# Probing the topology of density matrices

## TL;DR

This paper introduces the ensemble geometric phase (EGP), a new observable that preserves topological invariants in mixed quantum states, enabling topological detection even at finite temperature.

## Contribution

It proposes the EGP as a measurable topological invariant for mixed states and demonstrates its robustness in one-dimensional Gaussian quantum systems.

## Key findings

- EGP remains quantized in mixed Gaussian states
- EGP detects spectral singularities in density matrices
- Proposed interferometric measurement setup for ultracold atoms

## Abstract

The mixedness of a quantum state is usually seen as an adversary to topological quantization of observables. For example, exact quantization of the charge transported in a so-called Thouless adiabatic pump is lifted at any finite temperature in symmetry-protected topological insulators. Here, we show that certain directly observable many-body correlators preserve the integrity of topological invariants for mixed Gaussian quantum states in one dimension. Our approach relies on the expectation value of the many-body momentum-translation operator, and leads to a physical observable --- the "ensemble geometric phase" (EGP) --- which represents a bona fide geometric phase for mixed quantum states, in the thermodynamic limit. In cyclic protocols, the EGP provides a topologically quantized observable which detects encircled spectral singularities ("purity-gap" closing points) of density matrices. While we identify the many-body nature of the EGP as a key ingredient, we propose a conceptually simple, interferometric setup to directly measure the latter in experiments with mesoscopic ensembles of ultracold atoms.