Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Classifying the simplest Bell inequalities beyond qubits and their applications towards self-testing

Palash Pandya, Shubhayan Sarkar, Remigiusz Augusiak

TL;DR

This work extends the study of Bell nonlocality beyond qubits by focusing on the (2,2,3) scenario and constructing a general Bell operator with a sum-of-squares decomposition that is maximally violated by the maximally entangled qutrit state. It proves a robust self-testing statement: when the maximal quantum value is attained, there exist local unitaries that map the observables to a 3-dimensional Weyl-Heisenberg structure and certify the maximally entangled qutrit state up to local isometries, with Tsirelson bound $T_Q=4$. The authors derive a broad class of CGLMP/SATWAP-inspired inequalities, show the classical bound is strictly below 4, and reveal that SATWAP is a special case within their framework, highlighting an optimal 3-level self-testing point. The results advance device-independent certification and high-dimensional self-testing by providing explicit SOS-based constraints and a minimal-setting, two-measurement construction for $d=3$.

Abstract

Bell inequalities reveal the fundamentally nonlocal character of quantum mechanics. In this regard, one of the interesting problems is to explore all possible Bell inequalities that demonstrate a gap between local and nonlocal quantum behaviour. This is useful for the geometric characterisation of the set of nonlocal correlations achievable within quantum theory. Moreover, it provides a systematic way to construct Bell inequalities that are tailored to specific quantum information processing tasks. This characterisation is well understood in the simplest $(2,2,2)$ scenario, namely two parties performing two binary outcome measurements. However, beyond this setting, relatively few Bell inequalities are known, and the situation becomes particularly scarce in scenarios involving a greater number of outcomes. Here, we consider the $(2,2,3)$ scenario, or two parties performing two three-outcome measurements, and characterise all Bell inequalities that can arise from the simplest sum-of-squares decomposition and are maximally violated by the maximally entangled state of local dimension three. We then utilise them to self-test this state, along with a class of three-outcome measurements.

Classifying the simplest Bell inequalities beyond qubits and their applications towards self-testing

TL;DR

This work extends the study of Bell nonlocality beyond qubits by focusing on the (2,2,3) scenario and constructing a general Bell operator with a sum-of-squares decomposition that is maximally violated by the maximally entangled qutrit state. It proves a robust self-testing statement: when the maximal quantum value is attained, there exist local unitaries that map the observables to a 3-dimensional Weyl-Heisenberg structure and certify the maximally entangled qutrit state up to local isometries, with Tsirelson bound . The authors derive a broad class of CGLMP/SATWAP-inspired inequalities, show the classical bound is strictly below 4, and reveal that SATWAP is a special case within their framework, highlighting an optimal 3-level self-testing point. The results advance device-independent certification and high-dimensional self-testing by providing explicit SOS-based constraints and a minimal-setting, two-measurement construction for .

Abstract

Bell inequalities reveal the fundamentally nonlocal character of quantum mechanics. In this regard, one of the interesting problems is to explore all possible Bell inequalities that demonstrate a gap between local and nonlocal quantum behaviour. This is useful for the geometric characterisation of the set of nonlocal correlations achievable within quantum theory. Moreover, it provides a systematic way to construct Bell inequalities that are tailored to specific quantum information processing tasks. This characterisation is well understood in the simplest scenario, namely two parties performing two binary outcome measurements. However, beyond this setting, relatively few Bell inequalities are known, and the situation becomes particularly scarce in scenarios involving a greater number of outcomes. Here, we consider the scenario, or two parties performing two three-outcome measurements, and characterise all Bell inequalities that can arise from the simplest sum-of-squares decomposition and are maximally violated by the maximally entangled state of local dimension three. We then utilise them to self-test this state, along with a class of three-outcome measurements.
Paper Structure (13 sections, 4 theorems, 106 equations, 1 figure)

This paper contains 13 sections, 4 theorems, 106 equations, 1 figure.

Table of Contents

  1. Preliminaries
  2. The Bell Operator and the Quantum Bound
  3. Self-testing statement for the Bell operator
  4. Classical Value of the Bell Inequality
  5. Conclusions
  6. Acknowledgments
  7. Theorem 1: Sum-of-Squares and Tsirelson bound
  8. Theorem 2: Self-testing statement
  9. Three dimensional Hilbert space
  10. Tracelessness of Bob's observables
  11. Characterising Bob's observables
  12. Alice's observables and the State
  13. Genuine Incompatibility of two observables

Key Result

Theorem 1

If the Bell parameters satisfy $\alpha\beta^* + \gamma\delta^*=0$ and with $L_i$s defined above, then the Bell operator $W$ in eq:BI admits the following sum-of-squares decomposition,

Figures (1)

  • Figure 1: The plot depicts the maximum classical value under the constraints $\theta_\alpha = \theta_\beta +\pi/6$, $\theta_\gamma = \theta_\delta +\pi/6$ and $\theta_\delta = \theta_\beta +\pi/6$. The plot is for the interval $0\leq\theta_\beta\leq\pi/6$ as the maximum classical value is periodic in nature and the period is $\pi/6$. The red dot denotes the maximum classical value of the SATWAP inequality for $d=3$.

Theorems & Definitions (8)

  • Theorem 1
  • proof
  • Corollary 1.1
  • proof
  • Theorem 2
  • proof
  • Lemma 3
  • proof