increasing scattering length the spectral density first changes from sub-Ohmic to Ohmic and finally to super- Ohmic which then allows the appearance of memory effects.

2. Open systems as non-Markovian quantum probes FIG. 6 (Color online) The non-Markovianity measure N (Φ) of Eq. (34) for a qubit coupled to a one-dimensional Ising In addition to detecting, quantifying and controlling chain in a transverse field as a function of field strength λ∗ and particle number N. The measure vanishes along the black memory effects, it turns out to be fruitful to ask whether line and is nonzero everywhere else (Haikka et al., 2012). the presence or absence of such effects in the dynamics of an open quantum system allows to obtain important information about characteristic features of a complex environment the open system is interacting with. To il- The system-environment interaction (78) leads to the lustrate this point we present a physical scenario, where ∗ Markovian behavior indicates the presence of a quan- effective transverse field λ = λ + δ when the system is in the state e . At the critical point of the spin environ- tum phase transition in a spin environment (Sachdev, ∗ | i 2011). Consider a system qubit, with states g and e , ment (λ = 1) the spin-spin interaction and the external which interacts with an environment described| i by a one-| i field are of the same size and the environment is most dimensional Ising model in a transverse field (Quan et al., sensitive to external perturbations. This is reflected in a 2006). The Hamiltonians of the environment and the quite remarkable manner in the dynamics of the system et al. qubit-environment interaction are given by qubit as can be seen from Fig. 6 (Haikka , 2012). At the point λ∗ = 1 the system qubit experiences strong X H = J σzσz + λσx , (77) decoherence. As a matter of fact, it turns out that the dy- E − j j+1 j j namics of the system qubit always shows non-Markovian X behavior except at the critical point of the environment H = Jδ e e σx, (78) I − | ih | j where the dynamics is Markovian. Therefore, the sys- j tem qubit acts as a probe for the spin environment and where J is a microscopic energy scale, λ a dimensionless Markovian dynamics represents a reliable indicator of en- coupling constant describing the strength of the trans- vironment criticality. It is important to note that even x,z verse field, and σj are Pauli spin operators. The sys- though, strictly speaking, quantum phase transitions re- tem qubit is coupled with strength δ to all environmental quire to take the thermodynamic limit, the above results spins. The model yields pure dephasing dynamics de- hold for any number of environmental spins. scribed by a decoherence function G(t) which, for a pure environmental initial state Φ , is given by | i G(t) = Φ eiHg te−iHet Φ , (79) h | | i where the effective environmental Hamiltonians are IV. IMPACT OF CORRELATIONS AND EXPERIMENTAL Hα = HE + α HI α with α = g, e. We note that ac- REALIZATIONS cording to Eq.h | (79)| thei decoherence function is directly linked to the concept of the Loschmidt echo L(t) (Cuc- chietti et al., 2003) which characterizes how the environ- In this section we study the effect of two fundamental ment responds to perturbations by the system (Cucchi- kinds of correlations on the non-Markovian behavior of etti et al., 2004; Gorin et al., 2006; Goussev et al., 2008; open systems, namely correlations between the open sys- Peres, 1984). In fact, we have L(t) = G(t) 2 and, hence, tem and its environment and correlations within a com- there is also a direct connection between| the| Loschmidt posite environment. We conclude the section by giving echo and the evolution of the trace distance for optimal an overview of the experimental status of non-Markovian 1 2 p state pairs, i.e. D(ρS(t), ρS(t)) = L(t). open systems. 19

A. Initial system-environment correlations reveals one very important point: The information in the open system int(t) can increase above its initial value 1. Initial correlations and dynamical maps (0) only ifI there is some information initially outside Iint the system, i.e. ext(0) > 0. Obviously, the contraction Up to this point we have discussed the theory of open property (28) forI completely positive maps is a special quantum systems via the concept of a dynamical map, case of the inequality (80) which occurs when the system which in Sec. II.A.1 we wrote down in terms of the initial and the environment are initially uncorrelated and the state of the environment and of the total Hamiltonian. environmental state is not influenced by the system state In doing so we assumed that the open system and the 1,2 1,2 preparation, i.e. ρSE = ρS ρE. This leads to ext(0) = environment