Entanglement revival can occur only when the system-environment state is not a Markov state
aa r X i v : . [ qu a n t - ph ] A p r Entanglement revival can occur only when the system-environment state is not aMarkov state
Iman Sargolzahi ∗ Department of Physics, University of Neyshabur, Neyshabur, Iran
Markov states have been defined for tripartite quantum systems. In this paper, we generalize thedefinition of the Markov states to arbitrary multipartite case and find the general structure of animportant subset of them, which we will call strong Markov states. In addition, we focus on animportant property of the Markov states: If the initial state of the whole system-environment is aMarkov state, then each localized dynamics of the whole system-environment reduces to a localizedsubdynamics of the system. This provides us a necessary condition for entanglement revival in anopen quantum system: Entanglement revival can occur only when the system-environment state isnot a Markov state. To illustrate (a part of) our results, we consider the case that the environmentis modeled as classical. In this case, though the correlation between the system and the environmentremains classical during the evolution, the change of the state of the system-environment, from itsinitial Markov state to a state which is not a Markov one, leads to the entanglement revival inthe system. This shows that the non-Markovianity of a state is not equivalent to the existence ofnon-classical correlation in it, in general.
I. INTRODUCTION
A famous and important relation in quantum informa-tion theory is the strong subadditivity relation , i.e., foreach tripartite quantum state ρ ABE , the following in-equality holds: S ( ρ AB ) + S ( ρ BE ) − S ( ρ ABE ) − S ( ρ B ) ≥ , (1)where ρ AB = Tr E ( ρ ABE ), ρ BE = Tr A ( ρ ABE ) and ρ B = Tr AE ( ρ ABE ) are the reduced states and S ( ρ ) ≡− Tr( ρlogρ ) is the von Neumann entropy [1].Markov states have been defined, in Ref. [2], as tripar-tite quantum states which satisfy the strong subadditiv-ity relation with equality. Recently, Markov states havebeen applied in studying the dynamics of open quantumsystems [3, 4].In this paper, we generalize the definition of theMarkov states to arbitrary multipartite case. Our def-initions will be given in two forms, a weak one and a strong one. The strong form is more restricted than theweak form. In addition, we find the general structure ofthe strong Markov (SM) states.The above results will be given during our study ofthe role of the Markov states in entanglement dynamicsof open quantum systems. This will help us to give thedefinitions, and so the subsequent related results, suchthat they have clear physical meanings and applications.The dynamics of the entanglement in open quantumsystems, both in bipartite and multipartite cases, hasbeen studied widely [5]. Entanglement may decrease oreven experience revivals during the interaction of the sys-tem with the environment [6].Consider a bipartite system S = AB such that eachpart interacts with its local environment. One may ex-pect that in this case only entanglement decrease (sudden ∗ [email protected], [email protected] death) will occur, since entanglement does not increaseunder local operations. But, interestingly, it has beenshown, both theoretically and experimentally, that theentanglement revival can occur under such circumstances(see, e.g., [7–9]). More unexpectedly, entanglement re-vival can occur even when the environment is classical[9–14].We question when entanglement revival can occur, un-der local interactions, and find a necessary condition forthis phenomenon. We show that the entanglement re-vival (increase) can occur only when the whole state ofthe system-environment is not Markov state. This neces-sary condition is valid for both bipartite and multipartitecases.Usually, the initial state of the system-environment ischosen factorized, which is a Markov state. In addition,the dynamics of the system-environment is given by localunitary operators. So, the entanglement of the system S , initially, starts to decrease and if, e.g., at time t theentanglement of the system starts to revive (increase),then we conclude that ρ SE ( t ), the state of the system-environment at time t , is not a Markov state.In the next section, we consider the simplest case:When only the part B of our bipartite system S = AB interacts with the environment E . We recall the originaldefinition of the Markov states from Ref. [2], and we willsee that the entanglement revival can occur only when ρ ABE , the whole state of the system-environment, is nota Markov state.As stated before, the original definition of the Markovstates in Ref. [2] is for the tripartite case. In Sect. III, weextend the definition of the Markov states to the quadri-partite case. We give two definitions, a weak one and,a more restricted form, a strong one. We generalize theresult of Sect. II, about entanglement revival, to the casethat each part of the system S = AB interacts with itslocal environment. This will be done, using the weakdefinition. In addition, we give our first main result asTheorem 3. This theorem gives us the general structureof the quadripartite strong Markov (SM) states.In Sect. IV, we generalize our results to arbitrary multi-partite case. We give our second main result as Theorems4 and 5. In these theorems, we find the general structureof the SM states for arbitrary multipartite case. In addi-tion, we show that, as the previous sections, if the initialstate of the system-environment is a weak Markov (WM)state, then each localized dynamics of the whole system-environment reduces to a localized subdynamics of thesystem. Therefore, also for the multipartite case, entan-glement revival can occur only when the initial state ofthe system-environment is not a WM state.To illustrate (a part of) our results, we consider thecase that the environment is classical in Sect. V. Thoughthe correlation between the system and the environ-ment remains classical during the evolution, entangle-ment revival can occur. So, the whole state of thesystem-environment changes from its initial Markov stateto a state which is not a Markov one. This impliesthat non-Markovianity of the whole state of the system-environment is not equivalent to the existence of non-classical correlation between the system and the environ-ment.Finally, we end our paper in Sect. VI, with a summaryof our results. II. WHEN ONLY THE PART B INTERACTSWITH THE ENVIRONMENT
Consider a bipartite quantum system S = AB , suchthat the part A is isolated from the environment, andonly the part B interacts with the environment E . So,we have ρ ′ ABE = id A ⊗ Ad U BE ( ρ ABE ) ≡ I A ⊗ U BE ρ ABE I A ⊗ U † BE , (2)where ρ ABE ( ρ ′ ABE ) is the initial (final) state of thesystem- environment, id A ( I A ) is the identity map (op-erator) on the part A and U BE is a unitary operator onthe both B and E .Now, assume that we have ρ ABE = id A ⊗ Λ B ( ρ AB ) , (3)where ρ AB = Tr E ( ρ ABE ) is the initial state of the system S = AB and Λ B is a completely positive (CP) map from B to BE . [A completely positive map, on a state ρ , isa map which can be written as P i K i ρK † i , where K i arelinear operators such that P i K † i K i = I ( I is the identityoperator) [1].] If Eq. (3) holds, the tripartite state ρ ABE is called a Markov state [2]:
Definition 1.
