Freedman's theorem for unitarily invariant states on the CCR algebra

TL;DR

This paper characterizes the structure of unitarily invariant states on the CCR algebra, revealing they form a Bauer simplex with specific extreme points, extending Freedman's theorem to a non-commutative setting.

Contribution

It provides a complete description of unitarily invariant states on CCR algebras, identifying their convex structure and extreme points, thus generalizing Freedman's theorem.

Findings

The set of invariant states is a Bauer simplex.

Extreme states include the canonical trace and Gaussian states with variance ≥ 1.

The structure extends classical Freedman's theorem to the CCR algebra context.

Abstract

The set of states on ${\rm CCR}(\ch)$, the CCR algebra of a separable Hilbert space $\ch$, is here looked at as a natural object to obtain a non-commutative version of Freedman's theorem for unitarily invariant stochastic processes. In this regard, we provide a complete description of the compact convex set of states of ${\rm CCR}(\ch)$ that are invariant under the action of all automorphisms induced in second quantization by unitaries of $\ch$. We prove that this set is a Bauer simplex, whose extreme states are either the canonical trace of the CCR algebra or Gaussian states with variance at least $1$.