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arXiv:1510.06722 · Physical Review A

Incompatible quantum measurements admitting a local-hidden-variable model

Marco Túlio Quintino, Joseph Bowles, Flavien Hirsch, Nicolas Brunner

Published May 23, 2016• 31 citations
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Abstract

The observation of quantum nonlocality, i.e. quantum correlations violating a Bell inequality, implies the use of incompatible local quantum measurements. Here we consider the converse question. That is, can any set of incompatible measurements be used in order to demonstrate Bell inequality violation? Our main result is to construct a local hidden variable model for an incompatible set of qubit measurements. Specifically, we show that if Alice uses this set of measurements, then for any possible shared entangled state, and any possible dichotomic measurements performed by Bob, the resulting statistics are local. This represents significant progress towards proving that measurement incompatibility does not imply Bell nonlocality in general.

DOI
10.1103/physreva.93.052115

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The observation of quantum nonlocality, i.e. quantum correlations violating a Bell inequality, implies the use of incompatible local quantum measurements. Here we consider the converse question. That is, can any set of incompatible measurements be used in order to demonstrate Bell inequality violation? Our main result is to construct a local hidden variable model for an incompatible set of qubit measurements. Specifically, we show that if Alice uses this set of measurements, then for any possible shared entangled state, and any possible dichotomic measurements performed by Bob, the resulting statistics are local. This represents significant progress towards proving that measurement incompatibility does not imply Bell nonlocality in general.
We discuss the connection between the incompatibility of quantum measurements, as captured by the notion of joint measurability, and the violation of Bell inequalities. Specifically, we explicitly present a given set of non-jointly-measurable positive-operator-value measures (POVMs) MA{\mathcal{M}}_{A}MA​ with the following property. Considering a bipartite Bell test where Alice uses MA{\mathcal{M}}_{A}MA​, then for any possible shared entangled state $\ensuremath{\rho}$ and any set of (possibly infinitely many) POVMs NB{\mathcal{N}}_{B}NB​ performed by Bob, the resulting statistics admits a local model and can thus never violate any Bell inequality. This shows that quantum measurement incompatibility does not imply Bell nonlocality in general.
We investigate the relation between the incompatibility of quantum measurements and quantum nonlocality. We show that a set of measurements is not jointly measurable (i.e., incompatible) if and only if it can be used for demonstrating Einstein-Podolsky-Rosen steering, a form of quantum nonlocality. Moreover, we discuss the connection between Bell nonlocality and joint measurability, and give evidence that both notions are inequivalent. Specifically, we exhibit a set of incompatible quantum measurements and show that it does not violate a large class of Bell inequalities. This suggests the existence of incompatible quantum measurements which are Bell local, similarly to certain entangled states which admit a local hidden variable model.
In the case of a pair of two-outcome measurements, incompatibility is equivalent to Bell nonlocality. Indeed, any pair of incompatible two-outcome measurements can violate the Clauser–Horne–Shimony–Holt Bell inequality, which has been proven by Wolf et al (2009 Phys. Rev. Lett. 103 230402). In the case of more than two measurements the equivalence between incompatibility and Bell nonlocality is still an open problem, though partial results have recently been obtained. Here we show that the equivalence breaks for a special choice of three measurements. In particular, we present a set of three incompatible two-outcome measurements, such that if Alice measures this set, independent of the set of measurements chosen by Bob and the state shared by them, the resulting statistics cannot violate any Bell inequality. On the other hand, complementing the above result, we exhibit a set of N measurements for any that is -wise compatible, nevertheless it gives rise to Bell violation.
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.
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.
Bell nonlocality is a fundamental phenomenon of quantum physics as well as an essential resource for various tasks in quantum information processing. It is known that for the observation of nonlocality the measurements on a quantum system have to be incompatible, but the question of which incompatible measurements are useful, remained open. Here we prove that any set of incompatible measurements on qubits leads to a violation of a suitable Bell inequality in a multiparticle scenario, where all parties perform the same set of measurements. Since there exists incompatible measurements on qubits which do not lead to Bell nonlocality for two particles, our results demonstrate a fundamental difference between two-particle and multiparticle nonlocality, pointing at the superactivation of measurement incompatibility as a resource. In addition, our results imply that measurement incompatibility for qubits can always be certified in a device-independent manner.