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arXiv:1710.08517

Distribution of coherence in multipartite systems under entropic coherence measure

Kaifeng Bu, Lu Li, Arun Kumar Pati, Shao-Ming Fei, Junde Wu

Published Oct 23, 2017• 3 citations
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Abstract

The distribution of coherence in multipartite systems is one of the fundamental problems in the resource theory of coherence. To quantify the coherence in multipartite systems more precisely, we introduce new coherence measures, incoherent-quantum (IQ) coherence measures, on bipartite systems by the max- and min- relative entropies and provide the operational interpretation in certain subchannel discrimination problem. By introducing the smooth max- and min- relative entropies of incoherent-quantum (IQ) coherence on bipartite systems, we exhibit the distribution of coherence in multipartite systems: the total coherence is lower bounded by the sum of local coherence and genuine multipartite entanglement. Besides, we find the monogamy relationship for coherence on multipartite systems by incoherent-quantum (IQ) coherence measures. Thus, the IQ coherence measures introduced here truly capture the non-sharability of quantumness of coherence in multipartite context.

DOI
10.48550/arxiv.1710.08517

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The distribution of coherence in multipartite systems is one of the fundamental problems in the resource theory of coherence. To quantify the coherence in multipartite systems more precisely, we introduce new coherence measures, incoherent-quantum (IQ) coherence measures, on bipartite systems by the max- and min- relative entropies and provide the operational interpretation in certain subchannel discrimination problem. By introducing the smooth max- and min- relative entropies of incoherent-quantum (IQ) coherence on bipartite systems, we exhibit the distribution of coherence in multipartite systems: the total coherence is lower bounded by the sum of local coherence and genuine multipartite entanglement. Besides, we find the monogamy relationship for coherence on multipartite systems by incoherent-quantum (IQ) coherence measures. Thus, the IQ coherence measures introduced here truly capture the non-sharability of quantumness of coherence in multipartite context.
The distribution of quantum coherence in multipartite systems is one of the basic problems in the resource theory of coherence. While the usual coherence measures are defined on a single system and cannot capture the nonlocal correlation between subsystems, in order to deal with the distribution of coherence it is crucial to quantify the coherence in bipartite systems properly. Here, we introduce incoherent-quantum (IQ) coherence measures on bipartite systems, which can characterize the correlations between systems. According to the IQ coherence measures on bipartite systems, we find the distribution of coherence of formation and assistance in bipartite systems: the total coherence of formation is lower bounded by the sum of coherence of formation in each subsystem and the entanglement of formation between the subsystems, while the total coherence of assistance is upper bounded by the sum of coherence of assistance in each subsystem and the entanglement of assistance between subsystems. Besides, we also obtain the tradeoff relation between the coherence cost and entanglement cost, distillable coherence and distillable entanglement in bipartite systems. Thus, the IQ coherence measures introduced here truly capture the nonlocal correlation between subsystems and reveal the distribution of coherence in bipartite systems.
The operational characterization of quantum coherence is the cornerstone in the development of the resource theory of coherence. We introduce a new coherence quantifier based on maximum relative entropy. We prove that the maximum relative entropy of coherence is directly related to the maximum overlap with maximally coherent states under a particular class of operations, which provides an operational interpretation of the maximum relative entropy of coherence. Moreover, we show that, for any coherent state, there are examples of subchannel discrimination problems such that this coherent state allows for a higher probability of successfully discriminating subchannels than that of all incoherent states. This advantage of coherent states in subchannel discrimination can be exactly characterized by the maximum relative entropy of coherence. By introducing a suitable smooth maximum relative entropy of coherence, we prove that the smooth maximum relative entropy of coherence provides a lower bound of one-shot coherence cost, and the maximum relative entropy of coherence is equivalent to the relative entropy of coherence in the asymptotic limit. Similar to the maximum relative entropy of coherence, the minimum relative entropy of coherence has also been investigated. We show that the minimum relative entropy of coherence provides an upper bound of one-shot coherence distillation, and in the asymptotic limit the minimum relative entropy of coherence is equivalent to the relative entropy of coherence.
Abstract We introduce a notion of genuine correlated coherence. Such a notion is based on the possibility of concentrating on individual systems the coherence present in a distributed system, by making use of incoherent unitary transformations. We define an entropic quantifier of genuine correlated multipartite coherence for generic mixed states, and we focus on the bipartite pure-state case. In the latter case we derive necessary and sufficient conditions for the possibility of fully localizing the coherence, hence identifying the conditions for genuine correlated bipartite coherence. We analyze in detail the quantitative problem for the case of two-qubit pure states, identifying the states with the largest amount of genuine correlated coherence. Interestingly, such states do not have maximal global coherence nor maximal coherence rank.
Abstract This paper examines the coherence in multipartite systems. We first discuss the distribution of total coherence in a given multipartite quantum state into discord between subsystems and coherent dissonance in each individual subsystem, using the relative entropy as a distance measure. Then we give some trade-off relations between various types of coherence and discord within a bipartite system, and extend these results to the multipartite setting. Finally, the change of coherence in entanglement distribution is studied.
The distribution of coherence in multipartite systems is examined. We use a new coherence measure with entropic nature and metric properties, based on the quantum Jensen-Shannon divergence. The metric property allows for the coherence to be decomposed into various contributions, which arise from local and intrinsic coherences. We find that there are trade-off relations between the various contributions of coherence, as a function of parameters of the quantum state. In bipartite systems the coherence resides on individual sites or is distributed among the sites, which contribute in a complementary way. In more complex systems, the characteristics of the coherence can display more subtle changes with respect to the parameters of the quantum state. In the case of the XXZ Heisenberg model, the coherence changes from a monogamous to a polygamous nature. This allows us to define the shareability of coherence, leading to monogamy relations for coherence.
Abstract Quantum coherence is a fundamental property of quantum systems that can be studied within the framework of a quantum resource theory. From this resource-theoretic approach, many similarities emerge between coherence and quantum entanglement, the latter being another prominent quantum resource. In this paper, we study the relationship between coherence and entanglement from the perspective of resource loss/gain by unitary dynamics. Specifically we consider how much coherence can be either generated or destroyed on a bipartite system when the action is restricted to local unitary (LU) operations. For pure states, we find that the relative entropy of entanglement and the robustness of entanglement provide tight lower bounds on the amount of coherence that can be destroyed by LUs, as measured by the relative entropy and measures of coherence, respectively. This provides new operational interpretations of the entanglement entropy and robustness of entanglement in pure states as the minimal amount of coherence that persists when the system is subjected to LU transformations. We then study the amount of bipartite pure entanglement that can be either maximized or minimized when the action is restricted to a global incoherent operation. For two-qubit pure states, maximum and minimum are shown in terms of coefficients of given state with incoherent basis. Finally we consider a generalized version of the recently introduced CCP of a quantum channel.