are independent at the initial time of prepa- 0 and, thus, Eq. (80) reduces⊗ to Eq. (29). I ration [see Eqs. (6) and (7)]. However, experimentalists often study fast processes in strongly coupled systems, where this independence is rarely the case, as originally 2. Local detection of initial correlations pointed out by Pechukas (1994). Going beyond this approximation brings us up against the problem of describ- We found that initial information outside the open sys- ing the dynamical properties of open quantum systems tem can lead to an increase of trace distance. But how without dynamical maps. can this be used to develop experimental methods to de- 1 The theory of completely positive maps plays a crucial tect correlations in some unknown initial state ρSE? To role in many fields of quantum physics and thus a gen- this end, let us combine (80) and the inequality (33) for eral map description for open systems, even in the pres- the initial time in order to reveal the role of initial cor- ence of initial correlations, would be very useful. Thus, relations more explicitly (Laine et al., 2010b): naturally, a lot of effort has been put in trying to ex- (t) (0) D(ρ1 (0), ρ1 (0) ρ1 (0)) tend the formalism along these lines (Alicki, 1995; Sha- Iint − Iint 6 SE S ⊗ E bani and Lidar, 2009; Stelmachovic and Buzek, 2001). +D(ρ2 (0), ρ2 (0) ρ2 (0)) SE S ⊗ E In this framework the aim has been to determine un- 1 2 +D(ρE(0), ρE(0)). (81) der which conditions the system dynamics can be described by means of completely positive maps if initial This inequality clearly shows that an increase of the trace correlations are present. Unfortunately, no generally ac- distance of the reduced states implies that there are ini- 1 2 knowledged answer to this question has been found and tial correlations in ρSE(0) or ρSE(0), or that the initial the topic is still under discussion (Brodutch et al., 2013; environmental states are different. Rodr´ıguez-Rosario et al., 2010; Shabani and Lidar, 2009). As useful as a description in terms of completely pos- 1 2 itive maps would be, one may as well ask the question: D(⇢S(t), ⇢S(t)) How do the initial correlations influence the dynamics? Especially, since an experimentalist is often restricted to no sign of! looking only at a subset of system states, he would want initial! correlations to conclude something about the system based on the dynamics of this subset. It is not possible to construct a map, but perhaps something else could be concluded. In the following we present an approach, in which the dynamics of information flow between the system and the initial! correlations t environment is studied. We will see that the initial correlations modify the information flow and consequently can be witnessed from the dynamical features of the open FIG. 7 (Color online) Schematic picture of the behavior of system. the trace distance with and without initial correlations. If Let us have a closer look at the dynamics of the infor- the trace distance between two states, prepared such that ρ2 (0) = (Λ⊗I)ρ1 (0), never exceeds its initial value (dashed mation inside the open system (t), defined in Eq. (30), SE SE Iint black line), the dynamics does not witness initial correlations. for a general open system, where initial correlations could If, on the other hand, the trace distance for such pair of states be present. Suppose there are two possible preparations increases above the initial value (solid red line), we can con- giving rise to distinct system-environment initial states clude the presence of initial correlations. 1 2 ρSE(0) and ρSE(0). Since the total system and the environment evolve under unitary dynamics, we have Let us now assume that one can perform a state tomography on the open system at the initial time zero int(t) int(0) = ext(0) ext(t) 6 ext(0), (80) I − I I − I I and at some later time t, to determine the reduced states 1 1 where ext(t) is the information outside the open system ρS(0) and ρS(t). In order to apply inequality (81) to definedI in Eq. (31). This inequality, as simple as it is, detect initial correlations we need a second reference 20

