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Perfect Photon-Number Pairing Bounds Homodyne CHSH Violation

Muhammet Ali Güz

Published Aug 25, 2026• 0 citations
Publisher

Abstract

A bound on homodyne Bell violation set by the state class, not by the binning. For over two decades, attempts have been made to violate a CHSH inequality using homodyne detection with sign binning alone, since this measurement is nearly lossless and would permit a loophole-free test. The values achieved remain small: 2.048 and 2.076. Two causes are possible — the coarseness of sign binning, or the class of states used. This work separates the two exactly. Because the sign-binned quadrature measurement is a Hermitian operator with spectrum in [-1,1], the maximum over all states is the largest eigenvalue of the Bell operator, and the restriction to perfectly photon-number-correlated states |ψ⟩ = Σ cₙ |n,n⟩ is a projection onto the diagonal subspace. Both maxima are compared at identical angles, so no optimisation over states is involved. Result: the perfectly paired class saturates at S = 2.10198 ± 0.00001, established by convergence to Fock truncation N = 1150, while relaxed pairing reaches 2.22 under the same measurement prescription and continues to rise. The bottleneck is the pairing, not the binning. The gain originates from a targeted admixture of asymmetric photon number at |0,4⟩ and |4,0⟩. Three supporting computations explain the bound: the exact bound a₁ ≤ 0.811876 on the fundamental harmonic of the correlator, which the ideal paired state exhausts to 99.84 %; the completeness of the pairing criterion as an entanglement witness under local symplectic optimisation; and the ineffectiveness of complex amplitudes and Kerr nonlinearity. An attempted proof via subadditivity is documented as having failed, together with the reason no term-by-term estimate can succeed. The bound is presented explicitly as a numerically supported conjecture, not a theorem, and is falsifiable: a state of the form Σ cₙ |n,n⟩ with S > 2.1021 refutes it. Practical consequence for experiment: within the perfectly paired class, further optimisation cannot exceed 2.10198. Higher values require breaking the pairing deliberately. This record contains the manuscript and supplementary material in English and German (PDF and LaTeX sources), and the script reproduce_all.py, which regenerates every tabulated value in eight individually callable sections. Deutsche Fassung des Manuskripts und des Ergänzenden Materials sind in diesem Record enthalten.

