To quantify non-Markovianity of tripartite quantum states from an operational viewpoint, we introduce a class Ω° of operations performed by three distant parties. A tripartite quantum state is a free state under Ω° if and only if it is a quantum Markov chain. We introduce a function of tripartite quantum states that we call the non-Markovianity of formation, and prove that it is a faithful measure of non-Markovianity, which is continuous and monotonically nonincreasing under a subclass Ω of Ω°. We consider a task in which the three parties generate a non-Markov state from scratch by operations in Ω, assisted with quantum communication from the third party to the others, which does not belong to Ω. We prove that the minimum cost of quantum communication required therein is asymptotically equal to the regularized non-Markovianity of formation. Based on this result, we provide a direct operational meaning to a measure of bipartite entanglement called the c-squashed entanglement.
To quantify non-Markovianity of tripartite quantum states from an operational viewpoint, we introduce a class Ω° of operations performed by three distant parties. A tripartite quantum state is a free state under Ω° if and only if it is a quantum Markov chain. We introduce a function of tripartite quantum states that we call the non-Markovianity of formation, and prove that it is a faithful measure of non-Markovianity, which is continuous and monotonically nonincreasing under a subclass Ω of Ω°. We consider a task in which the three parties generate a non-Markov state from scratch by operations in Ω, assisted with quantum communication from the third party to the others, which does not belong to Ω. We prove that the minimum cost of quantum communication required therein is asymptotically equal to the regularized non-Markovianity of formation. Based on this result, we provide a direct operational meaning to a measure of bipartite entanglement called the c-squashed entanglement.
In order to analyze non-Markovianity 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 "non-Markovianity dilution", and prove that the optimal rate for the task, namely the "non-Markovianity cost", is bounded from above by entanglement of purification in the case of pure states. We also propose a classical resource theory of non-Markovianity.
Quantum non-Markovianity in tripartite quantum states ρABC represents a correlation between systems A and C when conditioned on the system B and is known to have both classical and quantum contributions. However, a systematic characterization of the latter is missing. To address this, we propose a faithful measure for non-Markovianity of genuine quantum origin called squashed quantum non-Markovianity (sQNM). It is based on the quantum conditional mutual information and is defined by the left-over non-Markovianity after squashing out all non-quantum contributions. It is lower bounded by the squashed entanglement between non-conditioning systems in the reduced state and is delimited by the extendibility of either of the non-conditioning systems. We show that the sQNM is monogamous, asymptotically continuous, convex, additive on tensor-product states, and generally super-additive. We characterize genuine quantum non-Markovianity as a resource via a convex resource theory after identifying free states with vanishing sQNM and free operations that do not increase sQNM in states. We use our resource-theoretic framework to bound the rate of state transformations under free operations and to study state transformation under non-free operations; in particular, we find the quantum communication cost from Bob ( B ) to Alice ( A ) or Charlie ( C ) is lower bounded by the change in sQNM in the states. The sQNM finds operational meaning; in particular, the optimal rate of private communication in a variant of conditional one-time pad protocol is twice the sQNM. Also, the minimum deconstruction cost for a variant of quantum deconstruction protocol is given twice the sQNM of the state.
We introduce and analyze a task that we call Markovianization, in which a tripartite quantum state is transformed to a quantum Markov chain by a randomizing operation on one of the three subsystems. We consider cases where the initial state is the tensor product of n copies of a tripartite state ρ ABC, and is transformed to a quantum Markov chain conditioned by Bn with a small error, using a random unitary operation on An. In an asymptotic limit of infinite copies and vanishingly small error, we analyze the Markovianizing cost, that is, the minimum cost of randomness per copy required for Markovianization. For tripartite pure states, we derive a singleletter formula for the Markovianizing costs. Counterintuitively, the Markovianizing cost is not a continuous function of states, and can be arbitrarily large even if the state is close to a quantum Markov chain. Our results have an application in analyzing the cost of resources for simulating a bipartite unitary gate by local operations and classical communication.
