Complexity of mixed Gaussian states from Fisher information geometry

TL;DR

This paper investigates the complexity of preparing mixed Gaussian states in harmonic lattices using Fisher information geometry, providing explicit measures and numerical results for thermal and reduced states.

Contribution

It introduces a geometric approach to quantify the complexity of mixed Gaussian states, including spectrum and basis complexity, and analyzes purification and specific lattice cases.

Findings

Explicit formulas for complexity measures in Gaussian states

Numerical results for thermal states and reduced density matrices

Analysis of purification process for mixed states

Abstract

We study the circuit complexity for mixed bosonic Gaussian states in harmonic lattices in any number of dimensions. By employing the Fisher information geometry for the covariance matrices, we consider the optimal circuit connecting two states with vanishing first moments, whose length is identified with the complexity to create a target state from a reference state through the optimal circuit. Explicit proposals to quantify the spectrum complexity and the basis complexity are discussed. The purification of the mixed states is also analysed. In the special case of harmonic chains on the circle or on the infinite line, we report numerical results for thermal states and reduced density matrices.