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arXiv:0707.0848 · Physical Review Letters

No-Local-Broadcasting Theorem for Multipartite Quantum Correlations

Marco Piani, Paweł Horodecki, Ryszard Horodecki

Published Mar 6, 2008• 425 citations
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Abstract

We prove that the correlations present in a multipartite quantum state have an operational quantum character even if the state is unentangled, as long as it does not simply encode a multipartite classical probability distribution. Said quantumness is revealed by the new task of local broadcasting, i.e., of locally sharing preestablished correlations, which is feasible if and only if correlations are stricly classical. Our operational approach leads to natural definitions of measures for quantumness of correlations. It also reproduces the standard no-broadcasting theorem as a special case.

DOI
10.1103/physrevlett.100.090502

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We prove that the correlations present in a multipartite quantum state have an operational quantum character even if the state is unentangled, as long as it does not simply encode a multipartite classical probability distribution. Said quantumness is revealed by the new task of local broadcasting, i.e., of locally sharing preestablished correlations, which is feasible if and only if correlations are stricly classical. Our operational approach leads to natural definitions of measures for quantumness of correlations. It also reproduces the standard no-broadcasting theorem as a special case.
Genuine multpartite quantum nonlocality can be quantified by the communication cost needed to reproduce the nonlocal correlation by classical communication models. This prompts the question as to how one may provide such an operational characterization for the nonclassicality of local multipartite correlations arising from genuinely quantum states possessing quantum discord. To answer this question, we consider a classical simulation protocol where one of the parties pre-share a random variable with the other parties who may also share arbitrary randomness. We demonstrate that genuine nonclassicality of local multipartite correlations is necessary for implying genuine superlocality which is the requirement for a larger dimension of the random variable in the above classical simulation protocol than that of quantum state that can generate the nonclassical correlations.
Entanglement and Bell nonlocality are known to be inequivalent: there exist entangled states that admit a local hidden-variable model for all local measurements. Here we show that this gap disappears in a minimal broadcast extension of the Bell scenario. Assuming only the validity of quantum theory, we prove that for every entangled state ρABρ_{AB}ρAB​ there exist local broadcasting maps and local measurements such that the resulting four--partite correlations cannot be reproduced by any broadcast network whose source is separable across the A∣BA|BA∣B cut. Thus, all entangled states are broadcast nonlocal in quantum theory. In addition, we show that all (also mixed) multipartite states can be broadcast-self-tested, according to a natural operational definition.
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.
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We provide a unified framework for nonsignalling quantum and classical multipartite correlations, allowing all to be written as the trace of some local (quantum) measurements multiplied by an operator. The properties of this operator define the corresponding set of correlations. We then show that if the theory is such that all local quantum measurements are possible, one obtains the correlations corresponding to the extension of Gleason's Theorem to multipartite systems. Such correlations coincide with the quantum ones for one and two parties, but we prove the existence of a gap for three or more parties.
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.