Small violations of full correlation Bell inequalities for multipartite pure random states

TL;DR

This paper investigates the likelihood of random multipartite pure states violating Bell inequalities, finding that such violations are rare for qubits and unlikely to grow with system size, especially considering experimental imperfections.

Contribution

It provides a probabilistic analysis showing that violations of full-correlation Bell inequalities are typically small or unlikely in random multipartite states, especially for qubits.

Findings

Violations grow exponentially with N are rare in random states.

Probability of any violation in higher dimensions is small.

Experimental imperfections further reduce the likelihood of observing violations.

Abstract

We estimate the probability of random $N$-qudit pure states violating full-correlation Bell inequalities with two dichotomic observables per site. These inequalities can show violations that grow exponentially with $N$, but we prove this is not the typical case. For many-qubit states the probability to violate any of these inequalities by an amount that grows linearly with $N$ is vanishingly small. If each system's Hilbert space dimension is larger than two, on the other hand, the probability of seeing \emph{any} violation is already small. For the qubits case we discuss furthermore the consequences of this result for the probability of seeing arbitrary violations (\emph i.e., of any order of magnitude) when experimental imperfections are considered.