On the largest Bell violation attainable by a quantum state

TL;DR

This paper investigates the maximum Bell violation achievable by quantum states using the projective tensor norm, providing bounds and showing the necessity of a logarithmic factor for maximally entangled states.

Contribution

It introduces a truncated SDP relaxation to bound Bell violations and proves the optimality of the logarithmic factor for maximally entangled states.

Findings

Provides an upper bound for the difference between classical and relaxed quantum values.

Shows the best complementation constant of ^n in (_) is of order ( n).

Demonstrates the logarithmic factor cannot be removed for maximally entangled states.

Abstract

We study the projective tensor norm as a measure of the largest Bell violation of a quantum state. In order to do this, we consider a truncated version of a well-known SDP relaxation for the quantum value of a two-prover one-round game, one which has extra restrictions on the dimension of the SDP solutions. Our main result provides a quite accurate upper bound for the distance between the classical value of a Bell inequality and the corresponding value of the relaxation. Along the way, we give a simple proof that the best complementation constant of $\ell_2^n$ in $\ell_1(\ell_\infty)$ is of order $\sqrt{\ln n}$. As a direct consequence, we show that we cannot remove a logarithmic factor when we are computing the largest Bell violation attainable by the maximally entangled state.