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IEEE International Symposium on Information Theory (ISIT)

Distillation of local purity from quantum states

Igor Devetak

Published Jan 1, 2005• 2 citations
Publisher

Abstract

Recently Horodecki et al. [Phys. Rev. Lett. 90, 100402 (2003)] introduced an important quantum information processing paradigm in which two parties sharing many copies of the same bipartite quantum state distill local pure states by means of local unitary operations assisted by a one-way (two-way) completely dephasing channel. Local pure states are a valuable resource from a thermodynamical point of view since they allow thermal energy to be converted into work by local quantum heat engines. The work deficit was defined as the decrease in the work extractable from the joint system due to this process of localization, and was proposed as a measure of the quantumness of correlations. We give a simple information theoretical characterization of the one-way work deficit, which turns out to be complementary to a previously known operational measure of classical correlations, the one-way distillable common randomness

DOI
10.1109/isit.2005.1523642

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Recently Horodecki et al. [Phys. Rev. Lett. 90, 100402 (2003)] introduced an important quantum information processing paradigm in which two parties sharing many copies of the same bipartite quantum state distill local pure states by means of local unitary operations assisted by a one-way (two-way) completely dephasing channel. Local pure states are a valuable resource from a thermodynamical point of view since they allow thermal energy to be converted into work by local quantum heat engines. The work deficit was defined as the decrease in the work extractable from the joint system due to this process of localization, and was proposed as a measure of the quantumness of correlations. We give a simple information theoretical characterization of the one-way work deficit, which turns out to be complementary to a previously known operational measure of classical correlations, the one-way distillable common randomness
Recently Horodecki et al. [Phys. Rev. Lett. 90, 100402 (2003)] introduced an important quantum information processing paradigm, in which two parties sharing many copies of the same bipartite quantum state distill local pure states by means of local unitary operations assisted by a one-way (two-way) completely dephasing channel. Local pure states are a valuable resource from a thermodynamical point of view, since they allow thermal energy to be converted into work by local quantum heat engines. We give a simple information-theoretical characterization of the one-way distillable local purity, which turns out to be closely related to a previously known operational measure of classical correlations, the one-way distillable common randomness.
Local pure states are an important resource for quantum computing. The problem of distilling local pure states from mixed ones can be cast in an information theoretic paradigm. The bipartite version of this problem where local purity must be distilled from an arbitrary quantum state shared between two parties, Alice and Bob, is closely related to the problem of separating quantum and classical correlations in the state and in particular, to a measure of classical correlations called the one-way distillable common randomness. In Phys. Rev. A 71, 062303 (2005), the optimal rate of local purity distillation is derived when many copies of a bipartite quantum state are shared between Alice and Bob, and the parties are allowed unlimited use of a unidirectional dephasing channel. In the present paper, we extend this result to the setting in which the use of the channel is bounded. We demonstrate that in the case of a classical-quantum system, the expression for the local purity distilled is efficiently computable and provide examples with their tradeoff curves.
We investigate the problem of producing local pure states by performing local noisy operations assisted by one-way classical communication on a given bipartite mixed state in the one-shot setting. We consider the following two scenarios:1)Scenario I: A party, say Alice, is provided with a single copy of some quantum state ρAon system A. The task for Alice is to extract pure qubit states using only noisy operations on A. We call this task purity concentration.2)Scenario II: Two parties, Alice and Bob possess the A and B sub-systems, respectively, of a given bipartite quantum state ρAB. They are allowed to perform any local noisy operations and communicate via a one-way dephasing (i.e., classical) channel. The task for them is to design a protocol using these resources such that together they can extract pure local qubit states from the shared state ρAB. We call this task local purity distillation.
We consider the amount of work which can be extracted from a heat bath using a bipartite state rho shared by two parties. In general it is less then the amount of work extractable when one party is in possession of the entire state. We derive bounds for this "work deficit" and calculate it explicitly for a number of different cases. In particuar, for pure states the work deficit is exactly equal to the distillable entanglement of the state. A form of complementarity exists between physical work which can be extracted and distillable entanglement. The work deficit is a good measure of the quantum correlations in a state and provides a new paradigm for understanding quantum nonlocality.
We consider the task of distilling local purity from a noisy quantum state ρABC, wherein we provide a protocol for three parties, Alice, Bob and Charlie, to distill local purity (at a rate P) from many independent copies of a given quantum state ρABC. The three parties have access to their respective subsystems of ρABC, and are only allowed to use local unitary operations. In addition, Alice and Bob can communicate with Charlie using a one-way multiple-access dephasing channel of link rates R1and R2, respectively. The objective of the protocol is to minimize the usage of the dephasing channel (in terms of rates R1and R2) while maximizing the asymptotic purity that can be jointly distilled from ρABC. To achieve this, we employ ideas from distributed measurement compression protocols, and in turn, characterize a set of sufficient conditions on (P, R1,R2) in terms of quantum information theoretic quantities such that P amount of purity can be distilled using rates R1and R2.
In spite of many results in quantum information theory, the complex nature of compound systems is far from clear. In general the information is a mixture of local and nonlocal (``quantum'') information. It is important from both pragmatic and theoretical points of view to know the relationships between the two components. To make this point more clear, we develop and investigate the quantum-information processing paradigm in which parties sharing a multipartite state distill local information. The amount of information which is lost because the parties must use a classical communication channel is the deficit. This scheme can be viewed as complementary to the notion of distilling entanglement. After reviewing the paradigm in detail, we show that the upper bound for the deficit is given by the relative entropy distance to so-called pseudoclassically correlated states; the lower bound is the relative entropy of entanglement. This implies, in particular, that any entangled state is informationally nonlocal---i.e., has nonzero deficit. We also apply the paradigm to defining the thermodynamical cost of erasing entanglement. We show the cost is bounded from below by relative entropy of entanglement. We demonstrate the existence of several other nonlocal phenomena which can be found using the paradigm of local information. For example, we prove the existence of a form of nonlocality without entanglement and with distinguishability. We analyze the deficit for several classes of multipartite pure states and obtain that in contrast to the GHZ state, the Aharonov state is extremely nonlocal. We also show that there do not exist states for which the deficit is strictly equal to the whole informational content (bound local information). We discuss the relation of the paradigm with measures of classical correlations introduced earlier. It is also proved that in the one-way scenario, the deficit is additive for Bell diagonal states. We then discuss complementary features of information in distributed quantum systems. Finally we discuss the physical and theoretical meaning of the results and pose many open questions.