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arXiv · 2510.22542

Krylov Complexity and Mixed-State Phase Transition

Hung-Hsuan Teh·Takahiro Orito
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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.

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Hung-Hsuan Teh, Takahiro Orito. 2025-10-26. Krylov Complexity and Mixed-State Phase Transition. https://arxiv.org/abs/2510.22542

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