Near-Optimal Bell Nonlocality with GKP and Entangled Cat States

Abstract

Gottesman-Kitaev-Preskill (GKP) states are widely studied as a bosonic encoding for fault-tolerant quantum computing because small displacement errors can be identified and corrected through syndrome measurements. However, fault-tolerant operation requires substantially greater GKP squeezing than is currently available in optical platforms. It is therefore important to identify quantum-information tasks that can be realised with finite-squeezing GKP states. Here, we study Bell nonlocality in finite-energy GKP Bell states under experimentally motivated measurement constraints. Although finite-energy GKP Bell states support Bell nonlocality even at low squeezing, logical Pauli measurements alone cannot reveal it. Accessing the maximum CHSH violation supported by the state generally requires logical non-Clifford rotations. We instead show that the required non-Gaussian resource can reside in the measurement itself. Photon-number-resolving detection combined with Gaussian preprocessing and adaptive feed-forward closely approaches the maximum violation of the CHSH Bell inequality without implementing logical non-Clifford gates. The same measurement principle strongly enhances Bell violations for cat-code Bell pairs and entangled coherent states, demonstrating that its usefulness extends beyond the GKP lattice. Our results identify non-Gaussian measurements as a practical route to revealing Bell nonlocality in finite-energy bosonic states.

The paper asks how much CHSH nonlocality finite-energy GKP Bell states reveal under restricted measurements, and whether logical non-Clifford gates are necessary. It compares logical-rotation-assisted homodyne benchmarks with physical receivers using Gaussian preprocessing, photon-number-resolving detection (PNRD), and two-stage adaptive feed-forward, then applies the receivers to cat-code Bell pairs and even entangled coherent states. The GKP pair has two-dimensional local support despite nonorthogonal finite-energy codewords. Its unrestricted CHSH maximum follows from its Schmidt coefficients: $S_{\max}=2\sqrt{1+4\lambda_0^2\lambda_1^2}$. At $0\,\mathrm{dB}$, the state is approximately a vacuum–even-cat entangled state with $S_{\max}\simeq2.177$. Within benchmarks that allow ideal setting-dependent logical rotations, optimising homodyne binning lowers the violation threshold from approximately $2.6$ to $1\,\mathrm{dB}$; these rotations generally require non-Clifford control. A bilateral adaptive PNRD receiver instead uses squeezing, a vacuum tap, and a second displacement conditioned on whether the first count is zero. Numerically optimised receivers achieve approximately $95\%$–$98\%$ of the unrestricted CHSH value over $0$–$6\,\mathrm{dB}$, with a gap below $0.125$. For a cat-code Bell pair at amplitude $\alpha\simeq0.40$, the receiver gives $S=2.816$, versus $2\sqrt2\simeq2.828$. Binary outcome assignments are optimised analytically; continuous receiver parameters use multistart see-saw optimisation. Reported receiver values are achieved lower bounds, not certified global optima. Detector models include inefficiency and dark counts, with violations surviving several tested settings, but inefficiency increasingly degrades performance at high squeezing. The study is theoretical and numerical; receiver parameters are restricted, and logical-rotation benchmarks omit magic-state preparation and injection errors.