A tripartite state ρ ABE is called aMarkov state, if it can be written as Eq. (3).
It can be shown that, for a tripartite state ρ ABE , thestrong subadditivity inequality, i.e., Eq. (1), holds withequality, if and only if, Eq. (3) holds (if and only if, ρ ABE is a Markov state) [2]. From Eqs. (2) and (3), we have ρ ′ AB = Tr E ( ρ ′ ABE )= Tr E ◦ [ id A ⊗ Ad U BE ] ◦ [ id A ⊗ Λ B ]( ρ AB )= id A ◦ [Tr E ◦ Ad U BE ◦ Λ B ]( ρ AB )= id A ◦ E B ( ρ AB ) , (4)where ρ ′ AB is the final state of the system S = AB and E B = Tr E ◦ Ad U BE ◦ Λ B is a CP map on the part B . E B isCP since it is a composition of three CP maps: Λ B is CPby assumption, Ad U BE is obviously CP and the CP-nessof Tr E can be shown easily [1].In other words, if the initial state of the whole system-environment is a Markov state as Eq. (3), then eachlocalized dynamics, as Eq. (2), reduces to a localizedsubdynamics as Eq. (4). In fact, we have the followingtheorem, which is proven in Ref. [3]. We give it in theform introduced in Ref. [4], which is appropriate for ourpurpose in this paper. Theorem 1.
If, for a tripartite state ρ ABE , each lo-calized dynamics as ρ ′ ABE = id A ⊗ F BE ( ρ ABE )= X j (cid:16) I A ⊗ F ( j ) BE (cid:17) ρ ABE (cid:16) I A ⊗ F ( j ) † BE (cid:17) , X j F ( j ) † BE F ( j ) BE = I BE , (5) reduces to a localized subdynamics as ρ ′ AB = id A ⊗ E B ( ρ AB )= X i (cid:16) I A ⊗ E ( i ) B (cid:17) ρ AB (cid:16) I A ⊗ E ( i ) † B (cid:17) , X i E ( i ) † B E ( i ) B = I B , (6) then ρ ABE is a Markov state as Eq. (3), and vice versa.In Eq. (5), F ( j ) BE are linear operators on BE and, in Eq.(6), E ( i ) B are linear operators on B . The inverse part of Theorem 1 states that if ρ ABE is a Markov state, then each localized dynamics as Eq.(5) reduces to a localized subdynamics as Eq. (6).Let M be an entanglement monotone (measure). So, M ( ρ ′ AB ) ≤ M ( ρ AB ), since entanglement does not in-crease under local operations as Eq. (6) (see, e.g., Ref.[5]). Therefore, Corollary 1.
If for a localized dynamics of the wholesystem-environment as Eq. (5), the entanglement of thesystem S = AB increases: M ( ρ ′ AB ) > M ( ρ AB ) , thenwe conclude that the initial state of the whole system-environment, ρ ABE , is not a Markov state as Eq. (3).
The following point is also worth noting. Assume thatfor all t ∈ [ t , t ′ ], the entanglement of the system S = AB increases monotonically. Since we have considered thetime evolution of the system-environment as Eq. (2), thetime evolution operator from t to t , U ABE ( t , t ), t < t ,is also localized as I A ⊗ U BE ( t , t ). Therefore, for each t ∈ [ t , t ′ ), the state of the system-environment ρ ABE ( t )is not a Markov state as Eq. (3).Let’s end this section with a theorem, proven in Ref.[2], which gives the general structure of the Markovstates, for the tripartite case. Theorem 2.
A tripartite state ρ ABE is a Markov stateas Eq. (3), if and only if, there exists a decompositionof the Hilbert space of the subsystem B , H B , as H B = L k H b Lk ⊗ H b Rk such that ρ ABE = M k λ k ρ Ab Lk ⊗ ρ b Rk E , (7) where { λ k } is a probability distribution ( λ k ≥ , P k λ k =1 ), ρ Ab Lk is a state on H A ⊗ H b Lk and ρ b Rk E is a state on H b Rk ⊗ H E . ( H A and H E are the Hilbert spaces of A and E , respectively.) Remark 1.
Theorem 2 is valid for the case that H B is finite dimensional, but H A and H E can be infinite di-mensional. The condition that H B is finite dimensionalcomes from the fact that the proof of Theorem 2 in Ref.[2] is based on a result, proven in Ref. [15], which is forthe finite dimensional case. III. WHEN EACH PART OF THE SYSTEMINTERACTS WITH ITS LOCAL ENVIRONMENT
Now, let’s consider the case that the two parts A and B of our bipartite system are separated from each otherand each part interacts with its own local environment.Let’s denote the local environment of A as E A , the lo-cal environment of B as E B and the whole state of thesystem-environment as ρ AE A BE B .First, we generalize the definition of the Markov statesto the quadripartite case. The original definition, in Eq.(3), is for the tripartite case. Definition 2.
