Operational Resource Theory of NonMarkovianity
Abstract
In order to analyze nonMarkovianity of tripartite quantum states from a resource theoretical viewpoint, we introduce a class of quantum operations performed by three distant parties, and investigate an operational resource theory (ORT) induced it. A tripartite state is a free state if and only if it is a quantum Markov chain. We prove monotonicity of functions such as the conditional quantum mutual information, intrinsic information, squashed entanglement, a generalization of entanglement of purification, and the relative entropy of recovery. The ORT has five bound sets, each of which corresponds to one of the monotone functions. We introduce a task of “nonMarkovianity dilution”, and prove that the optimal rate for the task, namely the “nonMarkovianity cost”, is bounded from above by entanglement of purification in the case of pure states. We also propose a classical resource theory of nonMarkovianity.
I Introduction
Markovianity of a physical system composed of three subsystems is a property that the first and the second subsystems are statistically independent when conditioned by the third one. In classical information theory, the system is modeled by random variables , and , which take values in finite sets , and , respectively, according to a joint probability distribution . The three systems are referred to as being a Markov chain in the order of if the probability is decomposed as , or equivalently, if the conditional mutual information is equal to zero. In analogy, a quantum Markov chain is defined as a tripartite quantum states for which the conditional quantum mutual information is zero [1]. Classical probability distributions and quantum states for which the conditional mutual information is not equal to zero have the property of nonMarkovianity.
NonMarkov classical and quantum states can be used as a resource for several information processing tasks that are impossible without using such states: Ref. [2, 3, 4] proved that classical random variables that obey nonMarkov distribution may be used in secret key agreement protocols, and Ref. [5] proved that nonMarkov quantum states are used as a resource for a task called the conditional quantum onetime pad. This situation resembles the one that entangled quantum states shared between distant parties can be used as a resource for tasks that cannot be accomplished without using entanglement (see e.g. [6, 7]). Hence, it would be natural to apply concepts and tools, developed in entanglement theory[6, 7], to an analysis of nonMarkovianity of classical and quantum states.
In this paper, we apply the concept of operational resource theory (ORT)[8, 9, 10, 11], which originates in entanglement theory, for analyzing nonMarkovianity of tripartite quantum states from an operational point of view. We consider a scenario in which three distant parties Alice, Bob and Eve perform operations on their systems by communicating classical messages, quantum messages and applying local operations. We restrict the class of free operations to be one consisting of (i) local operations by Alice and Bob, (ii) local reversible operations by Eve, (iii) broadcasting of classical messages by Alice and Bob, (iv) quantum communication from Alice and Bob to Eve, and their compositions. We analyze an ORT induced by . It turns out that a tripartite quantum state is a free state if and only if it is a quantum Markov chain. Thus the obtained ORT is regarded as an operational resource theory of nonMarkovianity.
One of the principal goals of an ORT in general is to identify 1) necessary and sufficient condition for one state to be convertible to another by free operations (singleshot convertibility), and 2) the optimal ratio of number of copies at which one state is asymptotically convertible to another by free operations (asymptotic convertibility). These are, however, often a highly complex problem as in the case for an ORT of multipartite entanglement (see e.g. [6, 12, 13, 7, 14]). In such cases, a key milestone would be to identify 3) subsets of states that are closed under free operations and tensor product (bound sets), and 4) realvalued functions of states that are monotonically nonincreasing under free operations (monotones).
In this paper, we mainly address 3) and 4) above for an ORT induced by . We also address 1) by considering particular examples of qubit systems, and 2) by introducing and analyzing a task of nonMarkovianity dilution, in which copies of a unit state is transformed to copies of another state by a free operation. We note that our approach is different from that of [15], which addresses quantification of nonMarkovianity of processes.
This paper is organized as follows. Section II provides the setting of the problem.
In Section III, we prove that the conditional quantum mutual information is monotonically nonincreasing under free operations, and that a state is a free state if and only if it is a quantum Markov chain.
In Section IV, we identify functions that are monotonically nonincreasing under free operations, including a quantum analog of intrinsic information[2, 3, 4], a generalization of entanglement of purification[16], the squashed entanglement[17] and the relative entropy of recovery[18]. In Section V, we find five bound sets, each of which have a clear correspondence to the monotone functions mentioned above, and analyze inclusion relations among them. In Section VI, we prove that the monotone functions and the bound sets are connected with each other by a type of “duality”, which is analogous to the duality of CQMI for fourpartite pure states[19]. Examples of singleshot convertibility are provided in Section VII. In Section VIII, we introduce a task of nonMarkovianity dilution, and prove that the nonMarkovianity cost, namely the optimal achievable rate in nonMarkovianity dilution, is bounded from above by the entanglement of purification for the case of pure states. In Section IX, we consider a classical resource theory of nonMarkovianity. Conclusions are given in Section X. Some of the proofs of the main results are provided in appendices.
Notations. A system composed of two subsystems and is denoted by . We abbreviate as . We denote as . When is a quantum operation on , we denote as or . For , represents . We denote simply as . The von Neumann entropy of a state is interchangeably denoted by and . represents the base logarithm of .
Ii Settings
Suppose three distant parties Alice, Bob and Eve have quantum systems , and , respectively. A quantum state on system is specified by finitedimensional Hilbert spaces , , and a density operator on . We denote the set of all quantum states on by . An operation on system is specified by a linear CPTP map from to , where and are Hilbert spaces corresponding to the input and the output of , respectively.
Consider the following classes of operations performed by Alice, Bob and Eve (see Figure 1):
: local quantum operations by Alice  
: local quantum operations by Bob  
: local reversible quantum operation by Eve  
: broadcasting of classical messages by Alice  
: broadcasting of classical messages by Bob  
: quantum communication from Alice to Eve  
: quantum communication from Bob to Eve 
By “reversible”, we mean that for any operation , there exists an operation by Eve such that is the identity operation on , with denoting the Hilbert space corresponding to the input of . We require that Eve cannot refuse to receive anything that is sent to her in the above operations, i.e., quantum communication from Alice in , one from Bob in , classical messages from Alice in and one from Bob in . This condition, together with the reversibility of Eve’s operation, imposes a strong restriction on what the three parties can perform by the above classes of operations. We denote by the set of operations that can be represented as a composition of operations belonging to the above classes.
In this paper, we consider a scenario in which Alice, Bob and Eve are only allowed to perform operations in . We analyze conditions under which a state on is convertible to another by an operation in . That is, we consider an operational resource theory induced by . Due to the condition of reversibility of Eve’s operations, it is too restrictive to define convertibility of a state to by the existence of an operation such that . Thus we relax the definition of state convertibility as follows.
Definition 1
A state is convertible to under if there exists an operation and a reversible operation on such that .
It should be noted that, if we ignore Eve’s system, and are regarded as subclasses of and , respectively, and and as equivalent to classical communication between Alice and Bob. Hence any operation in can be identified with local operations and classical communication (LOCC) between Alice and Bob, if we ignore Eve.
Iii Basic Properties
In this section, we analyze basic properties of an operational resource theory (ORT) induced by . We prove that the degree of nonMarkovianity of quantum states, measured by the conditional quantum mutual information, is monotonically nonincreasing under . We also prove that a state is a free state if and only if it is a quantum Markov chain. Thus an ORT proposed in this paper is regarded as that of nonMarkovianity. The “maximally nonMarkovian state”, from which any other state of a given dimension is generated by an operation in , is identified.
Iiia Monotonicity of Conditional Quantum Mutual Information
The conditional quantum mutual information (CQMI) of a tripartite quantum state on system is defined by
where is the von Neumann entropy. The CQMI is nonnegative due to the strong subadditivity of the von Neumann entropy[20], and has operational meanings in terms e.g. of quantum state redistribution[21, 22, 23], deconstruction protocols[24] and the conditional quantum onetime pad[5]. For simplicity, we denote by . The following lemma states that is monotonically nonincreasing under .
Lemma 2
For any and , we have .
Proof:
It suffices to prove that is monotonically nonincreasing under any class of operations that comprises .

