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arXiv:2607.29012

Bell violation with path-entangled number states under realistic detection

Christoph F. Wildfeuer

quant-phSubmitted Jul 31, 2026• Announced Aug 3, 2026• 0 citations
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Abstract

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.

Comments
16 pages, 3 figures, 2 tables

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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 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.
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.
Summary form only given. In a Bell experiment, one prepares pairs of entangled particles and sends them to two observers, Alice and Bob, for measurement and detection [1]. For some choices of their measurement settings, Alice and Bob may observe strong correlations between their results in accordance with the predictions of quantum mechanical theory. Conversely, any physical theory that assumes no physical influence can be faster than the speed of light and that properties of physical systems are elements of reality [2] - a local realistic theory - can predict only a limited amount of correlation between Alice's and Bob's measurement outcomes. Upon observing correlations sufficient to violate Bell's inequality, Alice and Bob must abandon local realism.It is common that in an experiment, some particles emitted by the source will not be detected. Then the subset of detected particles might display correlations that violate the Bell inequality although the entire ensemble can be described by a local realistic theory [3]. To achieve a conclusive Bell violation without assuming that the detected particles are a “fair” sample, a highly efficient setup is necessary. Due to experimental limitations, fair sampling has been assumed in most Bell experiments performed to date, and it has never been possible to avoid this assumption with photons due to the absence of efficient sources and detectors. Here we report the first Bell experiment with photons that does not rely on any fair-sampling assumption. In our experiment we employ the Eberhard inequality, a Bell inequality that by construction does not rely on the fair-sampling assumption [4]. Our source of entangled photon pairs utilizes bulk-crystal spontaneous parametric downconversion in a Sagnac configuration, which has shown to be very efficient [5, 6]. For photon detection, we employ superconducting transition-edge sensors, which not only offer high efficiency but also are intrinsically free of dark counts [7]. This combination of features is imperative for an experiment in which no correction of count rates can be tolerated. We achieve uncorrected coupling efficiencies over 73% in each arm of our source, facilitating a violation of the inequality by nearly 70 standard deviations after five minutes of measurement. Since the first experimental Bell test, a satisfactory laboratory realization of the motivating gedankenexperiment has remained an outstanding challenge. The two other main assumptions include “locality” [8, 9] and “freedom of choice” [10]. Invoking any of these renders an experiment vulnerable to explanation by a local realistic theory. The realization of an experiment that is free of all three assumptions-a so-called loopholefree Bell test-remains an important outstanding goal for the physics community with strong implications for quantum technologies. We note that with our experiment, photons are the first physical system for which each of these three assumptions has been successfully addressed, albeit in different experiments. This represents promise for practical applications like one-sided device-independent quantum key distribution [11], and in an additional test, we also demonstrate that our experimental apparatus is capable of implementing this protocol on both sides.
Abstract Optical hybrid entanglement can be created between two qubits, one encoded in a single photon and another one in coherent states with opposite phases. It opens the path to a variety of quantum technologies, such as heterogeneous quantum networks, merging continuous- and discrete-variable encoding, and enabling the transport and interconversion of information. However, reliable characterization of the non-local nature of this quantum state is limited so far to full quantum state tomography. Here, we perform a thorough study of Clauser–Horne–Shimony–Holt Bell inequality tests, enabling practical verification of quantum nonlocality for optical hybrid entanglement. We show that a practical violation of this inequality is possible with simple photon number on/off measurements if detection efficiencies stay above 82%. Another approach, based on photon-number parity measurements, requires 94% efficiency but works well in the limit of higher photon populations. Both tests use no postselection of the measurement outcomes and they are free of the fair-sampling hypothesis. Our proposal paves the way to performing loophole-free tests using feasible experimental tasks such as coherent state interference and photon counting.
We consider a subgroup of unitary transformations on a mode of light induced by a Mach-Zehnder Interferometer and an algebra of observables describing a photon-number detector preceded by the interferometer. We explore the uncertainty principles between such observables and their usefulness in performing a Bell-like experiment to show the violation of the Clauser-Horne-Shimony-Holt (CHSH) inequality under the physical assumption that the detector distinguishes between only zero photons and a nonzero number of photons. We show the local settings of the interferometer which lead to a maximal violation of the CHSH inequality.
We show that nonlocal correlation experiments on the two spatially separated modes of a maximally path-entangled number state may be performed. They lead to a violation of a Clauser-Horne Bell inequality for any finite photon number NNN. We also present an analytical expression for the two-mode Wigner function of a maximally path-entangled number state and investigate a Clauser-Horne-Shimony-Holt Bell inequality for such a state. We test other Bell-type inequalities. Some are violated by a constant amount for any NNN.