Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Correlation Self-Testing of Quantum Theory against Generalised Probabilistic Theories with Restricted Relabelling Symmetry

Kuntal Sengupta, Mirjam Weilenmann, Roger Colbeck

TL;DR

The paper investigates correlation self-testing of quantum theory within Generalised Probabilistic Theories by focusing on GPTs with restricted bipartite relabelling symmetry and introducing minimal $k$-preservability as a compositional-consistency criterion. Through detailed analysis of state- and effect-polytopes for various bipartite constructions (including PR-box mixtures and party-symmetric, non-relabellable cases), it shows that while many GPTs admit entanglement-swapping structures only under stringent constraints, quantum theory still outperforms these GPTs in the Adaptive CHSH game, enabling correlation self-testing for several classes of state spaces. A key insight is the link between minimal 2-preservability and Tsirelson’s bound, suggesting that compositional criteria beyond the no-restriction hypothesis can constrain nonlocal correlations in a way that supports quantum uniqueness in the ACHSH task. The work also clarifies the role of couplers, proves no-go results for many configurations, and discusses generalizations and connections to prior results, with implications for foundational understanding and experimental tests of quantum theory. Overall, the findings strengthen confidence in quantum theory as the correct description of nonlocal correlations under a broad set of GPT-based alternatives and open avenues for further exploration of compositional principles in GPTs.

Abstract

Correlation self-testing of quantum theory involves identifying a task or set of tasks whose optimal performance can be achieved only by theories that can realise the same set of correlations as quantum theory in every causal structure. Following this approach, previous work has ruled out various classes of generalised probabilistic theories whose joint state spaces have a certain regularity in the sense of a (discrete) rotation symmetry of the bipartite state spaces. Here we consider theories whose bipartite state spaces lack this regularity. We form them by taking the convex hull of all the local states and a finite number of non-local states. We show that a criterion of compositional consistency is needed in such theories: for a measurement effect to be valid, there must exist at least one measurement that it is part of. This goes beyond previous consistency criteria and corresponds to a strengthening of the no-restriction hypothesis. We show that quantum theory outperforms these theories in a task called the adaptive CHSH game, which shows that they can be ruled out experimentally. We further show a connection between compositional consistency and Tsirelson's bound.

Correlation Self-Testing of Quantum Theory against Generalised Probabilistic Theories with Restricted Relabelling Symmetry

TL;DR

The paper investigates correlation self-testing of quantum theory within Generalised Probabilistic Theories by focusing on GPTs with restricted bipartite relabelling symmetry and introducing minimal -preservability as a compositional-consistency criterion. Through detailed analysis of state- and effect-polytopes for various bipartite constructions (including PR-box mixtures and party-symmetric, non-relabellable cases), it shows that while many GPTs admit entanglement-swapping structures only under stringent constraints, quantum theory still outperforms these GPTs in the Adaptive CHSH game, enabling correlation self-testing for several classes of state spaces. A key insight is the link between minimal 2-preservability and Tsirelson’s bound, suggesting that compositional criteria beyond the no-restriction hypothesis can constrain nonlocal correlations in a way that supports quantum uniqueness in the ACHSH task. The work also clarifies the role of couplers, proves no-go results for many configurations, and discusses generalizations and connections to prior results, with implications for foundational understanding and experimental tests of quantum theory. Overall, the findings strengthen confidence in quantum theory as the correct description of nonlocal correlations under a broad set of GPT-based alternatives and open avenues for further exploration of compositional principles in GPTs.

Abstract

Correlation self-testing of quantum theory involves identifying a task or set of tasks whose optimal performance can be achieved only by theories that can realise the same set of correlations as quantum theory in every causal structure. Following this approach, previous work has ruled out various classes of generalised probabilistic theories whose joint state spaces have a certain regularity in the sense of a (discrete) rotation symmetry of the bipartite state spaces. Here we consider theories whose bipartite state spaces lack this regularity. We form them by taking the convex hull of all the local states and a finite number of non-local states. We show that a criterion of compositional consistency is needed in such theories: for a measurement effect to be valid, there must exist at least one measurement that it is part of. This goes beyond previous consistency criteria and corresponds to a strengthening of the no-restriction hypothesis. We show that quantum theory outperforms these theories in a task called the adaptive CHSH game, which shows that they can be ruled out experimentally. We further show a connection between compositional consistency and Tsirelson's bound.

Paper Structure

This paper contains 55 sections, 7 theorems, 106 equations, 3 figures, 53 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Correlation Self-Testing and the Adaptive CHSH Game
  4. Generalised Probabilistic Theories
  5. Generalised Local Theories
  6. Box World
  7. The Set of Correlations
  8. Entanglement Swapping in GPTs
  9. Bipartite Compositions of and their Effect Polytopes
  10. State spaces with 1 noisy PR-box extremal state
  11. State spaces with 2 noisy PR-box extremal states
  12. General Algorithm for state spaces with g noisy PR-box extremal states
  13. Bipartite Compositions of and their Effect Polytopes
  14. Minimal k-Preservability Criterion
  15. CH Type Effects of
  16. ...and 40 more sections

Key Result

Lemma 1

Let $e\in\mathcal{E}_{\mathcal{S}^{\boxtimes m}}$ be a weakly minimally $k$-preserving effect. Then $e$ is minimally $k$-preserving if and only if $u-e$ is weakly minimally $k$-preserving.

Figures (3)

  • Figure 1: Causal structure for the Adaptive CHSH game. Bob shares the resource $s_{AB}$ with Alice and the resource $s_{B'C}$ with Charlie. A referee asks questions to Alice and Charlie labelled by random variables $X$ and $Z$ respectively. Bob performs a joint measurement on his share of resources, the outcomes of which are labelled by the random variable $B$. Alice and Charlie perform local measurements on their subsystems, the outcomes of which are labeled by random variables $A$ and $C$. The value of all the random variables determine the score in the game. There are no non-classical tripartite resources shared by all the three parties (shared tripartite randomness is allowed).
  • Figure 2: A two dimensional slice of the set of correlations generated when fiducial measurements are performed on the states of the bipartite state space characterised by the 16 local deterministic states and one PR box PhysRevLett.102.110402. The vertical axes represent a CHSH inequality and the horizontal axes represent one of its symmetries obtained by relabelling the inputs. Local correlations, denoted by the square $\mathcal{C}$, satisfy $1/4 \leqslant \mathrm{CHSH}[\mathrm{p}] \leqslant 3/4$ and $1/4 \leqslant \mathrm{CHSH^*}[\mathrm{p}] \leqslant 3/4$.
  • Figure 3: A plot of the CHSH scores of the renormalised states obtained when an effect $\Tilde{e}$ from each of the four types is applied in the middle half of the 4-partite state $\mathrm{PR}_{2,\alpha} \otimes \mathrm{PR}_{2,\alpha}$. The red line is obtained when $\Tilde{e} \in \mathrm{Type\ 4}$. The brown line is obtained for any $\Tilde{e} \in \mathrm{Type\ 3}$. The yellow and blue lines are obtained when any $\Tilde{e}$ is taken from Type 1 and Type 2 respectively. The straight horizontal black lines represent the classical score $3/4$ and Tsirelson's bound.

Theorems & Definitions (15)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Lemma 1
  • proof
  • Theorem 1
  • Lemma 2
  • Theorem 2
  • Theorem 3
  • ...and 5 more