Structure of dimension-bounded temporal correlations

TL;DR

This paper investigates the structure of temporal correlations in quantum systems with dimension constraints, revealing nonconvexity and providing dimension bounds, nonlinear inequalities, and algorithms for analysis.

Contribution

It establishes the necessary and sufficient quantum system dimension for convex correlation spaces and introduces tools to detect nonconvexity in temporal correlations.

Findings

Temporal correlation space can be nonconvex under dimension constraints

Dimension bounds are provided for generating convex correlation spaces

Nonlinear inequalities are derived to witness nonconvexity in qubits and qutrits

Abstract

We analyze the structure of the space of temporal correlations generated by quantum systems. We show that the temporal correlation space under dimension constraints can be nonconvex. For the general case, we provide the necessary and sufficient dimension of a quantum system needed to generate a convex correlation space for a given scenario. We further prove that this dimension coincides with the dimension necessary to generate any point in the temporal correlation polytope. As an application of our results, we derive nonlinear inequalities to witness the nonconvexity for qubits and qutrits in the simplest scenario, and present an algorithm which can help to find the minimum for a certain type of nonlinear expressions under dimension constraints.