Krylov Complexity and Mixed-State Phase Transition

TL;DR

The paper connects decoherence in mixed states to Krylov complexity by vectorizing the density matrix into a pure state in a double Hilbert space, turning decoherence into imaginary-time evolution generated by an effective Hamiltonian. Through analytic and numeric studies of two dephasing Ising models, it shows that Krylov complexity remains nonsingular across a strong-to-weak crossover (nearest-neighbor dephasing) but undergoes a genuine area-to-volume-law transition at a mixed-state SWSSB phase transition (infinite-range dephasing) with a critical point at $\tau_c=0.5$. The work derives explicit Lanczos coefficients, solves the Krylov dynamics analytically for both models, and confirms the transition signatures with Rényi-2 correlators, providing a complexity-based diagnostic for mixed-state phase structure and insights into error proliferation under decoherence. Overall, the framework reveals that Krylov complexity can quantify information loss and phase structure in decohered quantum systems, offering a bridge between symmetry, mixed-state phases, and dynamical complexity with potential applications to mixed topological and symmetry-protected phases.

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.

Figures (4)

• Figure 1: (a) Schematic illustration of the wave-packet dynamics in the Krylov basis $\vert K_{n}\rangle$, where each $\vert K_{n}\rangle$ represents an $n$-error state generated by decoherence. As time evolves, the wave packet $\psi_{n}(\tau)$ spreads toward higher-error sectors, reflecting the growth of quantum complexity. (b) Gram–Schmidt construction of the Krylov basis. The $(n+1)$-th basis state $\vert K_{n+1}\rangle$ is obtained by applying the noise operator $H$ to the $n$-error state $\vert K_{n}\rangle$, followed by projecting out all lower-error components $\vert K_{n}\rangle$, $\vert K_{n-1}\rangle$, ...
• Figure 2: (a) Time evolution of the wave-packets in the Krylov basis for $H^{\text{NN}}$ which exhibits an SWSSB crossover. The wave packet spreads smoothly and develops a Gaussian form as $\tau$ increases. The inset shows the corresponding Lanczos coefficients; $L=100$. (b) Normalized Krylov complexity $\mathcal{K}(\tau)/(L-1)$ for $H^{\text{NN}}$, displaying a nonsingular profile that indicates the absence of a phase transition. The inset shows the time evolution of the Rényi-2 correlator $\chi$ for different system sizes, further confirming the crossover behavior.
• Figure 3: (a) Time evolution of the wave packets in the Krylov basis for $H^{\text{IR}}$, which exhibits a genuine SWSSB phase transition. The packet remains localized within the low-error subspaces for $\tau\le0.5$ (dotted lines), and begins to spread when $\tau$ exceeds the critical point $\tau=0.5$. The inset shows the corresponding Lanczos coefficients; $L=100$. (b) Normalized Krylov complexity $\mathcal{K}(\tau)/L$ for $H^{\text{IR}}$. In the thermodynamic limit ($L=500$), the Krylov complexity shows a sharp transition at $\tau=0.5$. $\mathcal{K}$ follows an area-low scaling ($\mathcal{K}\sim\mathrm{constant}$) for $\tau<0.5$ and a volume-law scaling ($\mathcal{K}\sim\mathcal{O}(L^{1})$) in the long time limit. The inset compares numerical (colored) and analytical result in the thermodynamic limit (black) in the area-law regime, confirming excellent agreement.
• Figure E1: The truncation cutoff $\epsilon_{\rm trunc}$ dependence of $\chi$. (a) $H^{\text{NN}}$ with $L=60$. (b) $H^{\text{IR}}$ with $L=30$. For either case, one can see that $\chi$ successfully converges when the truncation cut-off is set to a sufficiently small value, such as $10^{-14}$. In the case of $H^{\text{IR}}$, the results obtained using the tensor-network method for small truncation cut-off align well with the result obtained by Eq. (E1).