2 state ρS(0), which has the same environmental state, i.e. The left-hand side of this inequality yields a locally acces- 2 1 ρE(0) = ρE(0). This can be achieved by performing a sible lower bound for a measure of the quantum discord. 1 local quantum operation on ρSE(0) to obtain the state Experimental realizations of the scheme are briefly described in Sec. IV.C.2. Most recent studies suggest that ρ2 (0) = (Λ I)ρ1 (0). (82) SE ⊗ SE the method could also be applied for detecting a critical point of a quantum phase transition via dynamical The operation Λ acts locally on the variables of the open monitoring of a single spin alone (Gessner et al., 2014a). system, and may be realized, for instance, by the measurement of an observable of the open system, or by a unitary transformation induced, e.g., through an exter- 1 B. Nonlocal memory effects nal control field. Now, if ρSE(0) is uncorrelated then also ρ2 (0) is uncorrelated since it has been obtained through SE We now return to study the case of initially uncor- a local operation. Therefore, for an initially uncorre- related system-environment states, for which the usual lated state the trace distance cannot increase according description in terms of dynamical maps can be used. So to inequality (81). Thus, any such increase represents a far, we have concentrated on examples with a single sys- witness for correlations in the initial state ρ1 (0), as is SE tem embedded in an environment. However, many rel- illustrated in Fig. 7. We note that this method for the evant examples in quantum information processing in- local detection of initial correlations requires only local clude multipartite systems for which nonlocal proper- control and measurements of the open quantum system, ties such as entanglement become important and it is which makes it feasible experimentally and thus attrac- therefore essential to study how non-Markovianity is in- tive for applications. In fact, experimental realizations of fluenced by scaling up the number of particles. the scheme have been reported recently, see Sec. IV.C.2. The question of additivity of memory effects with re- Furthermore, Smirne et al. (2010) have studied how the spect to particle number was recently raised by Addis scheme can be employed for detecting correlations in et al. (2013, 2014); and Fanchini et al. (2013) and it was thermal equilibrium states. found that the different measures for non-Markovianity have very distinct additivity properties. Indeed, the re- tr tr U(t) 1 E 1 E 1 search on multipartite open quantum systems is yet in its ⇢S(0) ⇢SE(0) ⇢S(t) infancy and a conclusive analysis remains undone. In the = ⇤ I = ⌦ 6 following, we will discuss a particular feature of bipartite trE trE U(t) open systems: the appearance of global memory effects ⇢2 (0) ⇢2 (0) ⇢2 (t) S SE S in the absence of local non-Markovian dynamics. This is at variance with the standard situation in which the FIG. 8 (Color online) Scheme for the local detection of enlargement of considered degrees of freedom leads from system-environment quantum correlations in some state a non-Markovian to a Markovian dynamics (Martinazzo 1 ρSE (0), employing a local dephasing operation Λ to gener- et al., 2011). 2 ate a second reference state ρSE (0). We will now take a closer look at nonlocal maps, for which memory effects may occur even in the absence of The above strategy can even be used in order to lo- local non-Markovian effects. In Laine et al. (2012, 2013) cally detect a specific type of quantum correlations, i.e. it is shown that such maps may be generated from a system-environment states with nonzero quantum dis- local interaction, when correlations between the environ- cord (Modi et al., 2012). This is achieved by taking the ments are present and that they may exhibit dynamics local operation Λ in Eq. (82) to be a dephasing operation with strong global memory effects although the local dy- leading to complete decoherence in the eigenbasis of the namics is Markovian. The nonlocal memory effects are 1 state ρS(0) (Gessner and Breuer, 2011, 2013). The ap- studied in two dephasing models: in a generic model of plication of this dephasing operation destroys all quan- qubits interacting with correlated multimode fields (Wiß- 1 tum correlations of ρSE(0), while leaving invariant its mann and Breuer, 2013; Wißmann and Breuer, 2014) and 1 1 marginal states ρS(0) and ρE(0) (see Fig. 8). Thus, one in an experimentally realizable model of down converted can use the quantity (ρ1 (0)) = D(ρ1 (0), ρ2 (0)) as a photons traveling through quartz plates, which we will C SE SE SE measure for the quantum correlations in the initial state discuss later in detail. 1 ρSE(0) (Luo, 2008). Since the trace distance is invariant Before we turn to discuss any experimental endeavor to under unitary transformations and since the partial trace detect non-Markovian dynamics, let us discuss in more is a positive map, the corresponding local trace distance rigor about the generation and possible applications of 1 2 1 D(ρS(t), ρS(t)) provides a lower bound of (ρSE(0)) for nonlocal memory effects. Consider a generic scenario, C all times t > 0. Hence, we have where there are two systems, labeled with indices i = 1, 2,

1 2 1 which interact locally with their respective environments. max D(ρS(t), ρS(t)) 6 (ρSE(0)). (83) t>0 C The dynamics of the two systems can be described via 21 the dynamical map C. Experiments on non-Markovianity and correlations

