Equivalence between non-Markovian dynamics and correlation backflows

The information encoded into an open quantum system that evolves under a Markovian dynamics is always monotonically decreasing. Nonetheless, given a quantifier for the information contained in the system, in general it is not clear if for any non-Markovian dynamics it is possible to observe a non-monotonic evolution of this quantity, namely a backflow. We address this problem by considering correlations of finite-dimensional bipartite systems. For this purpose, we introduce a class of correlation measures and we prove that there exists at least one element from this class that provides a correlation backflow if and only if the dynamics is non-Markovian. Moreover, we provide a set of initial probe states that accomplish this witnessing task. This result provides the first one-to-one relation between non-Markovian dynamics of finite-dimensional quantum systems and correlation backflows.

The study of open quantum systems dynamics [1,2] is of central interest in quantum mechanics. When the environment that surrounds a quantum system is included in the description of its evolution, we say that the quantum system is open. Since there are no experimental scenarios where a quantum system can be considered completely isolated, this approach provides a far more realistic description of quantum evolutions.
The interaction between an open quantum system S and its environment E leads to two possible regimes of evolution. We focus on the phenomenological and mathematical descriptions of these dynamical regimes. The phenomena associated with the Markovian regime are characterized by the monotonic contractivity of the information contained in the open system. Indeed, in this case we have a unidirectional flow of information from S to E and we say that the environment is memoryless. Instead, in the non-Markovian regime, this flow is not unidirectional and part of the information lost is recovered in one or more subsequent time intervals. These phenomena are called backflows of information. However, it is nontrivial what mathematical framework is better suited to reproduce this phenomenology. Recently, a definition based on a notion of divisibility of dynamical maps, namely the operators describing the dynamical evolution of the system, achieved a promising consensus [2][3][4][5][6][7][8][9]. More in details, this mathematical property requires that the evolution between any two times is represented by a completely positive and trace-preserving (CPTP) linear map.
Many effort is directed to test this mathematical definition by studying the characteristic backflows of information that different physical quantities show when the evolution is non-Markovian. Therefore, once we consider a quantity that is contractive for any Markovian evolution, we can study its "non-Markovian witnessing potential", namely its ability to show a backflow when the dynamics is not Markovian. Distinguishability between states [6][7][8], correlation measures [9][10][11][12] and channel capacities [13] are some examples of quantities that have been studied in this scenario. Moreover, while Markovian phenomena are reproduced correctly by definition, the non-trivial point that has to be analyzed is if it is possible to obtain one-to-one connections between backflows of these quantities and non-Markovian dynamical maps. In-deed, this result would imply a correspondence between the phenomenological and the mathematical description of non-Markovianity that we have presented.
This question has been considered in Ref. [7], where the guessing probability of ensembles of evolving states is considered as a witness. The authors prove that for any dynamics there exists at least one initial ensemble of an enlarged state space for which the corresponding guessing probability shows a backflow if and only if the evolution is non-Markovian.
In this work we focus on the behavior of the set of correlation measures of bipartite systems that evolve under Markovian and non-Markovian evolutions. In particular, we study the connection between revivals of correlations of bipartite systems when the evolution of one subsystem is non-Markovian. Several measures of correlations have already been considered in this scenario, such as quantum mutual information [9] and entanglement measures [10]. Recently, a correlation measure that is able to witness almost all non-Markovian dynamics has been introduced [11]. However, none of these correlations are able to witness all non-Markovian dynamics [12].
The main result of this work is the first proof of a one-toone relation between backflows of correlation measures and non-Markovian dynamics. In particular, we introduce a class of correlation measures for bipartite systems that provides backflows if and only if the evolution is not Markovian. To do so, we make use of supplementary ancillary systems that allow to define initial probe states that succeed in this witnessing task.
Non-Markovianity and divisibility properties. In order to introduce the mathematical structure needed, given a generic finite-dimensional Hilbert space H, we define B(H) to be the set of linear bounded operators that act on H and S(H) the set of positive semidefinite, hermitian and trace one operators on H, namely the state space of H.
