Flexible constrained de Finetti reductions and applications

TL;DR

This paper explores a flexible, constrained version of de Finetti reductions in quantum information, demonstrating its broad applicability to symmetries, linear and convex constraints, and infinite-dimensional systems, with implications for quantum complexity and entropic measures.

Contribution

It introduces a generalized constrained de Finetti reduction applicable to various symmetries and constraints, extending its use to infinite-dimensional quantum systems and complex quantum protocols.

Findings

Applicable to symmetries commuting with permutation actions

Effective with permutation-invariant linear and convex constraints

Useful for analyzing entanglement and complexity in quantum systems

Abstract

De Finetti theorems show how sufficiently exchangeable states are well-approximated by convex combinations of i.i.d. states. Recently, it was shown that in many quantum information applications a more relaxed de Finetti reduction (i.e. only a matrix inequality between the symmetric state and one of de Finetti form) is enough, and that it leads to more concise and elegant arguments. Here we show several uses and general flexible applicability of a constrained de Finetti reduction in quantum information theory, which was recently discovered by Duan, Severini and Winter. In particular we show that the technique can accommodate other symmetries commuting with the permutation action, and permutation-invariant linear constraints. We then demonstrate that, in some cases, it is also fruitful with convex constraints, in particular separability in a bipartite setting. This is a constraint particularly interesting in the context of the complexity class $\mathrm{QMA}(2)$ of interactive quantum Merlin-Arthur games with unentangled provers, and our results relate to the soundness gap amplification of $\mathrm{QMA}(2)$ protocols by parallel repetition. It is also relevant for the regularization of certain entropic channel parameters. Finally, we explore an extension to infinite-dimensional systems, which usually pose inherent problems to de Finetti techniques in the quantum case.