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Physical Review A

Relation between “no broadcasting” for noncommuting states and “no local broadcasting” for quantum correlations

Shunlong Luo, Nan Li, Xuelian Cao

Published May 14, 2009• 19 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.79.054305

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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.
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.
One of the classical results concerning quantum channels is the characterization of entanglement-breaking channels [M. Horodecki, P. W. Shor, and M. B. Ruskai, Rev. Math. Phys. 15, 629 (2003)]. We address the question whether there exists a similar characterization on the level of quantum correlations which may go beyond entanglement. The answer is fully affirmative in the case of breaking quantum correlations down to the, so-called, QC (quantum-classical) type, while it is no longer true in the CC (classical-classical) case. The corresponding channels turn out to be measurement maps. Our study also reveals an unexpected link between quantum state and local correlation broadcasting and finite Markov chains. We present a possibility of broadcasting via non von Neumann measurements, which relies on the Perron-Frobenius theorem. Surprisingly, this is not the typical generalized controlled-not (c-not) gate scenario appearing naturally in this context.
The no-broadcasting theorem is a fundamental result in quantum information theory. It guarantees that a class of attacks on quantum protocols, based on eavesdropping and indiscriminate copying of quantum information, are impossible. Due to its fundamental importance, it is natural to ask whether it is an intrinsic quantum property or whether it also holds for a broader class of non-classical theories. To address this question, one could use the framework of correlation scenarios. Under this standpoint, Joshi, Grudka, and the Horodeckis conjectured that one cannot locally broadcast nonlocal behaviours. In this paper, we prove their conjecture based on the monotonicity of the relative entropy for behaviours. Additionally, following a similar reasoning, we obtain an analogous no-go theorem for steerable assemblages.
It is well known that the non-broadcasting theorem proved by Barnum et al is a fundamental principle of quantum communication. As we are aware, optimal broadcasting (OB) is the only method to broadcast noncommuting mixed states approximately. In this paper, motivated by the probabilistic cloning of quantum states proposed by Duan and Guo, we propose a new way for broadcasting noncommuting mixed states—probabilistic broadcasting (PB), and we present a sufficient condition for PB of mixed states. To a certain extent, we generalize the probabilistic cloning theorem from pure states to mixed states, and in particular, we generalize the non-broadcasting theorem, since the case that commuting mixed states can be exactly broadcast can be thought of as a special instance of PB where the success ratio is 1. Moreover, we discuss probabilistic local broadcasting (PLB) of separable bipartite states.
Quantum theory can be regarded as a noncommutative generalization of classical probability. From this point of view, one expects quantum dynamics to be analogous to classical conditional probabilities. In this paper, a variant of the well-known isomorphism between completely positive maps and bipartite density operators is derived, which makes this connection much more explicit. This isomorphism is given an operational interpretation in terms of statistical correlations between ensemble preparation procedures and outcomes of measurements. Finally, the isomorphism is applied to elucidate the connection between no-cloning and no-broadcasting theorems and the monogamy of entanglement, and a simplified proof of the no-broadcasting theorem is obtained as a by-product.
Abstract No-broadcasting theorem is one of the most fundamental results in quantum information theory; it guarantees that the simplest attacks on any quantum protocol, based on eavesdropping and copying of quantum information, are impossible. Due to the fundamental importance of the no-broadcasting theorem, it is essential to understand the exact boundaries of this limitation. We generalize the standard definition of broadcasting by restricting the set of states which we want to broadcast and restricting the sets of measurements which we use to test the broadcasting. We show that in some of the investigated cases broadcasting is equivalent to commutativity, while in other cases commutativity is not necessary.