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arXiv:1307.1760 · Physical Review A

Bipartite concurrence and localized coherence

Chang‐shui Yu, He‐Shan Song

Published Aug 19, 2009• 40 citations
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Abstract

Based on a proposed coherence measure, we show that the local coherence of a bipartite quantum pure state (coherence of its reduced density matrix) is exactly the same as the minimal average coherence with all potential pure-state realizations under consideration. In particular, it is shown that bipartite concurrence of pure states just captures the maximal difference between local coherence and the average coherence of one subsystem induced by local operations on the other subsystem with the assistance of classical communications, which provides an alternative operational meaning for bipartite concurrence of pure states. The relation between concurrence and the proposed coherence measure can also be extended to bipartite mixed states.

DOI
10.1103/physreva.80.022324

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Based on a proposed coherence measure, we show that the local coherence of a bipartite quantum pure state (coherence of its reduced density matrix) is exactly the same as the minimal average coherence with all potential pure-state realizations under consideration. In particular, it is shown that bipartite concurrence of pure states just captures the maximal difference between local coherence and the average coherence of one subsystem induced by local operations on the other subsystem with the assistance of classical communications, which provides an alternative operational meaning for bipartite concurrence of pure states. The relation between concurrence and the proposed coherence measure can also be extended to bipartite mixed states.
We study partial coherence and its connection with entanglement. First, we provide a sufficient and necessary condition for bipartite pure-state transformation under partially incoherent operations: A bipartite pure state can be transformed to another one if and only if a majorization relationship holds between their partial coherence vectors. As a consequence, we introduce the concept of maximal partial coherent states in the sense that they can be used to construct any bipartite state of the same system via partial incoherent operations. Second, we provide a strategy to construct measures of partial coherence by use of symmetric concave functions. Third, we establish the relationship between partial coherence and entanglement. We show that the minimal partial coherence under local unitary transformations is also a measure of entanglement for bipartite pure states, which can be extended to all mixed states by convex-roof extension. Conversely, every entanglement measure can be constructed as a partial coherence measure, and we further show that partial coherence measures are induced through maximal entanglement under partially incoherent operations for bipartite pure states.
Coherence is the most fundamental quantum feature in quantum mechanics. For a bipartite quantum state, if a measurement is performed on one party, the other party, based on the measurement outcomes, will collapse to a corresponding state with some probability and hence gain the average coherence. It is shown that the average coherence is not less than the coherence of its reduced density matrix. In particular, it is very surprising that the extra average coherence (and the maximal extra average coherence with all the possible measurements taken into account) is upper bounded by the classical correlation of the bipartite state instead of the quantum correlation. We also find the sufficient and necessary condition for the null maximal extra average coherence. Some examples demonstrate the relation and, moreover, show that quantum correlation is neither sufficient nor necessary for the nonzero extra average coherence within a given measurement. In addition, the similar conclusions are drawn for both the basis-dependent and the basis-free coherence measure.
We propose a measure to quantify correlations in a bipartite quantum system of two quibits by assessing the minimum difference between outcome states of a subsystem by performing a local measurement on the other subsystem. This maximum similarity measure is a monotone function of the concurrence for pure states of two qubits; for mixed states it accounts for entanglement, dissonance, and classical correlations. Besides, we found a closed formula for evaluating the similarity degree of an arbitrary mix state of two two-dimensional systems.
Bipartite mixed-state quantum entanglement (QE) and its measures play a crucial role in both theoretical research and practical quantum applications. Its internal structure is far more complex and less well understood compared with bipartite pure-state QE. Some existing measures involve inherently intractable global optimizations, while others are only applicable to highly limited-dimensional quantum systems. Here based on the inherent feature that bipartite QE systems nonseparable necessarily implies that local reduced density matrix differs from its \textquotedblleft native\textquotedblright density matrix, we propose a more physical and intuitive measure termed Reduction-induced Minimal Partial Entropy Difference to quantify arbitrary bipartite mixed-state QE. Partial Von Neumann Entropy is only a pure-state special case of this method. This measure offers intrinsic structural insights into bipartite QE characterization, thereby establishing itself as a valuable complementary measure. Its intuitive and clear physical picture, combined with wide applicability and relatively low computational complexity, facilitates exploring its potential quantum information applications, hence its conceptual framework and line of thought deserve to be further developed to describe and quantify multipartite QE in the future.
Interaction with environment may lead to the transition of quantum system from pure state to the mixed one. In this case, the problem of definition of entanglement may arise. In particular, quantitative measure of entanglement concurrence is introduced differently for pure and mixed states. To solve this problem, alternative definition of concurrence of mixed states is introduced in present work. In particular, concurrence of mixed states is determined as an average of pure states concurrences. Averaging is carried out over the quantum ensemble, corresponding to all possible realization of random variable.