Complexity of Mixed States in QFT and Holography

TL;DR

This paper investigates the complexity of Gaussian mixed states in free scalar quantum field theories using purification complexity, comparing results with holographic proposals and analyzing divergence structures and mutual complexity.

Contribution

It introduces the concept of mode-by-mode purifications and explores their optimality, providing new insights into the complexity of mixed states in QFT and holography.

Findings

Purification complexity aligns qualitatively with holographic complexity.

Mode-by-mode purifications are often near-optimal.

Complexity exhibits divergence structures and volume law behavior.

Abstract

We study the complexity of Gaussian mixed states in a free scalar field theory using the 'purification complexity'. The latter is defined as the lowest value of the circuit complexity, optimized over all possible purifications of a given mixed state. We argue that the optimal purifications only contain the essential number of ancillary degrees of freedom necessary in order to purify the mixed state. We also introduce the concept of 'mode-by-mode purifications' where each mode in the mixed state is purified separately and examine the extent to which such purifications are optimal. We explore the purification complexity for thermal states of a free scalar QFT in any number of dimensions, and for subregions of the vacuum state in two dimensions. We compare our results to those found using the various holographic proposals for the complexity of subregions. We find a number of qualitative similarities between the two in terms of the structure of divergences and the presence of a volume law. We also examine the 'mutual complexity' in the various cases studied in this paper.