Computational Bell Inequalities

TL;DR

The paper addresses the challenge of certifying quantum behavior in device-independent, single-prover interactions under cryptographic assumptions. It develops a formal link between such protocols and Bell inequalities by introducing a computational space of correlations (CSoC) and a Bell-mapping framework that extracts virtual inputs from transcripts, then leverages measurement-dependent locality (MDL) and a one-sided AMDL refinement to bound classical strategies. It defines the computational classical and quantum sets, and introduces a computational NPA-like hierarchy (CNPA) to outer-approximate $\mathscr{Q}_{\kappa}^{\mathrm{comp}}$ under approximate nonsignaling. The framework is demonstrated through concrete showcases based on trapdoor claw-free functions and compiled games, yielding computable Tsirelson-type bounds and entropy-certification results, and it achieves a modular, protocol-agnostic toolkit for analyzing single-prover quantum certification under finite leakage $\kappa$. This methodology bridges cryptography and Bell-nonlocality, enabling precise, scalable assessments of quantum advantage in cryptographic and certification tasks while clarifying how computational assumptions substitute for nonsignaling constraints.

Abstract

We introduce a systematic approach for analyzing device-independent single-prover interactive protocols under computational assumptions. This is done by establishing an explicit correspondence with Bell inequalities and nonlocal games and constructing a computational space of correlations. We show how computational assumptions are converted to computational Bell inequalities, in their rigorous mathematical sense, a hyperplane that separates the sets of classical and quantum verifier-prover interactions. We reveal precisely how the nonsignaling assumption in standard device-independent setups interchanges with the computational challenge of learning a hidden input (that we define). We further utilize our fundamental results to study explicit protocols using the new perspective. We take advantage of modular tools for studying nonlocality, deriving tighter Tsirelson bounds for single-prover protocols and bounding the entropy generated in the interaction, improving on previous results. Our work thus establishes a modular approach to analyzing single-prover quantum certification protocols based on computational assumptions through the fundamental lens of Bell inequalities, removing many layers of technical overhead. The link that we draw between single-prover protocols and Bell inequalities goes far beyond the spread intuitive understanding or known results about "compiled nonlocal games"; Notably, it captures the exact way in which the correspondence between computational assumptions and locality should be understood also in protocols based on, e.g., trapdoor claw-free functions (in which there is no clear underlying nonlocal game).

Figures (10)

• Figure 1: A 2D slice of the SoC of conditional probability distributions for some nonlocal game. The local set $\mathscr{L}$ in lighter blue and the quantum set $\mathscr{Q}$ in darker blue. The Bell inequality is represented by the magenta dashed line.
• Figure 2: Comparison between abstract systems and a nonlocal game interaction.
• Figure 3: Canonical form protocol structure.
• Figure 4: MDL interpretation of the protocol template -- The verifier $\mathcal{V}$ and device $\mathcal{D}$ receive respective inputs $x,m$ and respectively output $a,b$.
• Figure 5: A schematic $2$-dimensional slice of correlation space for fixed leakage ${\kappa}$ in the CHSH Bell scenario. The teal polygon is the average measurement-dependent local (AMDL) polytope $\mathscr{L}^{\text{A}}_{{\kappa}}$, which contains the computational-local set $\mathscr{L}_{{\kappa}}^{\mathrm{comp}}$. The location of the computational-quantum set $\mathscr{Q}_{{\kappa}}^{\mathrm{comp}}$ relative to $\mathscr{L}^{\text{A}}_{{\kappa}}$ is not asserted; however, the two are disjoint, as a computational Bell inequality separates $\mathscr{Q}_{{\kappa}}^{\mathrm{comp}}$ from $\mathscr{L}^{\text{A}}_{{\kappa}}$ (and hence from $\mathscr{L}_{{\kappa}}^{\mathrm{comp}}$). The dashed magenta line illustrates a shifted-CHSH inequality adapted to input leakage. The dashed lime line illustrates a facet-defining AMDL inequality, violated by some $Q\in\mathscr{Q}_{{\kappa}}^{\mathrm{comp}}$.

Theorems & Definitions (58)

• Definition 2.1: Total Variation Distance
• Definition 2.2: Nonsignaling set $\mathscr{N}$
• Definition 2.3: Local set $\mathscr{L}$
• Definition 2.4: Quantum set $\mathscr{Q}$
• Definition 2.5: Bell inequality
• Remark
• Remark
• Definition 2.6: MDL set $\mathscr{L}^{\text{M}}_{(l,h)}$
• Remark
• Claim 2.7: CHSH MDL inequality putz2014