Monotonicity under and : For any , and , we have
due to the data processing inequality for the conditional quantum mutual information.

Monotonicity under : For any , we have
due to the data processing inequality. Hence we obtain
In the same way, we also have
Thus we have
which implies the monotonicity (actually the invariance) of under .

Monotonicity under and : The state before broadcasting of classical message by Alice is represented by a density operator
(1) where is a register possessed by Alice, is a probability distribution, is a set of orthonormal pure states, and is a density operator for each . In the same way, the state after broadcasting is represented by
(2) where and are registers possessed by Bob and Eve, respectively. We have
which implies monotonicity under . The monotonicity under follows along the same line.

Monotonicity under and : Let be a quantum system that is transmitted from Alice to Eve. For any quantum state on , we have
(3) which implies monotonicity under . The monotonicity under follows along the same line.
IiiB Quantum Markov Chains are Free States
A quantum Markov chain is defined as a tripartite quantum state for which the conditional quantum mutual information (CQMI) is zero[1]. For a tripartite system composed of systems , and , the condition is represented as
(4) 
We denote the set of quantum Markov chains satisfying (4) by . Ref. [1] proved that Equality (4) is equivalent to the condition that there exists a quantum operation satisfying
as well as to the condition that there exists a linear isometry from to such that
(5) 
Here, is a probability distribution, is an orthonormal basis of , and and are quantum states on composite systems and , respectively, for each .
In an ORT, a state is called a free state if it can be generated from scratch by a free operation. That is, a state is called a free state under if, for any , there exists such that . Due to the following proposition, an ORT induced by is regarded as that of of nonMarkovianity.
Proposition 3
A state is a free state under if and only if .
Proof:
To prove the “if” part, consider the following procedure: 1. Alice generates a random variable which takes values in according to a probability distribution , 2. Alice broadcasts to Bob and Eve, 3. Eve records on her register, 3. Alice locally prepares a state and sends to Eve, 4. Bob locally prepares a state and sends to Eve, and 5. Alice and Bob discards . It is straightforward to verify that any state in the form of (5) can be generated by this protocol.
To prove the “only if” part, suppose that is a free state. By definition, for any there exists an operation such that . From Lemma 2, it follows that , which yields and thus .
In an ORT, it would be natural to require that the set of free states is closed under tensor product [8, 9, 10]. The following lemma states that this condition is satisfied in an ORT induced by .
Lemma 4
For any , we have .
Proof:
It is straightforward to verify that for any on and on we have
which implies .
IiiC Maximally nonMarkovian state
Let be a dimensional maximally entangled state defined by
and consider a state defined by
Suppose that and are dimensional quantum systems. It is straightforward to verify that any state is created from by the following operation, which is an element of : 1. Alice locally prepares a state , 2. Alice sends system to Eve, and 3. Alice teleports system to Bob by using as a resource. Therefore, is regarded as a dimensional “maximally nonMarkovian state” on . Note that the classical message transmitted by Alice in Step 3 is decoupled from the state obtained at the end of the protocol.
IiiD Nonconvexity
An ORT is said to be convex if the set of free states is convex, or equivalently, if any probabilistic mixture of free states is also a free state. Most of the ORTs proposed so far satisfies convexity (see [9, 10, 11] and the references therein), except that of nonGaussianity[25]. We show that an ORT induced by is another example of nonconvex ORTs.
Consider states defined by
respectively. These states apparently satisfy . On the contrary, any probabilistic mixture of the two states, which takes the form of
does not belong to because
Hence is not a convex set.
Iv Monotones
In this section, we introduce several functions of tripartite quantum states that are monotonically nonincreasing under . We will refer to these functions as nonMarkovianity monotones, analogously to entanglement monotones for multipartite quantum states [26, 6]. Proofs of the monotonicity are provided in Appendix B.
Iva CQMIbased Monotones
We introduce three nonMarkovianity monotones based on the conditional quantum mutual information. The first one is a quantum mechanical generalization of intrinsic information [2, 3, 4, 17], defined as
where infimum is taken over all operations on . The socond one is defined as
where the infimum is taken over all extensions of on , i.e., over all state satisfying
