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Physical Review A

Decomposition of bipartite states with applications to quantum no-broadcasting theorems

Shunlong Luo, Wei Sun

Published Jul 30, 2010• 66 citations
Publisher

Abstract

Correlations in bipartite quantum states are fundamental objects in quantum information theory. A canonical framework for studying correlations is the entangled versus separable dichotomy in which the decompositions of separable states as convex combinations of product states play an instrumental role. In this paper, motivated by both the representation of separable states and quantum no-broadcasting considerations, we establish a constructive decomposition representation for any bipartite state. As applications, we prove the conjectures proposed by Luo [Lett. Math. Phys. 92, 143 (2010)] concerning no-unilocal broadcasting for quantum correlations and further provide a unified picture for the celebrated quantum no-broadcasting theorem for noncommuting states by Barnum et al. [Phys. Rev. Lett. 76, 2818 (1996)], and the elegant no-local-broadcasting theorem for quantum correlations by Piani et al. [Phys. Rev. Lett. 100, 090502 (2008)]. The results reveal some intrinsic relation between quantumness of correlations and noncommutativity of states, and in particular, provide a characterization for zero quantum discord introduced by Ollivier and Zurek [Phys. Rev. Lett. 88, 017901 (2001)] from the broadcasting perspective. Furthermore, it is indicated that the distinction between the decomposition for general bipartite states and that for separable states might be useful in studying entanglement versus separability.

DOI
10.1103/physreva.82.012338

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Correlations in bipartite quantum states are fundamental objects in quantum information theory. A canonical framework for studying correlations is the entangled versus separable dichotomy in which the decompositions of separable states as convex combinations of product states play an instrumental role. In this paper, motivated by both the representation of separable states and quantum no-broadcasting considerations, we establish a constructive decomposition representation for any bipartite state. As applications, we prove the conjectures proposed by Luo [Lett. Math. Phys. 92, 143 (2010)] concerning no-unilocal broadcasting for quantum correlations and further provide a unified picture for the celebrated quantum no-broadcasting theorem for noncommuting states by Barnum et al. [Phys. Rev. Lett. 76, 2818 (1996)], and the elegant no-local-broadcasting theorem for quantum correlations by Piani et al. [Phys. Rev. Lett. 100, 090502 (2008)]. The results reveal some intrinsic relation between quantumness of correlations and noncommutativity of states, and in particular, provide a characterization for zero quantum discord introduced by Ollivier and Zurek [Phys. Rev. Lett. 88, 017901 (2001)] from the broadcasting perspective. Furthermore, it is indicated that the distinction between the decomposition for general bipartite states and that for separable states might be useful in studying entanglement versus separability.
The no-broadcasting theorem, first established by Barnum et al. [Phys. Rev. Lett. 76, 2818 (1996)], states that a set of quantum states can be broadcast if and only if it constitutes a commuting family. Quite recently, Piani et al. [Phys. Rev. Lett. 100, 090502 (2008)] showed, by using an ingenious and sophisticated method, that the correlations in a single bipartite state can be locally broadcast if and only if the state is effectively a classical one (i.e., the correlations therein are classical). In this Brief Report, under the condition of nondegenerate spectrum, we provide an alternative and significantly simpler proof of the latter result based on the original no-broadcasting theorem and the monotonicity of the quantum relative entropy. This derivation motivates us to conjecture the equivalence between these two elegant yet formally different no-broadcasting theorems and indicates a subtle and fundamental issue concerning spectral degeneracy which also lies at the heart of the conflict between the von Neumann projection postulate and the L\"uders ansatz for quantum measurements. This relation not only offers operational interpretations for commutativity and classicality but also illustrates the basic significance of noncommutativity in characterizing quantumness from the informational perspective.
Quantum correlations in a physical system are usually studied with respect to a unique and fixed decomposition of the system into subsystems, without fully exploiting the rich structure of the state space. Here, we show several examples in which the consideration of different ways to decompose a physical system enhances the quantum resources and accounts for a more flexible definition of quantumness measures. Furthermore, we give a different perspective regarding how to reassess the fact that local operations play a key role in general quantumness measures that go beyond entanglement---as discordlike ones. We propose a family of measures to quantify the maximum quantumness of a given state. For the discord-based case, we present some analytical results for $2\ifmmode\times\else\texttimes\fi{}d$-dimensional states. Applying our definition to low-dimensional bipartite states, we show that different behaviors can be reported for separable and entangled states vis-\`a-vis those corresponding to the usual measures of quantum correlations. We show that there is a close link between our proposal and the criterion to witness quantum correlations based on the rank of the correlation matrix, proposed by Daki\ifmmode \acute{c}\else \'{c}\fi{}, Vedral, and Brukner [Phys. Rev. Lett. 105, 190502 (2010)].
Quantum discord, as introduced by Olliver and Zurek [Phys. Rev. Lett. 88, 017901 (2001)], is a measure of the discrepancy between two natural yet different quantum analogs of the classical mutual information. This notion characterizes and quantifies quantumness of correlations in bipartite states from a measurement perspective, and is fundamentally different from the various entanglement measures in the entanglement vs separability paradigm. The phenomenon of nonzero quantum discord is a manifestation of quantum correlations due to noncommutativity rather than due to entanglement, and has interesting and significant applications in revealing the advantage of certain quantum tasks. We will evaluate analytically the quantum discord for a large family of two-qubit states, and make a comparative study of the relationships between classical and quantum correlations in terms of the quantum discord. We furthermore compare the quantum discord with the entanglement of formation, and illustrate that the latter may be larger than the former, although for separable states, the entanglement of formation always vanishes and thus is less than the quantum discord.
Broadcasting quantum and classical information is a basic task in quantum information processing, and is also a useful model in the study of quantum correlations including quantum discord. We establish a full operational characterization of two-sided quantum discord in terms of bilocal broadcasting of quantum correlations. Moreover, we show that both the optimal fidelity of unilocal broadcasting of the correlations in an arbitrary bipartite quantum state and that of broadcasting an arbitrary set of quantum states can be formulized as semidefinite programs (SDPs), which are efficiently computable. We also analyze some properties of these SDPs and evaluate the broadcasting fidelities for some cases of interest.
A bipartite state is called classical (with regard to correlations) if it is left undisturbed by a certain local von Neumann measurement, and is called separable if it can be represented as a convex combination of product states. Due to the perfect distinguishability of orthogonal vectors, a classical state can essentially be identified with a convenient bivariate probability distribution, and moreover, it is separable, but not vice versa. The notion of separability plays a key role in quantum information theory because entanglement is defined via separability. However, the definition of separability is ad hoc and formal. In this paper, we present an intrinsic characterization of separable states via classical states from the measurement perspective: Separable states are precisely those states that are reductions of classical states in higher dimensions with the natural partitions. Consequently, entangled states are precisely those states that cannot be represented as such reductions of classical states. This observation highlights the hidden mutually exclusive and complementary relations between classicality and entanglement.
Quantum discord, as defined by Olliver and Zurek (2002 Phys. Rev. Lett. 88 017901) as the difference of two natural quantum extensions of the classical mutual information, plays an interesting role in characterizing quantumness of correlations. Inspired by this idea, we will study quantumness of bipartite states arising from different quantum analogs of the classical conditional entropy. Our approach is intrinsic, in contrast to the Olliver–Zurek method that involves extrinsic local measurements. For this purpose, we introduce two alternative variants of quantum conditional entropies via conditional density operators, which in turn are intuitive quantum extensions of equivalent classical expressions for the conditional probability. The significance of these quantum conditional entropies in characterizing quantumness of bipartite states is illustrated through several examples.