On the set of reduced states of translation invariant, infinite quantum systems

TL;DR

This paper investigates the mathematical structure of reduced states in infinite translation-invariant quantum systems, proving they cannot be fully characterized by finite algebraic descriptions, highlighting fundamental limitations in their approximation.

Contribution

It proves that the set of reduced states is not semialgebraic, indicating intrinsic complexity, and shows that elementary transcendental functions cannot simplify their description.

Findings

The set of reduced states is not semialgebraic.

Hierarchies of algebraic approximations become tight only with infinitely many variables.

Elementary transcendental functions cannot provide a finite description.

Abstract

The set of two-body reduced states of translation invariant, infinite quantum spin chains can be approximated from inside and outside using matrix product states and marginals of finite systems, respectively. These lead to hierarchies of algebraic approximations that become tight only in the limit of infinitely many auxiliary variables. We show that this is necessarily so for any algebraic ansatz by proving that the set of reduced states is not semialgebraic. We also provide evidence that additional elementary transcendental functions cannot lead to a finitary description.