Entanglement of bipartite reduced state is quantified by an entanglement measure for bipartite quantum states. The monotonicity of under simply follows from the fact that any operation in can be identified with LOCC between Alice and Bob, if we ignore Eve. We may choose for an arbitrary entanglement monotone [26, 6] for bipartite quantum states. In this paper, we adopt the squashed entanglement [17, 27] defined by
for a bipartite state on , where the infimum is taken over all extensions of . Note that if and only if is a separable state. In the following, we will use the notation
for .
IvB Generalized Entanglement of Purification
Consider a state and suppose we split system to a composite system by applying a linear isometry from to . The entanglement of state between and is quantified by the squashed entanglement. Let be an arbitrary quantum system, and be a quantum state on satisfying . Noting that , the squashed entanglement of between and is given by
where the infimum is taken over all extensions of . By taking the infimum over all splitting of , we define
This quantity could be regarded as a generalization of the entanglement of purification introduced in [16].
Let and be arbitrary quantum systems, and let be a state on that is a purification of , i.e., . The squashed entanglement of state between and is given by
where the infimum is taken over all operations on . Taking the infimum over all purifications of , we define
Due to the data processing inequality for the CQMI, we have
(6) 
as well as
(7) 
IvC Relative Entropy of Recovery
Let be the set of operations that can be represented as a composition of operations belonging to , , , , and (but not to ). Instead of , we may consider an ORT induced by . It follows from the proof of Proposition 3 (see Section IIIB) that a state is a free state under if and only if .
The relative entropy of recovery [18] (see Remark 6 therein) is defined as
where the infimum is taken over all quantum operations from to . The regularized version of the above function has an operational meaning in the context of quantum hypothesis testing[28]. We prove in Appendix BF that is monotonically nonincreasing under . It is left open whether is a monotone under , or equivalently, whether it is monotonically nonincreasing under .
V Bound sets
In an operational resource theory (ORT), a set of states that is closed under free operations and tensor product is called a bound set. That is, a set is a bound set under if, for any and , we have and . In this section, we will show that an ORT induced by has five bound sets in addition to . Based on the equivalence between separability of bipartite states and Markovianity of its extension, we analyze inclusion relations among those bound sets.
Va Definitions
One of the five bound sets is the set of states that can be transformed to a quantum Markov chain by an (possibly irreversible) operation by Eve, i.e.,
(8) 
where we denote by the sets of quantum operations on . Let be an ancillary quantum system, and denote by the set of quantum Markov chains on . Another bound set is the set of states on that has an extension on such that the reduced state is a quantum Markov chain, namely,
Since any operation in is regarded as a LOCC between Alice and Bob by ignoring Eve, the set of states on that are separable between and when we trace out is also a bound set:
By definition, it is straightforward to verify that the two bound sets defined above have a clear correspondence to the nonMarkovianity monotones introduced in Section IV. Namely, we have
(9)  
(10)  
(11) 
Suppose we split system to a composite system by applying a linear isometry from to . Depending on the state , we may choose so that the state after splitting, namely , is a separable state between and . The set of such states is proved to be a bound set:
Here, we denoted by the set of separable states between and . Let and be arbitrary quantum systems, and let be a state on that is a purification of , i.e., . By properly choosing , the reduced state on may be a separable state between and . We define
We will prove in the next subsection that similar relations as (9)(11) hold between , and , , respectively (see (18) and (19)).
A proof that the sets , , , and are bound sets is separately provided in Appendix B. We will prove in Section VC that the bound sets introduced above are connected with each other by the following inclusion relations (see Figure 2):
(12)  
(13)  
(14)  
(15) 
It is left open whether relations (15) are strict.
VB Equivalence of Separability and Markovianity
The condition that a bipartite quantum state is separable is equivalent to the condition that there exists a quantum Markov chain that is an extension of the state, and to the condition that its purification is mapped to a quantum Markov chain by a local operation on the purifying system, as stated in the following lemma. We will use this equivalence for analyzing relations among bound sets and monotones.
Lemma 5
The following three conditions are equivalent:

is separable.

There exists an extension of satisfying .

For any purification of , there exists an operation on such that .
Proof:
1)2) was proved in [27] (see the proof of Theorem 7 therein), and 2)1) was proved in [29] (see Equality (6) therein). Thus it suffices to prove 1)3) and 3)2).
1)3): Suppose that is separable. By definition, there exists states , and a probability distribution such that
Thus a purification of is given by
Due to Uhlmann’s theorem[30], for any purification of , there exists a linear isometry on such that . Let be the dephasing operation on with respect to the basis , and define . Then we have
which completes the proof of 1)3).
3)2): Suppose that, for a purification of , there exists an operation on such that . It is straightforward to verify that is an extension of . Hence 3) implies 2).
VC Proof of Inclusion Relations
In the following, we prove Relations (12), (13) and (15), except the strictness of their relations. Relation (14) and the strictness of (12), (13) will be proved in Section VII by construction.
Relation (12)
Note that any state is decomposed by a linear isometry from to in the form of (5). Denoting and by and , respectively, it follows that the state (5) is separable between and , which implies . Let , and be quantum systsmes, and and be purifications of and , respectively, for each . A purification of is given by
from which, by tracing out , we obtain
Denoting and by and , respectively, the above state is separable between and , which leads to .
Relation (13)
Relation (15)
Suppose . There exists a purification of such that . Due to Lemma 5, there exists an operation such that . Noting that
we obtain
by the data processing inequality, which implies and thus . Hence , which implies .
Suppose . There exists a linear isometry from to such that . Due to Lemma 5, there exists an extension of . It is straightforward to verify that is an extension of . In addition, due to we have