12 12 12 ρS (t) = Φt ρS (0) Up to this point we have not yet discussed any ex- h i perimental aspects of the detection of memory effects. = tr U 12 (t)ρ12(0) ρ12(0)U 12†(t) , (84) E SE S ⊗ E SE In the framework of open quantum systems the environment is in general composed of many degrees of freedom 12 1 2 i where USE(t) = USE(t) USE(t) with USE(t) describing and is therefore difficult to access or control. To perform the local interaction between⊗ the system i and its environ- experiments on systems where the environment induced 12 ment. If the initial environment state ρE (0) factorizes, dynamical features can be controlled is thus challenging. 12 1 2 12 i.e. ρE (0) = ρE(0) ρE(0), also the map Φt factorizes. However, in the past years clever schemes for modifying Thus, the dynamics⊗ of the two systems is given by a lo- the environment have been developed allowing the estab- 12 1 2 12 cal map Φt = Φt Φt . On the other hand, if ρE (0) lishment of robust designs for noise engineering. In this ⊗ 12 exhibits correlations, the map Φt cannot, in general, be section we will briefly review some of the experimental factorized. Consequently, the environmental correlations platforms where a high level of control over the envi- may give rise to a nonlocal process even though the in- ronment degrees of freedom has been accomplished and teraction Hamiltonian is purely local (see Fig. 9). non-Markovian dynamics observed and quantified.

open system 1+2 1. Control and quantification of memory effects in photonic 12 t non-Markovian systems open system 1 open system 2 1 2 Quantum optical experiments have for many decades t Markovian t Markovian been the bedrock for testing fundamental paradigms of quantum mechanics. The appeal for using photonic sys- 1 2 tems arises from the extremely high level of control allow- interaction HI interaction HI ing, for example, controlled interactions between different degrees of freedom, preparation of arbitrary polarization states and a full state tomography. Needless to say, pho- environment!1 environment!2 tons thus offer an attractive experimental platform also correlations for studying non-Markovian effects. Liu et al. (2011) introduced an all-optical experiment FIG. 9 (Color online) Schematic picture of a system with which allows through careful manipulation of the initial nonlocal memory effects. The open systems 1 and 2 locally environmental states to drive the open system dynam- interact with their respective environments. Initial correla- ics from the Markovian to the non-Markovian regime, tions between the local environments cause the occurrence of nonlocal memory effects. to control the information flow between the system and the environment, and to determine the degree of non- Markovianity. In the experiment the photon polariza- For a local dynamical process, all the dynamical prop- tion degree of freedom (with basis states H and V for erties of the subsystems are inherited by the global sys- horizontal and vertical polarization, respectively)| i | playsi tem, but naturally for a nonlocal process the global dy- the role of the open system. The environment is repre- namics can display characteristics absent in the dynamics sented by the frequency degree of freedom of the pho- of the local constituents. Especially, for a nonlocal pro- ton (with basis ω ), which is coupled to the sys- ω>0 cess, even if the subsystems undergo a Markovian evo- tem via an interaction{| i} induced by a birefringent mate- lution, the global dynamics can nevertheless be highly rial (quartz plate). The interaction between the polar- non-Markovian as was shown for the dephasing models ization and frequency degrees of freedom in the quartz by Laine et al. (2012, 2013). plate of thickness L is described by the unitary operator inλωt Thus, a system can globally recover its earlier lost U(t) λ ω = e λ ω , where nλ is the refrac- quantum properties although the constituent parts are tion index| i ⊗ | fori a photon| withi ⊗ | polarizationi λ = H,V and undergoing decoherence and this way initial environmen- t = L/c is the interaction time with the speed of light c. tal correlations can diminish the otherwise destructive The photon is initially prepared in the state ψ χ , effects of decoherence. This feature has been further with ψ = α H + β V and χ = R dωf(ω) |ω i,⊗ where | i deployed in noisy quantum information protocols, such f(ω)| givesi the| i amplitude| i for| thei photon to| i be in a as teleportation (Laine et al., 2014) and entanglement mode with frequency ω. The quartz plate leads to a distribution (Xiang et al., 2014), suggesting that non- pure decoherence dynamics of superpositions of polariza- Markovianity could be a resource for quantum information states (see Sec. III.A.1) described by the complex tion. decoherence function G(t) = R dω f(ω) 2eiω∆nt, where | | 22