We consider an open quantum system S described by states in a finite-dimensional Hilbert space H S . The evolution of S from the initial time t 0 to any t ≥ t 0 is given by a dynamical map: a CPTP linear operator Λ S (t, t 0 ) : S(H S ) → S(H S ). Therefore, in order to define the complete temporal evolution of an open quantum system S , the linear operator Λ S (t, t 0 ) has arXiv:1906.06269v1 [quant-ph] 14 Jun 2019 to be CPTP for any t ≥ t 0 .
The main concepts needed to introduce the mathematical structure we adopt to define Markovian and non-Markovian dynamical maps are the divisibility of the dynamical map Λ S (t, t 0 ) and the positive (P) and completely positive (CP) divisibility of Λ S (t, t 0 ) in terms of intermediate maps V S (t, t ).
CP-divisibility is the most common definition used to describe Markovian dynamics and it is the one we consider in this work. Hence, in the following, we say that a dynamical map is Markovian if and only if it is CP-divisible. In the same way, if we consider a non-Markovian dynamical map, either it is not divisible or for some t 0 < t < t the intermediate map Measurements with fixed output probability distributions. Any measurement process on a quantum state ρ ∈ S(H) is defined by a positive-operator valued measure (POVM), namely a set of hermitian and positive semi-definite operators is the identity operator on H and n is the number of possible measurement outcomes. Each operator P i represents a different outcome and p i = Tr ρP i is the corresponding occurrence probability. Now we consider a generic bipartite system, where the corresponding Hilbert space is the ensemble of states of B that we obtain applying on ρ AB the generic POVM {P A,i } n i=1 defined for the subsystem A, where We call {p i } n i=1 and {ρ B,i } n i=1 respectively the output probability distribution and the output states of the measurement. We call their combination E(ρ AB , {P A,i } n i=1 ) the output ensemble. We consider a generic probability distribution composed by a finite number n of positive elements, where n i=1 p i = 1. We define the set of n-output POVMs such that, if applied on ρ ∈ S(H), provide outcomes where the output probability distribution is P.
: Tr ρ P i = p i , ∀i = 1, ... , n} . Similarly, given the state ρ AB of a bipartite system, we define those measurement processes that, if applied on one side of ρ AB , produce output ensembles that are distributed as P.
Definition 4. Given the finite probability distribution P = {p i } n i=1 , the n-output POVM {P A,i } n i=1 on H A is a P-POVM on A for ρ AB ∈ S(H AB ) if and only if it belongs to the set Analogously, we can define Π P B (ρ AB ). We notice that for any given P and ρ AB , it is easy to show that Π P is a non-empty and convex set for any ρ AB ∈ S(H AB ).
be an ensemble of n states in S(H). Given the probability distribution {p i } n i=1 , we randomly extract a state from E, which subsequently we want to identify. The guessing probability P g (E) is the average probability to successfully identify the extracted state with an optimal measurement. This quantity is defined as where the maximization is performed over the n-output POVMs of B(H). Now we show how the guessing probability of ensembles of states can be used to witness non-Markovian dynamics. We consider a finite-dimensional system H S ⊗ H A , where the open quantum system S is evolved by a generic dynamical map Λ S (t, t 0 ) and A is an ancillary system. Given an initial where For any Λ S (t, t 0 ) and E S A (t 0 ), the quantity P g (E S A (t)) is nonincreasing: Given any dynamical map Λ S (t, t 0 ) and time interval [τ, τ + ∆τ], there exist an ancillary system A and an initial ensemble such that we have a backflow if and only if there is no CPTP intermediate map is finite and dim(H A ) ≤ d S ≡ dim(H S ). We underline that, even if we do not make it explicit, E S A (t 0 ) strictly depends on Λ S (t, t 0 ) and [τ, τ + ∆τ]. The approach shown in Ref. [7] is very general and applies to any dynamical map Λ S (t, t 0 ) defined on a finite-dimensional system. Indeed, it provides the proof of a one-to-one relation between backflows of the guessing probability of evolving ensembles and the mathematical definition of non-Markovianity.
