Krylov complexity of purification

TL;DR

This paper explores the relationship between the complexity of mixed quantum states and their purification, revealing new inequalities, bounds, and potential links to gravity duals across various quantum systems.

Contribution

It introduces novel inequalities connecting complexities of mixed and purified states, and proposes Krylov mutual complexity as a tool for understanding gravity duals.

Findings

The spread complexity of purification of a vacuum into a thermal state equals the average Rindler particles.

Krylov complexity adheres to Lloyd-like bounds, relating to quantum speed limits.

Subadditivity of Krylov complexities contrasts with holographic volume complexity results.

Abstract

In quantum systems, purification can map mixed states into pure states and a non-unitary evolution into a unitary one by enlarging the Hilbert space. We establish a connection between the complexities of mixed quantum states and their purification, proposing new inequalities among these complexities. By examining single qubits, two-qubit Werner states, eight-dimensional Gaussian random unitary ensembles, and infinite-dimensional systems, we demonstrate how these relationships manifest across a broad class of systems. We find that the spread complexity of purification of a vacuum state evolving into a thermal state equals the average number of Rindler particles. This complexity is also shown to adhere to the Lloyd-like bound, indicating a further relation to the quantum speed limit. Finally, using mutual Krylov complexity, we observe subadditivity of the Krylov complexities, which contrasts with known results from holographic volume complexity. We put forward Krylov mutual complexity as a diagnosis of a potential gravity dual of Krylov complexities.