We call a quadripartite state ρ AE A BE B a weak Markov (WM) state if there exist CP maps Λ A ,from A to AE A , and Λ B , from B to BE B , such that ρ AE A BE B = Λ A ⊗ Λ B ( ρ AB ) , (8) where ρ AB = Tr E A E B ( ρ AE A BE B ) . If Eq. (8) holds, then each localized dynamics as F AE A ⊗ F BE B , for the whole system-environment, re-duces to a localized subdynamics as E A ⊗ E B , for thesystem: ρ ′ AB = Tr E A E B ( ρ ′ AE A BE B )= Tr E A E B ◦ [ F AE A ⊗ F BE B ] ◦ [Λ A ⊗ Λ B ]( ρ AB )= [Tr E A ◦ F AE A ◦ Λ A ] ◦ [Tr E B ◦ F BE B ◦ Λ B ]( ρ AB )= E A ◦ E B ( ρ AB ) , (9)where ρ ′ AE A BE B ( ρ ′ AB ) is the final state of the system-environment (system). Therefore, M ( ρ ′ AB ) ≤ M ( ρ AB ).In other words, Corollary 2.
If for a localized dynamics of the wholesystem-environment as F AE A ⊗ F BE B , the entanglementof the system S = AB increases: M ( ρ ′ AB ) > M ( ρ AB ) ,then we conclude that the initial state of the wholesystem-environment, ρ AE A BE B , is not a WM state as Eq.(8). Consider a special case that the localized dynamics ofthe whole system-environment is as id AE A ⊗ F BE B . So,from Eq. (9), we have ρ ′ AB = Φ A ◦ E B ( ρ AB ) , (10)where Φ A = Tr E A ◦ Λ A is a CP map on A . Note that ρ A = Tr B ( ρ AB ) does not change during the evolution.So, a natural requirement, which we may want to add,is that, for arbitrary CP map E B on B , we must haveΦ A ◦ E B ( ρ AB ) = id A ◦ E B ( ρ AB ). Similarly, for arbitraryCP map E A on A , we must have E A ◦ Φ B ( ρ AB ) = E A ◦ id B ( ρ AB ), where Φ B = Tr E B ◦ Λ B is a CP map on B .The above discussion leads us to the following defini-tion: Definition 3.
We call a quadripartite state ρ AE A BE B a strong Markov (SM) state if1. there exist CP maps Λ A , from A to AE A , and Λ B ,from B to BE B , such that ρ AE A BE B = Λ A ⊗ Λ B ( ρ AB ) ,and2. for each arbitrary CP maps E A and E B , we have E A ◦ Φ B ( ρ AB ) = E A ◦ id B ( ρ AB ) and Φ A ◦ E B ( ρ AB ) = id A ◦ E B ( ρ AB ) , respectively. In the following of this section, we will prove our firstmain result: The general structure of the quadripartiteSM states.
Theorem 3.
A quadripartite state ρ AE A BE B is a SMstate, if and only if, there exist decompositions of theHilbert spaces of the subsystems A , H A , and B , H B , as H A = L j H a Lj ⊗ H a Rj and H B = L k H b Lk ⊗ H b Rk , respec-tively, such that ρ AE A BE B = M j,k λ jk ρ a Lj E A ⊗ ρ a Rj b Lk ⊗ ρ b Rk E B . (11) In Eq. (11), { λ jk } is a probability distribution, ρ a Lj E A isa state on H a Lj ⊗ H E A , ρ a Rj b Lk is a state on H a Rj ⊗ H b Lk and ρ b Rk E B is a state on H b Rk ⊗ H E B . ( H E A and H E B arethe Hilbert spaces of E A and E B , respectively.)Proof. Showing that the state given in Eq. (11) canbe written as Eq. (8) has been done in Ref. [4]. Inaddition, using the result of Ref. [4], showing that thesecond property in Definition 3 is also fulfilled, by a stateas Eq. (11), is simple. So, in the following, we focus onthe reverse: Each quadripartite SM state can be writtenas Eq. (11).First, note that the CP map Λ B , in Eq. (8), is a mapfrom B to BE B . To make the input and output spacesthe same, we redefine Λ B in the following way: If foran operator x on B we have Λ B ( x ) = X , where X is aoperator on BE B , we set Λ B ( x ⊗ | E B ih E B | ) = X where | E B i is a fixed state in H E B . This redefinition allowsus to write Λ B in the following form. One can find anancillary Hilbert space H C B , a fixed state | C B i ∈ H C B and a unitary operator V B on H B ⊗ H E B ⊗ H C B in sucha way that the CP map Λ B can be written as [1]:Λ B ( x ) = Λ B ( x ⊗ | E B ih E B | )= Tr C B (cid:16) V B ( x ⊗ | E B ih E B | ⊗ | C B ih C B | ) V † B (cid:17) . (12)Also, note that for the CP map Φ B , on the B , we haveΦ B ( x ) = Tr E B C B (cid:16) V B ( x ⊗ | E B ih E B | ⊗ | C B ih C B | ) V † B (cid:17) , (13)with the unitary V B introduced in Eq. (12). Similarresults can be driven for the CP maps Λ A and Φ A .Second, from Eq. (8), we haveΦ A ⊗ Φ B ( ρ AB ) = ρ AB . So, using the property 2 in Definition 3, we can rewritethe above equation as id A ⊗ Φ B ( ρ AB ) = ρ AB . (14)Now, from the proof of Theorem 2 in Ref. [2], we knowthat if Eq. (14) holds, then there exists a decompositionof the H B as H B = L k H b Lk ⊗ H b Rk such that:1. ρ AB can be decomposed