Robust CHSH Self-Testing with Finite-Energy GKP States

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=\sqrtπ), the CHSH value exceeds the local bound above (4.56) dB of per-peak squeezing, and Kaniewski's extractability bound becomes nontrivial above (5.02) dB. Calibrating only (d) using an independently characterized finite-energy parameter lowers these model thresholds to (4.21) dB and (4.58) dB, respectively; at (12) dB, it raises the score from (2.69486) to (2.78858) and the corresponding target-state overlap bound from (0.90758) to (0.97243). 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.

Source: arXiv:2608.11122v1 - http://arxiv.org/abs/2608.11122v1 PDF: https://arxiv.org/pdf/2608.11122v1 Original Link: http://arxiv.org/abs/2608.11122v1