Robust Bell Nonlocality from Gottesman-Kitaev-Preskill States

Abstract

Bell tests based on homodyne detection are strongly constrained in continuous-variable systems. Can Gottesman-Kitaev-Preskill (GKP) encoding turn homodyne detection into a practical tool for revealing Bell nonlocality? We consider a physically motivated model in which each party performs homodyne detection and digitizes the continuous outcome via a fixed periodic binning, corresponding to logical Pauli measurements. Within this framework, we derive a bipartite no-go: CHSH cannot be violated for Bell-pair states. Moving beyond two parties, we show that finitely squeezed GKP-encoded GHZ and W states nevertheless exhibit strong multipartite nonlocality, violating multipartite Bell inequalities with homodyne-only readout. We quantify the required squeezing thresholds and robustness to loss, providing a route toward homodyne-based Bell tests in continuous-variable systems.

Figures (5)

• Figure 1: Bell nonlocality with Gottesman–Kitaev–Preskill States. Nonlocal correlations can be pulled out of continuous-variable systems using GKP states and only homodyne detection.
• Figure 2: Ideal vs finite-energy GKP structure in phase space. Comparison between ideal and finite-energy ("realistic") square GKP encodings in phase space and in the position quadrature. (a) sketch of the position-space marginal distribution showing that the ideal GKP code words form an infinitely sharp Dirac-comb lattice, while finite-energy GKP states replace each spike by a narrow Gaussian peak and suppress distant peaks under a broad Gaussian envelope (dashed curve), giving a normalizable state with finite mean energy. (b) Phase-space dot/contour plots of the Wigner functions associated with the logical GKP Pauli operators $\mathbbm{1}, X, Y$ and $Z$. Red (blue) marks positive (negative) regions, with the dot size indicating the relative magnitude of each lattice peak.
• Figure 3: Multipartite Bell nonlocality with finitely squeezed GKP encoding under homodyne measurement. (a) MABK Bell parameter $S_N^{\textsf{M}}$ for encoded GHZ states. Left: lossless case for different numbers of parties $N$. Shaded regions indicate parameter regimes where the LHV bound $S_N^{\textsf{M}}\le 2^{\lfloor N/2\rfloor}$ is violated. Odd $N$ are shown with dotted lines and even $N$ with solid lines, highlighting that some consecutive system sizes share the same classical bound. Right: $S_3^{\textsf{M}}$ versus squeezing for different loss transmissivities $\eta$; the dashed horizontal line marks the LHV bound. (b) Cabello functional $S_N^{\textsf{C}}$ for encoded $W$ states. Left: lossless case for varying $N$. Right: $N=3$ with varying $\eta$; the dashed horizontal line marks the local bound $S_N^{\textsf{C}}\le 0$.
• Figure 4: Critical squeezing and robustness to excess noise. (a) Critical squeezing $r_{\mathrm{crit}}$ (dB) for observing a violation at $N=3$ versus loss transmissivity $\eta$, comparing GHZ/MABK (blue) and W/Cabello (red). (b) Robustness under thermal noise in the loss channel: the environment port is a thermal state with mean photon number $n_{\mathrm{th}}$. Heat maps show $S_3^{\textsf{M}}$ (left) and $S_3^{\textsf{C}}$ (right) as functions of $(\eta,n_{\mathrm{th}})$; dashed lines mark the local bounds ($S_3^{\textsf{M}}=2$ and $S_3^{\textsf{C}}=0$), and above them the local region is shown in blue hues.
• Figure 5: Nonlocality beyond Bell. Distance to the local polytope, $\mathcal{D}(\boldsymbol p_n)$, for the binned-homodyne behaviours generated by finitely squeezed GKP-encoded GHZ (solid lines) and $W$ (dashed lines) resources, as a function of the squeezing (dB). Left: $n=2$ inputs, corresponding to the effective Pauli settings $(X,Y)$ for GHZ and $(X,Z)$ for $W$. Right: $n=3$ inputs, corresponding to $(X,Y,Z)$. Colours indicate the channel transmissivity $\eta$ (from $\eta=1$ to $\eta=0.75$). Positive values of $\mathcal{D}$ certify Bell nonlocality.

Theorems & Definitions (12)

• Lemma 1: Invariance under local conjugation
• proof
• Lemma 2: Pauli set is invariant under Clifford conjugation
• proof
• Theorem 3: No CHSH violation with Pauli measurements on a local-Clifford Bell pair
• proof
• Lemma 4: Wigner marginal at $\theta=0$
• proof
• Corollary 5: Rotated Wigner marginal
• proof