A class of correlation measures. Let P ≡ {p i } i be a generic finite probability distribution and ρ AB ∈ S(H A ⊗H B ) a generic finite-dimensional bipartite system state. We introduce the correlation measure where the maximization is performed over the P-POVMs on A for ρ AB and we used the definitions ) and p max ≡ max i p i (see Fig 1). Therefore, we can consider a class of correlation measures where each element is defined by a different distribution P.
The operational meaning of this correlation measure for a given P is the following. Its value (modulo p max ) is the largest guessing probability of the ensembles {p i , ρ B,i } i that A can generate on B measuring its side of ρ AB with P-POVMs.
, it follows that the largest distinguishability of the output ensembles of B that A can generate measuring ρ (1) . To consider C P A a proper correlation measure for any P, we have to show that it is: zero-valued for product states, non-negative and monotonically decreasing under local operations [12]. In order to prove the first property, given a generic product state Similarly, we can define the class of measures of the form given by the bipartition between the subsystems A and B, where the open quantum system S undergoes the evolution defined by Λ S (t, t 0 ). symmetric class of measures Finally, we notice that the correlation measures given in Eqs. (7), (8) and (9) can be considered as generalizations for generic distributions P of the correlation measures introduced in Ref. [11], where only uniform distributions are considered.
The probe states. We want to show that for any non-Markovian dynamics there exist at least one correlation measure and one initial probe state that show a correlation backflow. In particular, similarly to Ref. [7], we consider the most general scenario where a dynamical map Λ S (t, t 0 ) defines the evolution for t ≥ t 0 and we focus on a generic time interval [τ, τ + ∆τ]. We provide an initial probe state and a distribution P for which the correlation measure C P A shows a backflow in the time interval [τ, τ + ∆τ] if and only if there is no CPTP intermediate map V S (τ + ∆τ, τ) for Λ S (t, t 0 ).
First, we introduce the bipartition and the state space needed to consider C P A and the initial probe state. We define the bipartite system S(H A ⊗ H B ) such that dim(H A ) = n and H B ≡ H S ⊗ H A ⊗ H A , where dim(H S ) = dim(H A ) = d S and dim(H A ) = n + 1. We fix orthonormal basis for H A and H A as follows: .. , |n+1 A } is a set of n+1 orthonormal states on H A . We notice that the ancillary systems A and A that we considered with S in B can be introduced as a single ancillary system H A ⊗ H A (see Fig. 2).
We define the states ρ B,i ≡ ρ S A ,i ⊗ |n + 1 n + 1| A of S(H B ), where we made use of the states that define the ensemble . We introduce the following class of initial probe states ρ (λ) AB (t 0 ) of the bipartite system S(H A ⊗ H B ) parametrized by λ ∈ [0, 1) where σ S A is a generic state of S(H S ⊗H A ) and we underline that the index i runs from 1 to n. Since the ancillary systems do not evolve, the action of the dynamical map of the evolution on the probe state, i.e., I A ⊗ Λ S (t, t 0 ) ⊗ I A A (ρ (λ) AB (t 0 )), preserves the initial classical-quantum separable structure for any t ≥ t 0 where ρ B,i (t) = Λ S (t, t 0 ) ⊗ I A A (ρ B,i ) and σ S A (t) = Λ S (t, t 0 ) ⊗ I A (σ S A ). We notice that each state ρ B,i (t), for i = 1, ... , n, is orthogonal to each σ S A (t) ⊗ | j j| A , for j = 1, ... , n. Finally, since Tr B [ρ (λ) AB (t)] does not depend on t, the collection of P- ) is time independent. Witnessing non-Markovianity with correlations. We want to provide a procedure that witnesses any non-Markovian evolution with backflows of correlations. In the case of bijective or pointwise non-bijective Λ S (t, t 0 ), this scenario has been studied in Refs. [10,11]. Moreover, the negativity entanglement measure is able to witness any non-Markovian qubit evolution [10].