as ρ AB = M k q k ρ Ab Lk ⊗ ρ b Rk , (15)where { q k } is a probability distribution, ρ Ab Lk is a stateon H A ⊗ H b Lk and ρ b Rk is a state on H b Rk , and2. the unitary operator V B , in Eq. (13), is as V B = M k I b Lk ⊗ V b Rk E B C B , (16)where I b Lk is the identity operator on H b Lk and V b Rk E B C B is a unitary operator on H b Rk ⊗ H E B ⊗ H C B .Also note that, since during the proof of Theorem 2 inRef. [2] a result of Ref. [15] has been used, H B is finitedimensional.Similarly, starting from Φ A ⊗ id B ( ρ AB ) = ρ AB , it canbe shown that there exists a decomposition of the finitedimensional Hilbert space H A as H A = L j H a Lj ⊗ H a Rj such that:1. ρ AB can be decomposed as ρ AB = M j p j ρ a Lj ⊗ ρ a Rj B , (17)where { p j } is a probability distribution, ρ a Lj is a state on H a Lj and ρ a Rj B is a state on H a Rj ⊗ H B , and2. the unitary operator V A is as V A = M j V a Lj E A C A ⊗ I a Rj , (18) where I a Rj is the identity operator on H a Rj and V a Lj E A C A is a unitary operator on H a Lj ⊗ H E A ⊗ H C A .Third, consider the projectionΠ j ≡ Π A j ⊗ I B = (Π a Lj ⊗ Π a Rj ) ⊗ I B , (19)where Π A j , Π a Lj and Π a Rj are the projectors onto H A j = H a Lj ⊗ H a Rj , H a Lj and H a Rj , respectively. So, from Eqs.(15) and (17), we haveΠ j ρ AB Π j = p j ρ a Lj ⊗ ρ a Rj B = M k q k σ A j b Lk ⊗ ρ b Rk , (20)where σ A j b Lk = ¯Π j ρ Ab Lk ¯Π j and ¯Π j ≡ Π A j ⊗ Π b Lk (whereΠ b Lk is the projection onto H b Lk ). σ A j b Lk is a positive oper-ator on H A j ⊗ H b Lk . Let p ′ jk = Tr( σ A j b Lk ); so 0 ≤ p ′ jk ≤ p ′ jk >
0, we define ρ A j b Lk = σ A j b Lk p ′ jk , otherwise, if p ′ jk = 0, we define ρ A j b Lk arbitrarily. So,Eq. (20) can be rewritten as p j ρ a Lj ⊗ ρ a Rj B = M k q k p ′ jk ρ A j b Lk ⊗ ρ b Rk . Tracing from both sides, with respect to a Lj , we get p j ρ a Rj B = M k λ jk ρ a Rj b Lk ⊗ ρ b Rk , (21)where ρ a Rj b Lk = Tr a Lj ( ρ A j b Lk ) and λ jk = q k p ′ jk . Therefore,Eq. (17) can be rewritten as ρ AB = M j,k λ jk ρ a Lj ⊗ ρ a Rj b Lk ⊗ ρ b Rk . (22)Fourth, combining Eqs. (12) and (16) gives usΛ B = M k id b Lk ⊗ Λ b Rk , (23)where id b Lk is the identity map on b Lk and Λ b Rk is a CPmap from b Rk to b Rk E B . Similarly, we haveΛ A = M j Λ a Lj ⊗ id a Rj , (24)where Λ a Lj is a CP map from a Lj to a Lj E A .Finally, using Eqs. (8), (22), (23) and (24), we achieveEq. (11), which completes the proof. (cid:3) Remark 2.
As Theorem 2, Theorem 3 is valid for thecase that H A and H B are finite dimensional, but H E A and H E B can be infinite dimensional. In other words,the system S = AB is finite dimensional but the envi-ronments E A and E B can be infinite dimensional. Note that, during the proof of Theorem 3, we onlyrequire that Φ A ⊗ Φ B ( ρ AB ) = id A ⊗ Φ B ( ρ AB ) = Φ A ⊗ id B ( ρ AB ) = ρ AB . So, we can give the definition of theSM states in a less restricted form, as follows: Definition ′ . We call a quadripartite state ρ AE A BE B a strong Markov (SM) state if1. there exist CP maps Λ A , from A to AE A , and Λ B ,from B to BE B , such that ρ AE A BE B = Λ A ⊗ Λ B ( ρ AB ) ,and2. Φ A ⊗ Φ B ( ρ AB ) = id A ⊗ Φ B ( ρ AB ) = Φ A ⊗ id B ( ρ AB ) = ρ AB . For each quadripartite state ρ AE A BE B , which possessesthe two properties in Definition 3 ′ , Theorem 3 is valid.Now, using Eqs. (22), (23) and (24), it can be shownsimply that the property 2 of Definition 3 holds for thestates ρ AB as Eq. (22). Therefore, Definitions 3 and 3 ′ are equivalent.Let’s end this section with examining the second prop-erty of Definition 3, for a special interesting case. Con-sider a quadripartite SM state ρ AE A BE B . We, e.g., havefor the CP map Λ A Λ A ⊗ id B ( ρ AB ) = Λ A ⊗ Φ B ( ρ AB )= Tr E B ◦ [Λ A ⊗ Λ B ]( ρ AB ) = ρ AE A B . (25)So, id AE A ⊗ Λ B ( ρ AE A B ) = Λ A ⊗ Λ B ( ρ AB ) = ρ AE A BE B ;(26)i.e., according to tripartition ( AE A ; B ; E B ), ρ AE A BE B isa tripartite Markov state and can be written as Eq. (7).This, also, can be shown directly from Eq. (11). IV. THE MULTIPARTITE CASE
Now, we consider the case that the system is N -partite, S = S S . . . S N . Different parts of the system are sepa-rated from each other and each part S i interacts with itslocal environment E i . We denote the whole state of thesystem-environment as ρ SE = ρ S E ...S N E N . Definition 4.
