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arXiv:1202.3049 · Physical Review A

Robust nonlocality tests with displacement-based measurements

Jonatan Bohr Brask, Rafael Chaves

Published Jul 31, 2012• 22 citations
arXivHTMLPDF

Abstract

Lately, much interest has been directed towards designing setups that achieve decisive tests of local realism. Here we present Bell tests with measurements based on linear optical displacements and single-photon detection. The scheme displays good tolerance to loss. In particular, for entangled squeezed states, we find thresholds compatible with current efficiencies of detectors and sources. Furthermore, the scheme is easily extendible to any number of observers, allowing observation of multipartite nonlocality for a single photon shared among multiple modes. We also consider the case of atom-photon entanglement, where the loss threshold can be lowered further, as well as local filters compensating transmission and coupling inefficiencies at the source.

DOI
10.1103/physreva.86.010103

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Lately, much interest has been directed towards designing setups that achieve decisive tests of local realism. Here we present Bell tests with measurements based on linear optical displacements and single-photon detection. The scheme displays good tolerance to loss. In particular, for entangled squeezed states, we find thresholds compatible with current efficiencies of detectors and sources. Furthermore, the scheme is easily extendible to any number of observers, allowing observation of multipartite nonlocality for a single photon shared among multiple modes. We also consider the case of atom-photon entanglement, where the loss threshold can be lowered further, as well as local filters compensating transmission and coupling inefficiencies at the source.
Recently, much interest has been directed toward designing setups that achieve realistic loss thresholds for decisive tests of local realism---in particular in the optical regime. We analyze the feasibility of such Bell tests based on a WWW state shared between multiple parties, which can be realized, for example, by a single photon shared between spatial modes. We develop a general error model to obtain thresholds for the efficiencies required to violate local realism and also consider two concrete optical-measurement schemes.
We investigate quantum nonlocality of a single-photon entangled state under feasible measurement techniques consisting of on-off and homodyne detections along with unitary operations of displacement and squeezing. We test for a potential violation of the Clauser-Horne-Shimony-Holt (CHSH) inequality, in which each of the bipartite party has a freedom to choose between two measurement settings, with each measurement yielding a binary outcome. We find that single-photon quantum nonlocality can be detected when two or less of the four total measurements are carried out by homodyne detection. The largest violation of the CHSH inequality is obtained when all four measurements are squeezed-and-displaced on-off detections. We test the robustness of violations against imperfections in on-off detectors and single-photon sources, finding that the squeezed-and-displaced measurement schemes perform better than the displacement-only measurement schemes.
A single photon delocalised over two modes is entangled, and whether that entanglement can violate a Bell inequality has been disputed for three decades. In principle it is settled: the surviving objection was the absence of a shared phase reference, and a local oscillator supplies one, acting as a shared frame rather than as a measurement setting. In practice it is open, and what remains is a question about the apparatus. Loss does not spoil a parity measurement so much as rescale it, carrying it into the same one-parameter family of detector operators that already contains on--off detection. Within that family we give closed correlators for path-entangled number states, or N00N states, at arbitrary efficiency, with dark counts, mode mismatch, phase noise and upstream loss. Because loss and displacement commute up to a rescaling of the displacement amplitude, loss before and after the displacement is interchangeable, and a symmetric experiment is governed by one overall efficiency: the probability that a heralded photon is detected. For a single photon the Clauser--Horne--Shimony--Holt threshold on that efficiency is 0.83 with on--off detection and 0.95 with parity, so the scheme that needs no photon-number resolution is the more robust, by twelve percentage points of loss and a factor of five in tolerable dark counts. Efficiency is no longer the obstacle; the mode overlap is. The best demonstrated telecom coupling, a buildable interferometer and today's detectors give an overall efficiency of 0.86, at which the required overlap is 0.92. We show that this overlap carries the heralded photon's single-mode weight as well as the local oscillator's mode match, so buying more of it costs heralding efficiency --- and no source has reported the two together.
We analyze an all-optical setup which enables Bell-inequality violation over long distances by exploiting probabilistic entanglement swapping. The setup involves only two-mode squeezers, displacements, beamsplitters, and on-off detectors. We describe how events must be arranged to close both the detection and locality loopholes. We analyze a scenario with dichotomic inputs and outputs, and check the robustness of the Bell inequality violation for up to six parties, with respect to phase, amplitude, and dark count noise, as well as loss.
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