The evolution of the initial state ρ AS A = n i=1 p i |i i| A ⊗ρ S A ,i is an intuitive choice for the study of C P A (ρ AS A (t)) as a witness of non-Markovian dynamics. Indeed, the projective mea- , which provides backflows for any non-Markovian dynamics (see Eq. (6)). Nonetheless, in general {|i i| A } n i=1 is not selected by the maximization required to evaluate C P A (ρ AS A (t)) [12]. Therefore, we need to make use of the probe states ρ (λ) AB (t 0 ). We present the main result of this paper, namely that the class of correlation measures C P A is able to witness any non-Markovian dynamics. Proof. We consider the ancillary system H = H A ⊗ H A , the correlation measure C AB = C P A and the set of initial probe states ρ (λ) AB (t 0 ) for λ ∈ [0, 1). We want to prove that, for wisely chosen values of the parameter λ, we have a backflow if and only if there is no CP intermediate map V S (τ + ∆τ, τ) for the dynamical map Λ S (t, t 0 ) studied. We notice, since Π P A (ρ (λ) AB (t)) = Π P (Tr B [ρ (λ) AB (t)]), the set Π P A (ρ (λ) AB (t)) does not depend on t and λ. Therefore, the pro- for any t and λ. In the following, if not specified otherwise, the index i runs from 1 to n. The output ensemble that we obtain measuring ρ (λ) The corresponding guessing probability is (see Appendix) Now we consider a P-POVMs {P A,i } on A for ρ (λ) AB (t) different from {|i i| A } i . In general, we obtain (see Appendix): . Similarly to Eq. (14), we obtain In order to understand in which occasions we have a backflow of C P A (ρ (λ) AB (t)) in the time interval [τ, τ + ∆τ], we write: Now we focus on the behavior of C P A (ρ (λ) AB (t)) for t = τ and different values of λ (we omit the dependence on τ of some quantities to increase readability). This quantity is given by the P-POVMs that provide the maximum value of Eq. (16) when ρ (λ) AB (τ) is considered. In general, this maximum may be obtained with more than one P-POVM. We choose one of them and we call it {P (λ) A,i } i , i.e., We consider Eq. (16) when {P (λ) A,i } i is chosen. We define the corresponding ensembles that appear in this expression E ⊥(λ) and E (λ) , namely such that We focus on the second term of Eq. (17) and we distinguish the two possible scenarios:

• (B): {P (λ)
A,i } i {|i i| A } i for any value of λ < 1. We start studying case (A). In Appendix we prove that if A,i } i = {|i i| A } i for some λ * , then the same is also true for any λ ≥ λ * . Considering Eqs. (6), (14) and (17), it is clear that in this case we obtain a backflow if and only if there is no intermediate CPTP map V S (τ + ∆τ, τ) for Λ S (t, t 0 ). The case given by the scenario (B) needs some additional analysis: Having in mind the implications of {P (λ) A,i } i {|i i| A } i , the latter inequality, together with Eqs. (14) and (19), implies that Thanks to Eqs. (14) and (19), the condition given by the inequality (21) can be written as: We deduce that, for λ that approaches its maximum value: This result leads us to the limits: lim λ→1 E (λ) = E S A (τ) and lim λ→1 P g (E (λ) ) = P g (E S A (τ)) . (26) Thanks to the last limit, the property given in Eq. (22) and the continuity property of the guessing probability, we have that: ∀δ > 0, ∃λ δ > 0 : P g (E (λ) ) − P g (E S A (τ)) < δ , ∀λ ∈ (λ δ , 1) .
To conclude, we consider inequalities (17) and (29) for λ ∈ (λ, 1) and we obtain if and only if there is no intermediate CPTP map V S (τ + ∆τ, τ) for Λ S (t, t 0 ), which concludes the proof for the case (B).
