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CHSH inequality always holds in bipartite qutrits with spin-1 observables

Hyunho Cha

Abstract

We resolve a conjecture of Hanotel and Loubenets concerning CHSH inequality in bipartite qutrits. It states that nonseparable pure states of two qutrits do not violate the CHSH inequality when each party is restricted to spin-1 observables. We prove a stronger result that \emph{all} bipartite states on $\mathbb{C}^3 \otimes \mathbb{C}^3$ satisfy the CHSH inequality under spin-1 measurements, regardless of whether the state is pure or mixed.

CHSH inequality always holds in bipartite qutrits with spin-1 observables

Abstract

We resolve a conjecture of Hanotel and Loubenets concerning CHSH inequality in bipartite qutrits. It states that nonseparable pure states of two qutrits do not violate the CHSH inequality when each party is restricted to spin-1 observables. We prove a stronger result that \emph{all} bipartite states on satisfy the CHSH inequality under spin-1 measurements, regardless of whether the state is pure or mixed.
Paper Structure (4 sections, 6 theorems, 52 equations)

This paper contains 4 sections, 6 theorems, 52 equations.

Table of Contents

  1. Setup and main result
  2. Reduction to a two-parameter model
  3. Exact spectrum of the reduced operator
  4. Proof of Theorem \ref{['thm:main']}

Key Result

Theorem 1

For every density operator $\rho$ on $\mathbb C^3\otimes \mathbb C^3$ and every choice of unit vectors $a,a',b,b'\in \mathbb{R}^3$, Equivalently, no two-qutrit state violates the CHSH inequality under spin-$1$ measurements.

Theorems & Definitions (12)

  • Theorem 1
  • Lemma 1: Rotational covariance
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Proposition 1
  • proof
  • proof : Proof of Theorem \ref{['thm:main']}
  • ...and 2 more