Stability of mixed-state quantum phases via finite Markov length

Abstract

For quantum phases of Hamiltonian ground states, the energy gap plays a central role in ensuring the stability of the phase as long as the gap remains finite. We propose Markov length, the length scale at which the quantum conditional mutual information (CMI) decays exponentially, as an equally essential quantity characterizing mixed-state phases and transitions. For a state evolving under a local Lindbladian, we argue that if its Markov length remains finite along the evolution, then it remains in the same phase, meaning there exists another quasi-local Lindbladian evolution that can reverse the former one. We apply this diagnostic to toric code subject to decoherence and show that the Markov length is finite everywhere except at its decodability transition, at which it diverges. CMI in this case can be mapped to the free energy cost of point defects in the random bond Ising model. This implies that the mixed state phase transition coincides with the decodability transition and also suggests a quasi-local decoding channel.

Figures (4)

• Figure 1: (left) Quantum conditional mutual information $I(A:C|B)$ quantifies how non-local is the correlation between $A$ and its complement. When $I(A:C|B)$ decays exponentially with $B$'s width $r$, we call the corresponding length scale the Markov length $\xi$. (right) The dark grey line is a path of mixed-states generated from local Lindbladian evolution, i.e.$\rho_t=\mathcal{T}\exp(\int_0^t\mathcal{L}(\tau)\mathrm{d}\tau)[\rho_0]$. For each segment in which $\xi$ remains finite, $e.g.$ below (above) the dashed line, we argue there exists a quasi-local Lindbladian $\widetilde{\mathcal{L}}_{1(2)}$ that reverses $\mathcal{L}$'s action. Thus states within each segment are in the same mixed-state phase.
• Figure 2: Reversal circuit for continuous evolution--- (top left) Two layers of the forward circuit $\mathcal{G}$ acting from bottom to top. Each box is a quantum channel $\mathcal{E}_{x,t}=\exp(\delta t\mathcal{L}_{x,t})$. (top right) Gates in each layer are reorganized into multiple layers so that any two gates in a layer are at least distance $2r$ separated (in the figure $r=2$). (bottom) The reversal circuit $\widetilde{\mathcal{G}}$ constructed from the reorganized $\mathcal{G}$ by replacing each $\mathcal{E}_{x,t}$ with its reversal $\widetilde{\mathcal{E}}_{x,t}$ (grey box) defined in Eq.\ref{['eq: reversal_circuit']}. The reversal circuit acts from top to bottom.
• Figure 3: CMI of dephased toric code state ---(a) Partition with $A$ fixed and varying $r$ (width of $B,C$). (b)$I(A:C|B)$ peaks around $p_c\approx 0.11$ and peak size decays with $r$. (c) Finite-size collapse with the scaling ansatz Eq.\ref{['eq: scaling_form']}, with $(p_c, \nu, \alpha)=(0.11, 1.8, 1.1)$. (d) Above ($p=0.15$) or below ($p=0.05$) the critical point, CMI decays exponentially with $r$, in contrast to power-law decay at the critical point $p_c \approx 0.11$.
• Figure 4: Illustration of a non-simply connected region Q. Only qubits (edges) that belong to $Q$ is drawn. Supports of the two non-local operators $A_{\widetilde{\square}}$ and $B_{\widetilde{+}}$ surrounding the hole are denoted with green and blue edges, respectively.