The robustness of this backflow is provided by following results that are valid for any Λ S (t, t 0 ) and [τ, τ+∆τ]: the guessing probability of ensembles of states is a continuous function, the set of P-POVMs for ρ (λ) AB (t) does not depend on λ and t and there exists a continuous interval of values of λ for which ρ (λ) AB (t 0 ) allows backflows of C P A (ρ (λ) AB (t)) when the evolution is non-Markovian. Therefore, if we add a small enough perturbation to the initial probe state, we still witness a correlation backflow. Hence, the state which provides a backflow of C P A in the scenario described above is not unique: there exists a set of initial states with the same dimension of S(H A ⊗ H B ) that accomplish this task.
Since there are no assumptions for the structure of E S A (t 0 ), it is straightforward to adapt our technique to any ensemble of states of S(H S ⊗ H A ). In particular, if the evolution of an initial ensemble {p i , φ S A ,i } n i=1 provides a backflow of the corresponding guessing probability in a time interval [τ, τ + ∆τ], we can consider C P A (ψ (λ) We make some examples of ensembles (different from E S A (t 0 )) that can be considered with this approach to witness particular classes of non-Markovian evolutions. A constructive method that provides ensembles of two equiprobable states that witness any bijective or pointwise non-bijective non-Markovian dynamics is given in Ref. [8]. Moreover, in Ref. [12] an explicit example that shows how to apply this method is studied. The existence of two states ensembles that detect any image nonincreasing dynamics, namely such that Im(Λ t ) ⊆ Im(Λ s ) for any s < t, is proven in Ref. [15]. Finally, in Ref. [14] is proven that two-states ensembles are sufficient to witness any qubit dynamics.
Discussion. In this work we showed that any non-Markovian dynamics can be witnessed through the backflows of the set of correlation measures C P A . For this purpose, we introduced a class of initial probe states ρ (λ) AB (t 0 ) that allows to accomplish this task. Hence, we proved the first one-to-one correspondence between Markovian evolutions, i.e., defined by CP-divisible dynamical maps, and monotonic decrease of the set of correlation measures, i.e., the absence of backflows of correlatons.
It would be useful to obtain a constructive method that provides the explicit form of the elements of the ensemble E S A (t 0 ) that we used for the definition of the probe state ρ (λ) AB (t 0 ). Indeed, while we prove the existence of probe states that are able to show backflows C P A for any dynamics that is not Markovian, their components are not known explicitly.
Finally, since the class of bipartite correlations that we studied does not consider the subsystems A and B in a symmetric way, an open question is to understand if also C P AB (see Eq. (9)) is able to witness any non-Markovian dynamics as C P A .

Monotonic behavior of C P A under local operations
We consider a general bipartite finite-dimensional quantum system defined on the Hilbert space H AB = H A ⊗ H B . Therefore, the finite-dimensional states that we consider are ρ AB ∈ S(H AB ). We consider a generic finite probability distribution P = {p i } n i=1 and we prove that C P A is monotone under local operations of the form Λ A ⊗ I B and In order to show the effect of the application of a local operation of the form Λ A ⊗ I B on C P A (ρ AB ), we look at Π P A (ρ AB ) in a different way. Each element of this collection is a P-POVM for ρ AB , i.e., they generate output ensembles where the output probability distribution is P = {p i } i . In fact, we can consider C P A (ρ AB ) (see Eq. (7)) as the maximization over all the possible output ensembles with output probability distribution P that we can generate measuring the subsystem A of ρ AB .