We call a N -partite state ρ S E ...S N E N a weak Markov (WM) state if there exist CP maps Λ i ,from S i to S i E i , such that ρ S E ...S N E N = Λ ⊗ Λ ⊗ · · · ⊗ Λ N ( ρ S S ...S N ) , (27) where ρ S S ...S N = Tr E ... E N ( ρ S E ...S N E N ) . Therefore, for a WM state, each localized dynamics as F S E ⊗ · · · ⊗ F S N E N , for the whole system-environment,reduces to a localized subdynamics as E S ⊗ · · · ⊗ E S N ,for the system. So, we readily conclude that: Corollary 3.
If for a localized dynamics of the wholesystem-environment as F S E ⊗ · · · ⊗ F S N E N , the entan-glement of the system S = S S . . . S N increases, thenwe conclude that the initial state of the whole system-environment, ρ S E ...S N E N , is not a WM state as Eq.(27). If we define the CP map Φ i ≡ Tr E i ◦ Λ i on the subsys-tem S i , then, from Eq. (27), we haveΦ ⊗ Φ ⊗ · · · ⊗ Φ N ( ρ S S ...S N ) = ρ S S ...S N . (28)Now, as the previous section, we define a 2 N -partiteSM state as the following: Definition 5.
We call a N -partite state ρ S E ...S N E N a strong Markov (SM) state if1. Eq. (27) holds for it, and2. in Eq. (28), we can replace one or more Φ i with id S i . Theorem 4. A N -partite state ρ S E ...S N E N is astrong Markov (SM) state, if and only if, there exist de-compositions of the Hilbert spaces of the subsystems S i , H S i , as H S i = L j i H ( s i ) Lji ⊗ H ( s i ) Rji , such that ρ S E ...S N E N = M j ,...,j N λ j ...j N ρ ( s ) Lj ... ( s N ) LjN ⊗ ρ ( s ) Rj E ⊗ · · · ⊗ ρ ( s N ) RjN E N . (29) In Eq. (29), { λ j ...j N } is a probability distribution, ρ ( s ) Lj ... ( s N ) LjN is a state on H ( s ) Lj ⊗ · · · ⊗ H ( s N ) LjN and ρ ( s i ) Rji E i is a state on H ( s i ) Rji ⊗ H E i . ( H E i is the Hilbertspace of E i .)Proof. Showing that a 2 N -partite state as Eq. (29)is a SM state, as Definition 5, is not difficult. It can bedone by noting that, for a state as (29), we haveΛ i = M j i id ( s i ) Lji ⊗ Λ ( s i ) Rji , (30)where id ( s i ) Lji is the identity map on ( s i ) Lj i and Λ ( s i ) Rji isa CP map from ( s i ) Rj i to ( s i ) Rj i E i .So, we focus on proving the reverse: Each 2 N -partiteSM state, as Definition 5, can be decomposed as Eq. (29).From the property 2 of Definition 5, we know that,according to the bipartition S ; S . . . S N , we haveΦ ⊗ id S ...S N ( ρ S ; S ...S N ) = ρ S ; S ...S N . It is similar to Eq. (14). So, we conclude that thereexists a decomposition of the H S as H S = L j H ( s ) Lj ⊗H ( s ) Rj such that ρ S ...S N can be decomposed as ρ S ; S ...S N = M j q j ρ ( s ) Lj S ...S N ⊗ ρ ( s ) Rj , (31)where { q j } is a probability distribution, ρ ( s ) Lj S ...S N isa state on H ( s ) Lj ⊗ H S ...S N and ρ ( s ) Rj is a state on H ( s ) Rj .Similarly, according to the bipartition S ; S S . . . S N ,we have Φ ⊗ id S S ...S N ( ρ S ...S N ) = ρ S ...S N and so ρ S ...S N = M j q j ρ S ( s ) Lj S ...S N ⊗ ρ ( s ) Rj , H S = M j H ( s ) Lj ⊗ H ( s ) Rj . (32)By defining the projector Π j = Π ( s ) Lj ⊗ Π ( s ) Rj ⊗ I S ...S N , from Eqs. (31) and (32), we haveΠ j ρ S ...S N Π j = q j ρ ( s ) Lj S ...S N ⊗ ρ ( s ) Rj M j q j σ ( S ) j ( s ) Lj S ...S N ⊗ ρ ( s ) Rj , where σ ( S ) j ( s ) Lj S ...S N = ¯Π j ρ S ( s ) Lj S ...S N ¯Π j , with¯Π j = Π ( s ) Lj ⊗ Π ( s ) Rj ⊗ Π ( s ) Lj ⊗ I S ...S N . So, by asimilar line of reasoning, as obtained from Eqs. (20)-(22), we achieve ρ S ...S N = M j ,j λ j j ρ ( s ) Lj ( s ) Lj S ...S N ⊗ ρ ( s ) Rj ⊗ ρ ( s ) Rj . (33)By continuing this method, we finally get ρ S ...S N = M j ,...,j N λ j ...j N ρ ( s ) Lj ... ( s N ) LjN ⊗ ρ ( s ) Rj ⊗ · · · ⊗ ρ ( s N ) RjN , H S i = M j i H ( s i ) Lji ⊗ H ( s i ) Rji . (34)In Eq. (34), { λ j ...j N } is a probability distribution, ρ ( s ) Lj ... ( s N ) LjN is a state on H ( s ) Lj ⊗ · · · ⊗ H ( s N ) LjN and ρ ( s i ) Rji is a state on H ( s i ) Rji .Next, note that, during the proof of Theorem 3, fromEq. (14), we have concluded Eq. (23). Here also, fromΦ i ⊗ id S ...S i − S i +1 ...S N ( ρ S ...S N ) = ρ S ...S N , we conclude Eq. (30). So, from Eqs. (27), (30) and (34),we achieve Eq. (29), and the proof is completed. (cid:3) Remark 3.
