Krylov Complexity and Mixed-State Phase Transition

TL;DR

This paper introduces a framework linking quantum decoherence and complexity via Krylov space, revealing phase transition behaviors in Krylov complexity related to mixed-state symmetry-breaking.

Contribution

It develops a unified approach connecting decoherence processes with quantum complexity growth through Krylov space analysis, highlighting phase transition signatures.

Findings

Krylov complexity remains nonsingular during certain symmetry-breaking crossovers.

Krylov complexity exhibits a singular transition in genuine phase transitions.

The framework applies to dephasing quantum channels and reveals new phase transition phenomena.

Abstract

We establish a unified framework connecting decoherence and quantum complexity. By vectorizing the density matrix into a pure state in a double Hilbert space, a decoherence process is mapped to an imaginary-time evolution. Expanding this evolution in the Krylov space, we find that the $n$-th Krylov basis corresponds to an $n$-error state generated by the decoherence, providing a natural bridge between error proliferation and complexity growth. Using two dephasing quantum channels as concrete examples, we show that the Krylov complexity remains nonsingular for strong-to-weak spontaneous symmetry-breaking (SWSSB) crossovers, while it exhibits a singular area-to-volume-law transition for genuine SWSSB phase transitions, intrinsic to mixed states.