∆n = nV nH . An optimal pair of states maximizing state tomography can further be performed thus allowing the non-Markovianity− measure (34) for this map is given a rigorous quantification of the memory effects. by ψ = √1 ( H V ) and the corresponding trace Also, a series of other experimental studies on non- | 1,2i 2 | i ± | i 1 2 Markovian dynamics in photonic systems have been per- distance is D(ρS(t), ρS(t)) = G(t) . Clearly, controlling the frequency spectrum f(ω)| 2 changes| the dynamical formed recently. Tang et al. (2012) report a measure- features of the map. | | ment of the non-Markovianity of a process with tunable system- environment interaction, Cialdi et al. (2011) observe controllable entanglement oscillations in an effective non-Markovian channel and in (Chiuri et al., 2012; a) Jin et al., 2014) simulation platforms for a wide class of non-Markovian channels are presented.

2. Experiments on the local detection of correlations

A number of experiments have been carried out in order to demonstrate the local scheme for the detection of correlations between an open system and its environment described in Sec. IV.A.2. Photonic realizations of this scheme have been reported in (Li et al., 2011; Smirne et al., 2011), where the presence of initial correlations between the polarization and the spatial degrees of freedom of photons is shown by the observation of an increase of the local trace distance between a pair of initial state. b) As explained in Sec. IV.A.2, it is also possible to reveal locally the presence of quantum correlations represented by system-environment states with nonzero discord if the local operation Λ in Eq. (82) is taken to induce complete decoherence in the eigenbasis of the open system state. The first photonic realizations of this strategy based on Eq. (83) is described in (Tang et al., 2014). Moreover, Cialdi et al. (2014) have extended the method to enable the discrimination between quantum and classical correlations, and applied this extension to a photonic experimental realization. All experiments mentioned so far employ photonic degrees of freedom to demonstrate non-Markovianity and FIG. 10 (Color online) Non-Markovian dynamics arising from system-environment correlations. The first experiment an engineered frequency spectrum. (a) The frequency spec- showing these phenomena for matter degrees of freedom trum of the initial state for various values of the tilting angle θ of the cavity. (b) The change of the trace distance as functions has been described by Gessner et al. (2014b). In this ex- of the tilting angle θ. The transition from the non-Markovian periment nonclassical correlations between the internal to the Markovian regime occurs at θ ≈ 4.1◦, and from the electronic degrees of freedom and the external motional Markovian to the non-Markovian regime at θ ≈ 8.0◦, corre- degrees of freedom of a trapped ion have been observed sponding to the occurrence of a double peak structure in the and quantified by use of Eq. (83). Important features of frequency spectrum (a) (Liu et al., 2011). the experiment are that the lower bounds obtained from the experimental data are remarkably close to the true The modification of the photon frequency spectrum is quantum correlations present in the initial state, and that realized by means of a Fabry-Pérotcavity mounted on a it also allows the study of the temperature dependence rotator which can be tilted in the horizontal plane. The of the effect. shape of the frequency spectrum is changed by changing the tilting angle θ of the Fabry-Pérotcavity, as can be seen in Fig. 10a. Further, the frequency spectrum 3. Non-Markovian quantum probes detecting nonlocal changes the dephasing dynamics such that the open sys- correlations in composite environments tem exhibits a reversed flow of information and thus allows to tune the dynamics from Markovian to non- In Sec. IV.B we demonstrated that initial correlations Markovian regime (see Fig. 10b). In the experiment, full between local parts of the environment can lead to non- 23 local memory effects. In Liu et al. (2013a) such nonlocal coefficient K of the photon pairs from measurements per- memory effects were experimentally realized in a pho- formed on the polarization degree of freedom. Thus by tonics system, where manipulating the correlations of the performing tomography on a small system we can ob- photonic environments led to non-Markovian dynamics of tain information on frequency correlations, difficult to the open system. The experimental