Theorem 4 is valid for the case that H S i are finite dimensional, but H E i can be infinite dimen-sional. In other words, the system S = S . . . S N is finitedimensional but the environments E i can be infinite di-mensional. As stated in Corollary 3, for a WM state, each localizeddynamics as F S E ⊗ · · · ⊗ F S N E N reduces to a localizedsubdynamics as E S ⊗ · · · ⊗ E S N . Now, for an SM state,from Eqs. (30) and (34), it can be shown that if F S i E i = id S i E i , then E S i = id S i .Till now, we have considered the case that our M -partite Markov state includes even subsystems: M = 2 N .In the following, we consider the case that M = 2 N − N = 3 , , . . . . The case that N = 2, and so M = 3, hasbeen considered in Sect. II.Consider the case that the system is N -partite, S = S S . . . S N . The part S is isolated and the other parts S i , i = 1, each interacts with its local environment E i .We denote the whole state of the system-environment as ρ SE = ρ S S E ...S N E N . Definition 6.
We call a (2 N − -partite state ρ S S E ...S N E N a strong Markov (SM) state if 1. there exist CP maps Λ i , from S i to S i E i , i = 1 ,such that ρ S S E ...S N E N = id S ⊗ Λ ⊗ · · · ⊗ Λ N ( ρ S S ...S N ) , (35) where ρ S S ...S N = Tr E ... E N ( ρ S S E ...S N E N ) , and2 in the relation id S ⊗ Φ ⊗ · · · ⊗ Φ N ( ρ S S ...S N ) = ρ S S ...S N , (36) where Φ i ≡ Tr E i ◦ Λ i , i = 1 , is a CP map on S i , we canreplace one or more Φ i with id S i . In addition, we call a (2 N − ρ S S E ...S N E N a weak Markov (WM) state, if it only pos-sesses the property 1, in the above definition. Obviously,for a WM state, each localized dynamics as id S ⊗F S E ⊗· · · ⊗ F S N E N , for the whole system-environment, reducesto a localized subdynamics as id S ⊗ E S ⊗ · · · ⊗ E S N , forthe system. Therefore, a result, similar to Corollary 3,can be obtained for this case, too.In the following of this section, we give our final mainresult: The structure of the (2 N − Theorem 5. A (2 N − -partite state ρ S S E ...S N E N is a strong Markov (SM) state, if and only if, there existdecompositions of the Hilbert spaces of the subsystems S i , H S i , i = 1 , as H S i = L j i H ( s i ) Lji ⊗ H ( s i ) Rji , such that ρ S S E ...S N E N = M j ,...,j N λ j ...j N ρ S ( s ) Lj ... ( s N ) LjN ⊗ ρ ( s ) Rj E ⊗ · · · ⊗ ρ ( s N ) RjN E N . (37) In Eq. (37), { λ j ...j N } is a probability distribution, ρ S ( s ) Lj ... ( s N ) LjN is a state on H S ⊗H ( s ) Lj ⊗· · ·⊗H ( s N ) LjN and ρ ( s i ) Rji E i is a state on H ( s i ) Rji ⊗ H E i .Proof. As Theorem 4, proving that a state given in Eq.(37) is a SM state, as the Definition 6, is not difficult,since, here also, the CP maps Λ i are as Eq. (30).The proof of the reverse, i.e. each (2 N − ρ S ...S N = M j ,...,j N λ j ...j N ρ S ( s ) Lj ... ( s N ) LjN ⊗ ρ ( s ) Rj ⊗ · · · ⊗ ρ ( s N ) RjN , H S i = M j i H ( s i ) Lji ⊗ H ( s i ) Rji ( i = 1) . (38)In Eq. (38), { λ j ...j N } is a probability distribution, ρ S ( s ) Lj ... ( s N ) LjN is a state on H S ⊗H ( s ) Lj ⊗· · ·⊗H ( s N ) LjN and ρ ( s i ) Rji is a state on H ( s i ) Rji .Then, using Eqs. (30), (35) and (38), we get Eq. (37),and the proof is completed. (cid:3)
Remark 4.