The effect of the first local operation that we consider is: is the set of Kraus operators that defines Λ A . Now we analyze the relation between Π P A (ρ AB ) and Π P A (ρ AB ). Given a P-POVM forρ AB , i.e., {P A,i } i ∈ Π P A (ρ AB ), the probabilities and the states of the output ensemble E ρ AB , {P A,i } i are Tr ρ AB P A,i = p i and ρ B,i = Tr A ρ AB P A,i /p i . Now we write p i = Tr A ρ AB P A,i = Tr Λ A ⊗ I B (ρ AB )P A,i as follows where we have defined the operatorsP A,i ≡ Λ * Moreover, we show that they are positive semi-definite operators. Indeed, for any |ψ A ∈ H A , we have ψ| APA, where each element of the last sum is non-negative because P A,i is positive semi-definite. It follows that {P A,i } i is a POVM and in particular a P-POVM for ρ AB , i.e., {P A,i } i ∈ Π P A (ρ AB ). Thus, for every P- A (ρ AB ) for ρ AB , such that the output ensembles are identical: Hence, any P-distributed ensemble of B that can be generated fromρ AB can also be obtained from ρ AB . Therefore, we obtain the following inclusion Finally, since as we said above C P A (ρ AB ) is the maximum guessing probability of the P-distributed output ensembles that can be generated from ρ AB , from Eq. (32) we conclude that C P A (ρ AB ), compared to C P A (ρ AB ), is defined as maximization over a larger set. Hence, for any state ρ AB and CPTP map Λ A , we obtain Next we show that C P A (ρ AB ) is monotonic under local operations of the form I A ⊗ Λ B . We find that the collection of the P-POVMs forρ AB = I A ⊗ Λ B (ρ AB ), namely Π P A (ρ AB ), coincides with Π P A (ρ AB ). In order to prove this, we apply a general POVM {P A,i } i both on ρ AB andρ AB and we show that the respective output ensembles are defined by the same probability distribution. Indeed, being Tr ρ AB P A,i (Tr I A ⊗ Λ B (ρ AB )P A,i ) the probability for the i-th output of the POVM considered when it is applied on ρ AB (ρ AB ), we have Tr I A ⊗ Λ B (ρ AB )P A,i = Tr ρ AB P A,i , where this identity uses the trace-preserving property of the superoperator I A ⊗ Λ B . Consequently, if {P A,i } i is a P-POVM for ρ AB , which means that Tr ρ AB P A,i = p i , in the same way Tr Given a P-POVM {P A,i } i both for ρ AB andρ AB , we compare the corresponding output states From Eq. (35) and the definition of the guessing probability, it follows that The consequence of the last relation is that for any P-distributed output ensemble ensemble that we can generate fromρ AB there exists at least one P-distributed output ensemble that we can generate from ρ AB for which the guessing probability is equal or greater. Hence, considering the definition of C P A (see Eq. (7)), Eqs. (34) and (36), we conclude that for any state ρ AB and CPTP map Λ B .
Performing P-POVMs on the probe state: the orthogonal and the parallel components We show how to obtain the results given in Eqs. (14) and (16). The projective measurement {|i i| A } n i=1 is a P-POVM on A for ρ (λ) AB (t) for any t and λ (see Eq. (11)). We consider E(ρ (λ) AB (t), {|i i| A } i ), namely the ensemble of states of B that we obtain measuring ρ (λ) AB (t) with {|i i| A } i : where ρ B,i = ρ S A ,i ⊗ |n + 1 n + 1| A (see Eq. (10)). We evaluate the guessing probability of this ensemble and we obtain P g (ρ (λ) We notice that, for any i = 1, . . . , n, every state that belongs to the set {σ S A (t) ⊗ |i i| A } i is orthogonal to every state of the set {ρ S A ,i (t) ⊗ |n + 1 n + 1| A } i . It follows that the elements of {P B,i } i that provide a non-zero product σ S A (t) ⊗ |i i| A P B,i belong to span({|i j| B } i j ), where |i B and | j B belong to the tensor product between the elements of M S A , i.e., an orthonormal basis of H S ⊗ H A , and {|k A } n k=1 (notice that dim(H A ) = n + 1). Similarly, the elements of {P B,i } i that provide a non-zero product ρ S A ,i (t) ⊗ |n + 1 n + 1| A P B,i belong to span({|i j | B } i j ), where |i B and | j B belong to the tensor product between the elements of M S A and |n + 1 A . Therefore, since span({|i j| B } i j ) is orthogonal to span({|i j | B } i j ), the maximization in Eq. (39) can be divided in two independent maximizations