Bipartite matrix-valued tensor product correlations that are not finitely representable

TL;DR

This paper investigates matrix-valued bipartite quantum correlations, demonstrating that for most input-output configurations, finite-dimensional correlations differ from their infinite-dimensional counterparts, revealing fundamental distinctions in quantum correlation sets.

Contribution

It establishes the existence of finite matrix size correlations that are not finitely representable, showing a separation between finite and infinite-dimensional quantum correlation sets for most input-output pairs.

Findings

Finite matrix size correlations differ from infinite-dimensional correlations.

Separation occurs for all input-output pairs except (2,2).

Finite-dimensional correlations are not always finitely representable.

Abstract

We consider the matrix-valued generalizations of bipartite tensor product quantum correlations and bipartite infinite-dimensional tensor product quantum correlations, respectively. These sets are denoted by $C_q^{(n)}(m,k)$ and $C_{qs}^{(n)}(m,k)$, respectively, where $m$ is the number of inputs, $k$ is the number of outputs, and $n$ is the matrix size. We show that, for any $m,k \geq 2$ with $(m,k) \neq (2,2)$, there is an $n \leq 4$ for which we have the separation $C_q^{(n)}(m,k) \neq C_{qs}^{(n)}(m,k)$.