scheme further pro- measure directly. Thus, in this context, the open system vided a controllable diagnostic tool for the quantification (polarization degrees of freedom) can serve as a quantum of these correlations by repeated tomographic measure- probe which allows us to gain nontrivial information on ments of the polarization. the correlations in the environment (frequency degrees of Let us take a closer look at the system under study in freedom). the experiment. A general pure initial polarization state of a photon pair can be written as ψ = a HH + | 12i | i b HV + c VH + d VV and all initial states of the V. SUMMARY AND OUTLOOK polarization| i | plusi the| frequencyi degrees of freedom are product states In this Colloquium we have presented recent advances Z in the definition and characterization of non-Markovian Ψ(0) = ψ12 dω1dω2 g(ω1, ω2) ω1, ω2 , (85) dynamics for an open quantum system. While in the clas- | i | i ⊗ | i sical case the very definition of Markovian stochastic pro- where g(ω1, ω2) is the probability amplitude for photon 1 cess can be explicitly given in terms of constraints on the to have frequency ω1 and for photon 2 to have frequency conditional probabilities of the process, in the quantum ω2, with the corresponding joint probability distribution realm the peculiar role of measurements prevents a direct P (ω , ω ) = g(ω , ω ) 2. If the photons pass through formulation along the same path and new approaches are 1 2 | 1 2 | quartz plates, the dynamics can be described by a gen- called for in order to describe memory effects. We have eral two-qubit dephasing map, where the different de- therefore introduced a notion of non-Markovianity of a coherence functions can be expressed in terms of Fourier quantum dynamical map based on the behavior in time of transforms of the joint probability distribution P (ω1, ω2). the distinguishability of different system states, as quan- For a Gaussian joint frequency distribution with identical tified by their trace distance, which provides an intrinsic single frequency variances C and correlation coefficient characterization of the dynamics. This approach allows K, the time evolution of the trace distance correspond- to connect non-Markovianity with the flow of informa- ing to the Bell-state pair ψ± = √1 ( HH VV ) is tion from the environment back to the system and nat- | 12i 2 | i ± | i found to be urally leads to the introduction of quantum information concepts. It is further shown to be connected to other 1 D(t) = exp ∆n2C t2 + t2 2 K t t , (86) approaches recently presented in the literature building −2 1 2 − | | 1 2 on divisibility properties of the quantum dynamical map. where ti denotes the time photon i has interacted with In particular, a generalization of the trace distance cri- its quartz plate up to the actual observation time t. For terion allows to identify Markovian time evolutions with uncorrelated photon frequencies we have K = 0 and the quantum evolutions which are P-divisible, thus leading trace distance decreases monotonically, corresponding to to a clear-cut connection between memory effects in the Markovian dynamics. However, as soon as the frequen- classical and quantum regimes. As a crucial feature the cies are anticorrelated, K < 0, the trace distance is non- trace distance approach to non-Markovianity can be ex- monotonic which signifies quantum memory effects and perimentally tested and allows for the study of system- non-Markovian behavior. On the other hand, the local environment correlations, as well as nonlocal memory ef- frequency distributions are Gaussian and thus for the sin- fects emerging from correlations within the environment. gle photons the trace distance monotonically decreases. We have illustrated the basic feature of the considered Therefore we can conclude, that the system is locally theoretical approaches to quantum non-Markovianity, Markovian but globally displays nonlocal memory effects. typically leading to a recovery of quantum coherence In the experiment two quartz plates act consecutively properties, by analyzing in detail simple paradigmatic for the photons, and the magnitude of the initial anti- model systems, and further discussing more complex correlations between the local reservoirs is tuned. After models which show important connections between non- the photon exits the quartz plates, full two-photon polar- Markovianity of the open system dynamics and features ization state tomography is performed and by changing of the environment. It is indeed possible to obtain infor- the quartz plate thicknesses, the trace distance dynam- mation on complex quantum systems via study of the dy- ics is recovered. 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