Theorem 5 is valid for the case that H S i , i = 1 , are finite dimensional, but H S and H E i can beinfinite dimensional. Till now, we have defined the Markov states for all M -partite cases for which M = 3 , , . . . . The generalizationto the cases that M = 1 , ρ S a Markov state,since there is a CP map, i.e., id S , such that ρ S = id S ( ρ S ).In addition, we can call each bipartite state ρ SE aMarkov state, too. It is so since one can find a CPmap Λ, from S to SE , such that ρ SE = Λ( ρ S ), where ρ S = Tr E ( ρ SE ). For example, Λ can be constructedas Λ = ¯Λ ◦ Ξ. The CP map Ξ is defined as Ξ( ρ S ) =( I S ⊗ | E i ) ρ S ( I S ⊗ h E | ), where | E i is a fixed statein H E . The completely positive map ¯Λ, which maps ρ S ⊗ | E ih E | to the ρ SE , can be found, e.g., using themethod introduced in Ref. [16]. V. EXAMPLE: THE CLASSICALENVIRONMENT
We end our paper with an example of the simplest case,i.e., the case studied in Sect. II. Some other examplesare also given in Ref. [17].Consider the case that the system S is bipartite, S = AB . The part A is isolated from the environment andonly the part B interacts with the environment E . Inaddition, assume that the effect of E on B can be mod-eled as acting random unitary operators U ( j ) B on B , eachwith the probability p j . Therefore, the whole dynamicsof the system can be written as ρ AB ( t ) = X j p j (cid:16) I A ⊗ U ( j ) B ( t ) (cid:17) ρ AB (0) (cid:16) I A ⊗ U ( j ) † B ( t ) (cid:17) , (39)where I A is the identity operator on A , ρ AB (0) is theinitial state of the system and ρ AB ( t ) is the state of thesystem at time t . In Eq. (39), U ( j ) B ( t ) = U ( j ) B ( t,
0) is aunitary time evolution, acting on B with the probabil-ity p j , from the initial moment to the time t . Note that U ( j ) B ( t ,
0) = U ( j ) B ( t , t ) U ( j ) B ( t , U ( j ) B ( t ) = e − iH j t/ ~ , with a time-independentHamiltonian H j .In Refs. [9, 12–14], some quantum systems, for whichthe time evolution is given by Eq. (39), are studied.An important example is when the subsystem B is cou-pled to a random external field and the subsystem A isisolated from this classical external field [12, 13]. Thecharacteristics of the classical external field are not af-fected by interaction with the B and so its state remainsunchanged during the evolution.We can model the whole system-environment evolu-tion as the following [11]. We get the initial state of thesystem-environment as ρ SE (0) = ρ AB (0) ⊗ X j p j | j E ih j E | , (40) where {| j E i} is an orthonormal basis for E . In addition,the system-environment undergoes the evolution given bythe unitary operator U SE ( t ) = X j I A ⊗ U ( j ) B ( t ) ⊗ | j E ih j E | . (41)From Eqs. (40) and (41), it can be shown simply thatthe reduced dynamics of the system S = AB is givenby Eq. (39). In addition, the reduced state of the en-vironment remains unchanged during the evolution. Wehave ρ E ( t ) = P j p j | j E ih j E | = ρ E (0), which is a classicalstate, i.e., it contains no superposition of the basis states | j E i .If the initial state of the system, ρ AB (0), be an en-tangled state, since the environment is classical, we mayexpect that, during the time evolution of the system,entanglement decreases monotonically. But, unexpect-edly, it has been shown, both theoretically and exper-imentally, that for a system which undergoes the timeevolution given by Eq. (39), entanglement revivals canoccur [9, 10, 12–14].Note that, ρ ABE (0) in Eq. (40) is a Markov state; thatis, it can be written in the form of Eq. (7). It is, in fact, afactorized state which is due to the case that H B = H b L ⊗H b R and H b R is a trivial one-dimensional Hilbert space.In addition, the dynamics of the system-environment inEq. (41) is localized as Eq. (2). Therefore, the reduceddynamics of the system in Eq. (39) is also localized as Eq.(4). So, M ( ρ AB ( t )) ≤ M ( ρ AB (0)), for all t >
0. This isin agreement with the results of Refs. [9, 10, 12–14].From Eq. (41), we see that the time evolution operatorof the system-environment, from t to t ( t < t ), is as U ABE ( t , t ) = I A ⊗ X j U ( j ) B ( t , t ) ⊗ | j E ih j E | , (42)which is in the form of Eq. (2). Therefore, if, at time t = t , entanglement starts to increase, it indicates that ρ ABE ( t ) is not a Markov state. Note that the state ofthis hybrid quantum-classical system SE changes fromits initial factorized state ρ ABE (0) in Eq. (40) to the state ρ ABE ( t ), which cannot be written as Eq. (7). So, al-though the reduced state of E remains unchanged duringthe evolution, the whole state of the system-environmentchanges from its initial factorized one to a state whichis not a Markov state and this change can lead to theentanglement revival.As we see in the following, during the evolution, thoughthe whole state of the system-environment changes fromits initial Markov state to a non-Markovian state, butthe correlation between the system S = AB and the en-vironment E remains classical. This implies that thenon-Markovianity of the ρ SE is not equivalent to exis-tence of non-classical correlation between S and E .If we define the one-dimensional projectors Π ( j ) E = | j E ih j E | , then, from Eq. (40), it can be seen that ρ SE (0) does not change under local projective measure-ment { I S ⊗ Π ( j ) E } ; that is, if we perform the measurement { I S ⊗ Π ( j ) E } on ρ SE (0) and then mix the results of dif-ferent outcomes, we achieve the pre-measurement state ρ SE (0). This can be interpreted as the existence of noquantum correlation between S and E [18].The above argument is also true for ρ SE ( t ). From Eqs.