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arXiv:quant-ph/0210053 · Physical Review Letters

Symmetric Extensions of Quantum States and Local Hidden Variable Theories

Barbara M. Terhal, Andrew C. Doherty, David Schwab

Published Apr 16, 2003• 98 citations
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Abstract

While all bipartite pure entangled states violate some Bell inequality, the relationship between entanglement and nonlocality for mixed quantum states is not well understood. We introduce a simple and efficient algorithmic approach for the problem of constructing local hidden variable theories for quantum states. The method is based on constructing a so-called symmetric quasiextension of the quantum state that gives rise to a local hidden variable model with a certain number of settings for the observers Alice and Bob.

DOI
10.1103/physrevlett.90.157903

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While all bipartite pure entangled states violate some Bell inequality, the relationship between entanglement and nonlocality for mixed quantum states is not well understood. We introduce a simple and efficient algorithmic approach for the problem of constructing local hidden variable theories for quantum states. The method is based on constructing a so-called symmetric quasiextension of the quantum state that gives rise to a local hidden variable model with a certain number of settings for the observers Alice and Bob.
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Abstract In this paper we consider the description by a general Bell-type non-local hidden variable theory of bipartite quantum states with two observables per sub-system. We derive Bell inequalities of the Collins-Gisin.-Liden-Massar-Popescu type which involve combinations of the probabilities of related outcomes for measurements for the four pairs of sub-system observables. It is shown that the corresponding quantum theory expressions violate the Bell inequalities in the case of the maximally entangled state of the bipartitite system. The CHSH Bell inequality is also derived from this general CGLMP Bell-type non-local hidden variable theory. This shows that quantum theory can not be underpinned by a Bell-type non-local hidden variable theory. So as a general Bell-type local hidden variable theory has already been shown to conflict with quantum theory, it follows that quantum theory can not be understood in terms of any CGLMP Bell-type hidden variable theory—local or non-local.
We consider Bell tests in which the distant observers can perform local filtering before testing a Bell inequality. Notably, in this setup, certain entangled states admitting a local hidden variable model in the standard Bell scenario can nevertheless violate a Bell inequality after filtering, displaying so-called hidden nonlocality. Here we ask whether all entangled states can violate a Bell inequality after well-chosen local filtering. We answer this question in the negative by showing that there exist entangled states without hidden nonlocality. Specifically, we prove that some two-qubit Werner states still admit a local hidden variable model after any possible local filtering on a single copy of the state.
Building upon the results of R. Augusiak et al. [Phys. Rev. Lett. 115, 030404 (2015)] we develop a general approach to the generation of genuinely entangled multipartite states of any number of parties from genuinely entangled states of a fixed number of parties, in particular, the bipartite entangled ones. In our approach, certain isometries whose output subspaces are either symmetric or genuinely entangled in some multipartite Hilbert spaces are applied to local subsystems of bipartite entangled or multipartite genuinely entangled quantum states. To prove that entanglement of the resulting states is indeed genuine we then introduce criteria allowing us to decide it efficiently. The construction is then exploited to provide examples of multipartite states that are genuinely entangled but not genuinely nonlocal, giving further illustration for the inequivalence between entanglement and nonlocality in the multiparticle scenario. It is also shown how to construct genuinely entangled states which are unsteerable across certain bipartite cuts.
We consider Bell tests in which the distant observers can perform local filtering before testing a Bell inequality. Notably, in this setup, certain entangled states admitting a local hidden variable model in the standard Bell scenario can nevertheless violate a Bell inequality after filtering, displaying so-called hidden nonlocality. Here we ask whether all entangled states can violate a Bell inequality after well-chosen local filtering. We answer this question in the negative by showing that there exist entangled states without hidden nonlocality. Specifically, we prove that some two-qubit Werner states still admit a local hidden variable model after any possible local filtering on a single copy of the state.