DOI
10.5281/zenodo.22093653

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A bound on homodyne Bell violation set by the state class, not by the binning. For over two decades, attempts have been made to violate a CHSH inequality using homodyne detection with sign binning alone, since this measurement is nearly lossless and would permit a loophole-free test. The values achieved remain small: 2.048 and 2.076. Two causes are possible — the coarseness of sign binning, or the class of states used. This work separates the two exactly. Because the sign-binned quadrature measurement is a Hermitian operator with spectrum in [-1,1], the maximum over all states is the largest eigenvalue of the Bell operator, and the restriction to perfectly photon-number-correlated states |ψ⟩ = Σ cₙ |n,n⟩ is a projection onto the diagonal subspace. Both maxima are compared at identical angles, so no optimisation over states is involved. Result: the perfectly paired class saturates at S = 2.10198 ± 0.00001, established by convergence to Fock truncation N = 1150, while relaxed pairing reaches 2.22 under the same measurement prescription and continues to rise. The bottleneck is the pairing, not the binning. The gain originates from a targeted admixture of asymmetric photon number at |0,4⟩ and |4,0⟩. Three supporting computations explain the bound: the exact bound a₁ ≤ 0.811876 on the fundamental harmonic of the correlator, which the ideal paired state exhausts to 99.84 %; the completeness of the pairing criterion as an entanglement witness under local symplectic optimisation; and the ineffectiveness of complex amplitudes and Kerr nonlinearity. An attempted proof via subadditivity is documented as having failed, together with the reason no term-by-term estimate can succeed. The bound is presented explicitly as a numerically supported conjecture, not a theorem, and is falsifiable: a state of the form Σ cₙ |n,n⟩ with S > 2.1021 refutes it. Practical consequence for experiment: within the perfectly paired class, further optimisation cannot exceed 2.10198. Higher values require breaking the pairing deliberately. This record contains the manuscript and supplementary material in English and German (PDF and LaTeX sources), and the script reproduce_all.py, which regenerates every tabulated value in eight individually callable sections. Deutsche Fassung des Manuskripts und des Ergänzenden Materials sind in diesem Record enthalten.
We analyze Bell inequalities violations in photonic experiments for which the measurement apparatuses are restricted to homodyne measurements. Through numerical optimization of the Clauser-Horne-Shimony-Holt inequality over homodyne measurements and binning choices, we demonstrate large violations for states with a bounded number of photons. When considering states defined within qubit local subspaces of two Fock states, such as NOON states, a violation is observed solely within the qubit Fock space spanned by zero and two photons. For more generic states, large violations are obtained. Significant violations are observed even for states containing three photons locally and under realistic values of noise and losses. We propose concrete implementations to achieve such violations, opening new avenues for Bell experiments with homodyne detectors.
Witnessing continuous-variable Bell nonlocality is a challenging endeavor, but Bell himself showed how one might demonstrate this nonlocality. Although Bell nearly showed a violation using the Clauser-Horne-Shimony-Holt (CHSH) inequality with sign-binned position-momentum statistics of entangled pairs of particles measured at different times, his demonstration is subject to approximations not realizable in a laboratory setting. Moreover, he does not give a quantitative estimation of the maximum achievable violation for the wave function he considers. In this article, we show how his strategy can be reimagined using the transverse positions and momenta of entangled photon pairs measured at different propagation distances, and we find that the maximum achievable violation for the state he considers is actually very small relative to the upper limit of 222\sqrt{2}22​. Although Bell's wave function does not produce a large violation of the CHSH inequality, other states may yet do so.
Entanglement, describing the inseparability of a quantum multiparty system, is one of the most intriguing features of quantum mechanics. Violation of Bell inequality, for ruling out the possibility of local hidden variable theories, is commonly used as a strong witness for quantum entanglement. In previous Bell test experiments with photonic entanglement based on two-photon coincidence measurement, the photon temporal wave packets are absorbed completely by the detectors. That is, the photon coherence time is much shorter than the detection time window. Here we demonstrate generation of frequency-bin entangled narrowband biphotons, and for the first time, test the Clauser-Horne-Shimony-Holt (CHSH) Bell inequality |S|<= 2 for their nonlocal temporal correlations with time-resolved detection. We obtain a maximum |S| value of 2.52+/-0.48 that violates the CHSH inequality. Our result will have applications in quantum information processing involving time-frequency entanglement.
We present a full-oscillator analysis of a finite-energy GKP CHSH test whose observed score yields robust Bell-pair self-testing. Periodically binned position and momentum give the Pauli settings, while a fixed binary coarse-graining of photon number modulo four and its displaced conjugate realize the tilted settings. For a number-filtered GKP source, we retain the finite codeword overlap, define the measurements on all photon-number sectors, and compute the physical correlations without logical post-corrections. With the canonical ideal-logical displacement d=πd=\sqrtπd=π​, the CHSH value exceeds the local bound above 4.564.564.56 dB of per-peak squeezing, and Kaniewski's extractability bound becomes nontrivial above 5.025.025.02 dB. Calibrating only ddd using an independently characterized finite-energy parameter lowers these model thresholds to 4.214.214.21 dB and 4.584.584.58 dB, respectively; at 121212 dB, it raises the score from 2.694862.694862.69486 to 2.788582.788582.78858 and the corresponding target-state overlap bound from 0.907580.907580.90758 to 0.972430.972430.97243. This calibration is fixed before Bell-test data are collected. The displacement activates the odd modulo-four sectors, so their fixed a priori assignments are a genuine finite-energy component. The large gain is specific to the deterministic phase-bit coarse-graining; independently calibrating the one-bit POVM with randomized odd-sector outcomes gives only a much smaller improvement. These are honest-model predictions, not loss, detection-efficiency, or finite-sample thresholds. In an experiment, a device-independent guarantee for an extracted Bell pair follows by inserting a confidence lower bound on the observed CHSH score into the self-testing theorem.
We theoretically and experimentally investigate the optimal conditions for the Bell experiment using spontaneous parametric down-conversion (SPDC) sources. In theory, we show that a relatively large average photon number (typically $\ensuremath{\sim}0.5$) is desirable to observe the maximum violation of the Clauser-Horne-Shimony-Holt (CHSH) inequality. Moreover, we show that the violation converges to 0.27 in the limit of large average photon number. In experiment, we perform the Bell experiment without postselection using polarization-entangled photon pairs at 1550-nm telecommunication wavelength generated from SPDC sources. While the violation of the CHSH inequality is not directly observed due to the overall detection efficiencies of our system, the experimental values agree well with those obtained by the theory with experimental imperfections. Furthermore, in the range of small average photon numbers ($\ensuremath{\le}0.1$), we propose and demonstrate a method to estimate the ideal CHSH value intrinsically contained in the tested state from the lossy experimental data without assuming the input quantum state.
Motivated by very recent experiments, we consider a scenario ``\`a la Bell'' in which two protagonists test the Clauser-Horne-Shimony-Holt (CHSH) inequality using a photon-pair source based on spontaneous parametric down conversion and imperfect photon detectors. The conventional wisdom says that (i) if the detectors have unit efficiency, the CHSH violation can reach its maximum quantum value of 222\sqrt{2}22​. To obtain the maximal possible violation, it suffices that the source emits (ii) maximally entangled photon pairs (iii) in two well-defined single modes. Through a nonperturabive calculation of nonlocal correlations, we show that none of these statements are true. By providing the optimal pump parameters, measurement settings and state structure for any detection efficiency and dark count probability, our results give the recipe to close all the loopholes in a Bell test using photon pairs.