(40) and (41), we have ρ SE ( t ) = X j p j ρ ( j ) AB ( t ) ⊗ | j E ih j E | , (43)where ρ ( j ) AB ( t ) = I A ⊗ U ( j ) B ρ AB (0) I A ⊗ U ( j ) † B . (44)So, ρ SE ( t ) is also unchanged under the measurement { I S ⊗ Π ( j ) E } and the correlation between the system S = AB and the environment E remains classical duringthe evolution.Note that, in Eq. (43), each ρ ( j ) AB ( t ) is coupled to afixed unchanged state of the environment | j E ih j E | . Asexpected, there is no correlation between the ρ ( j ) AB ( t )and | j E ih j E | , since the environment is classical and un-changed during the evolution. So, any classical cor-relation in Eq. (43) is due to the mixing different ρ ( j ) AB ( t ) ⊗ | j E ih j E | , each with the probability p j .In fact, we are encountered with an ensemble of thestates as { p j , ρ ( j ) AB ( t ) ⊗ | j E ih j E |} . Therefore, it can beargued [14] that the real amount of entanglement presentbetween A and B is X j p j M ( ρ ( j ) AB ( t ))= X j p j M ( ρ AB (0)) = M ( ρ AB (0)) , (45)where we have used this fact that under local operation,in Eq. (44), entanglement between A and B does notchange. The only reason which prevent us to achieve allof this amount is the mixing in Eq. (43). So, one candefine the hidden entanglement as [14]: M H ( t ) = X j p j M ( ρ ( j ) AB ( t )) − M ( ρ AB ( t ))= M ( ρ AB (0)) − M ( ρ AB ( t )) , (46) which gives the amount of entanglement, though presentbetween A and B , is hidden (inaccessible) for us (see alsoRef. [17]). VI. SUMMARY
Markov states has been defined for the tripartite case[2]. In this paper, we have generalized the definition ofthe Markov state to arbitrary M -partite case.We have given two forms of definitions: weak Markov (WM) states and strong Markov (SM) states. The setof SM states is a subset of the set of WM states. For M ≤
3, the two sets are the same. For
M >
3, though itseems that the set of SM states is a proper subset of theset of WM states, a careful treatment is needed to provewhether these two sets are the same or not.For WM states, we have seen that each localized dy-namics for the whole system-environment reduces to alocalized subdynamics of the system. This provides usa necessary (but, in general, insufficient) condition, forentanglement increase: Entanglement revival can occuronly when the initial state of the system-environmentstate is not a WM state.Our main results, in this paper, are for SM states. Wehave found the general structure of the SM states, forarbitrary M -partite case, in Theorems 3, 4 and 5.If the initial state of the whole system-environment ρ S E ...S N E N is a SM sate, then, since each SM state is,in addition, a WM state, each localized dynamics for thesystem-environment as F S E ⊗ · · · ⊗ F S N E N reduces to alocalized subdynamics as E S ⊗ · · · ⊗ E S N for the system.Also, if F S i E i = id S i E i , then E S i = id S i .According to the two above interesting properties, itseems that the SM states can play an important role instudying open quantum systems.We have ended our paper by studying an example ofthe simplest case, i.e., the tripartite case ρ ABE . We haveconsidered the case that though the environment E isclassical, entanglement revival can occur in the system S = AB . Entanglement revival can occur only whenthe whole state of the system-environment changes fromits initial Markov state to a non-Markovian state. But,during this change, the correlation between the system S and the environment E remains classical. This impliesthat the non-Markovianity of a state is not equivalent toexistence of non-classical correlation between the systemand the environment. [1] M. A. Nielsen and I. L. Chuang, Quantum Computationand Quantum Information (Cambridge University Press,Cambridge, 2000).[2] P. Hayden, R. Jozsa, D. Petz and A. Winter, Structureof states which satisfy strong subadditivity of quantumentropy with equality, Commun. Math. Phys. , 359(2004). [3] F. Buscemi, Complete positivity, Markovianity, and thequantum data-processing inequality, in the presence ofinitial system-environment correlations, Phys. Rev. Lett. , 140502 (2014).[4] I. Sargolzahi and S. Y. Mirafzali, Structure of states forwhich each localized dynamics reduces to a localized sub-dynamics, Int. J. Quantum Inf. , 1750043 (2017). [5] L. Aolita, F. de Melo and L. Davidovich, Open-systemdynamics of entanglement: a key issues review, Rep.Prog. Phys. , 042001 (2015).[6] T. Yu and J. H. Eberly, Sudden death of entanglement,Science , 598 (2009).[7] B. Bellomo, R. Lo Franco and G. Compagno, Non-Markovian effects on the dynamics of entanglement,Phys. Rev. Lett. , 160502 (2007).[8] B. Bellomo, R. Lo Franco and G. Compagno, Entan-glement dynamics of two independent qubits in envi-ronments with and without memory, Phys. Rev. A ,032342 (2008).[9] A. Orieux, A. D’Arrigo, G. Ferranti, R. Lo Franco, G.Benenti, E. Paladino, G. Falci, F. Sciarrino and P. Mat-aloni, Experimental on-demand recovery of entanglementby local operations within non-Markovian dynamics, Sci.Rep. , 8575 (2015).[10] R. Lo Franco and G. Compagno, Overview on the phe-nomenon of two-qubit entanglement revivals in classicalenvironments, arXiv:1608.05970 (2016).[11] R. Lo Franco, B. Bellomo, E. Andersson and G. Com-pagno, Revival of quantum correlations without system-environment back-action, Phys. Rev. A , 032318(2012).[12] J.-S. Xu, K. Sun, C.-F. Li, X.-Y. Xu, G.-C. Guo, E. An- dersson, R. Lo Franco and G. Compagno, Experimentalrecovery of quantum correlations in absence of system-environment back-action, Nat. Commun. , 2851 (2013).[13] B. Leggio, R. Lo Franco, D. O. Soares-Pinto, P.Horodecki and G. Compagno, Distributed correlationsand information flows within a hybrid multipartitequantum-classical system, Phys. Rev. A , 032311(2015).[14] A. D’Arrigo, R. Lo Franco, G. Benenti, E. Paladino andG. Falci, Recovering entanglement by local operations,Ann. Phys. , 211 (2014).[15] M. Koashi and N. Imoto, Operations that do not disturbpartially known quantum states, Phys. Rev. A , 022318(2002).[16] D. M. Tong, L. C. Kwek, C. H. Oh, J.-L. Chen and L. Ma,Operator-sum representation of time-dependent densityoperators and its applications, Phys. Rev. A , 054102(2004).[17] I. Sargolzahi and S. Y. Mirafzali, Entanglement increasefrom local interaction in the absence of initial quantumcorrelation in the environment and between the systemand the environment, Phys. Rev. A , 022331 (2018).[18] H. Ollivier and W. H. Zurek, Quantum discord: a mea-sure of the quantumness of correlations, Phys. Rev. Lett.88