We introduce and analyze a task that we call Markovianization, in which a tripartite quantum state is transformed to a quantum Markov chain by a randomizing operation on one of the three subsystems. We consider cases where the initial state is a tensor product of n copies of a tripartite state ρABC, and is transformed to a quantum Markov chain conditioned by Bnwith a small error, by a random unitary operation on An. In an asymptotic limit of infinite copies and vanishingly small error, we analyze the Markovianizing cost, that is, the minimum cost of randomness per copy required for Markovianization. For tripartite pure states, we derive a single-letter formula for the Markovianizing costs. Counterintuitively, the Markovianizing cost is not a continuous function of states, and can be arbitrarily large even if the state is an approximate quantum Markov chain. Our results have an application for distributed quantum computation.
Distributed quantum computation is a task in which many distant parties perform a large quantum computation in collaboration with each other, by using classical communication, quantum communication or shared entanglement as resources.In this thesis, we consider one of the simplest nontrivial task of distributed quantum computation, that is, implementation of bipartite unitary by LOCC (local operations and classical communication) assisted by shared entanglement.Here, two input states are distributed over two distant parties.Contrary to previous researches, we consider an asymptotic scenario in which the two parties perform the same bipartite unitary on infinitely many independent input pairs.We analyze the minimum amount of resources required for the task by exploiting concepts and techniques developed in the field of quantum Shannon theory, which has originally been developed to analyze quantum communication.As a tool for analyzing distributed quantum computation, we first introduce a task that we call Markovianization.Markovianization is a task in which a tripartite pure state is transformed to a state called quantum Markov chain, by a randomizing operation on one of the three subsystems.We derive the minimum cost of randomness required for Markovianization (Markovianizing cost) by exploiting a novel decomposition of a Hilbert space introduced in the context of quantum cryptography.We then apply the obtained result to distributed quantum computation.We mainly consider implementation of bipartite unitaries by entanglement-assisted LOCC protocols consisting of three steps, called one-round protocols.This is the first nontrivial case because it is not possible to implement bipartite unitaries by protocols consisting of only one or two steps.Our main result is that we derive the minimum costs of entanglement and classical communication.We show that the minimum costs are given by the Markovianizing cost of a tripartite state associated with the unitary.The result indicates that information about input states of computation is divided into three components: one of which only information of amplitude in a basis is required for computation (information of phase is not), another of which both amplitude and phase information are required, and the other of which no information is involved in computation.Our construction of a protocol that requires the smallest amount of resources is based on this decomposition.We also propose a method to compute the Markovianizing cost of a state associated with the unitary.We also consider two-round protocols, that is, protocols consisting of concatenation of two one-round protocols, for two-qubit controlled unitaries.We show that, for a particular class of two-qubit controlled unitaries, the minimum entanglement cost in two-round protocols is strictly smaller than that in one-round protocols.This result is the first reported case that indicates a property of a trade-off relation between entanglement requirement and the number of turns in entanglement-assisted LOCC tasks.The research in this thesis opens a possibility to build a new bridge between quantum computation theory and quantum communication theory, which are the two largest subfield of quantum information theory, by analyzing a problem in the former using concepts and techniques developed in the latter.
The quantification and characterization of non-Markovian dynamics in quantum systems is an essential endeavor both for the theory of open quantum systems and for a deeper understanding of the effects of non-Markovian noise on quantum technologies. Here, we introduce the robustness of non-Markovianity, an operationally-motivated, optimization-free measure that quantifies the minimum amount of Markovian noise that can be mixed with a non-Markovian evolution before it becomes Markovian. We show that this quantity is a bonafide non-Markovianity measure, since it is faithful, convex, and monotonic under composition with Markovian channels. A two-fold operational interpretation of this measure is provided, with the robustness measure quantifying an advantage in both a state discrimination and a channel discrimination task. Furthermore, we provide a closed-form analytical expression for this measure and show that, quite remarkably, the robustness measure is exactly equal to half the Rivas-Huelga-Plenio (RHP) measure [Phys. Rev. Lett. \textbf{105}, 050403 (2010)]. As a result, we provide a direct operational meaning to the RHP measure while endowing the robustness measure with the physical characterizations of the RHP measure.