Information Critical Phases under Decoherence

TL;DR

This work defines an information critical phase in mixed-state quantum memory by identifying a divergent Markov length $\xi_M=\infty$ while the conventional correlation length remains finite, and shows its realization in decohered $\mathbb{Z}_{N}$ Toric codes for $N>4$. By mapping the decohered density matrices to partition functions of disordered $\mathbb{Z}_{N}$ clock models along the Nishimori line, the authors connect information-theoretic diagnostics—coherent information $I^{(c)}$ and conditional mutual information $I(A{:}C|B)$—to phases of these classical statistical systems, including an intermediate quasi-long-range-ordered (QLRO) regime with a fractional memory plateau. They reveal that the information critical phase corresponds to a Coulombic pure-state decomposition with gapless photon-like modes and, in the ungauged dual, to a superfluid-like SWSSB phase with emergent $U(1)$ symmetry. A novel decoding protocol combining minimum-cost flow with worm refinement is proposed, shown to be optimal at the decodability threshold, while it fails in the information-critical phase, highlighting the need for new decoding strategies to harness fractional protection. Overall, the work broadens quantum memory theory by introducing gapless mixed-state phases that preserve a finite fraction of logical information and by linking information theory to disordered clock-model criticality.

Abstract

Quantum critical phases are extended regions of phase space characterized by a diverging correlation length. By analogy, we define an information critical phase as an extended region of a mixed state phase diagram where the Markov length, the characteristic length scale governing the decay of the conditional mutual information (CMI), diverges. We demonstrate that such a phase arises in decohered $\mathbb{Z}_{N}$ Toric codes by assessing both the CMI and the coherent information, the latter quantifying the robustness of the encoded logical qudits. For $N>4$, we find that the system hosts an information critical phase intervening between the decodable and non-decodable phases where the coherent information saturates to a fractional value in the thermodynamic limit, indicating that a finite fraction of logical information is still preserved. We show that the density matrix in this phase can be decomposed into a convex sum of Coulombic pure states, where gapped anyons reorganize into gapless photons. We further consider the ungauged $\mathbb{Z}_{N}$ Toric code and interpret its mixed state phase diagram in the language of strong-to-weak spontaneous symmetry breaking. We argue that in the dual model, the information critical phase arises because the spontaneously broken off-diagonal $\mathbb{Z}_{N}$ symmetry gets enhanced to a U(1) symmetry, resulting in a novel superfluid phase whose gapless modes involve coherent excitations of both the system and the environment. Finally, we propose an optimal decoding protocol for the corrupted $\mathbb{Z}_{N}$ Toric code and evaluate its effectiveness in recovering the fractional logical information preserved in the information critical phase. Our findings identify a gapless analog for mixed-state phases that still acts as a fractional topological quantum memory, thereby extending the conventional paradigm of quantum memory phases.

Figures (7)

• Figure 1: (a) Information critical phase. The Markov length $\xi_M$ sets the length scale for the spatial decay of the conditional mutual information $I(A\,{:}\,C|B)$, analogous to how the correlation length $\xi$ controls the decay of the mutual information. An information critical phase is an extended region where $\xi_M=\infty$. (b) Phase Diagram. We map the mixed-state phase diagram of the $\mathbb{Z}_{N>4}$ toric code as a function of effective temperatures $T^x$ and $T^z$, which parametrize the strengths of $X$- and $Z$-dephasing, respectively. Throughout the diagram $\xi=0$ while the Markov length can diverge. The ungauged description admits a strong-to-weak SSB interpretation. The dual spin model evolves from a strongly $\mathbb{Z}_N$-symmetric phase into an intermediate regime where the broken axial $\mathbb{Z}_N$ symmetry effectively enhances to an emergent $U(1)$ symmetry, giving rise to a quasi-long range ordered superfluid phase. Further increasing the decoherence strength eventually causes full SWSSB of the $\mathbb{Z}_N$ symmetry. (c) Coherent information. For $N=8$, we plot the normalized $I^{(c)}$ as a function of $T$ across different system sizes. It approaches its maximal value for $T<0.185$, and remains finite in the information-critical regime $0.185<T<0.38$, indicating fractionally recoverable logical information. For $T>0.38$, it decays to zero. (d) Logical error rate. We devise an optimal decoder that combines a minimum-integer-cost-flow solver with a subsequent worm-algorithm refinement. The decoder has a threshold that coincides with the point at which the Toric code exits the perfectly decodable phase. Within the fractionally decodable phase, we find that the decoder is unsuccessful, indicating the need for a different type of decoder.
• Figure 2: The plaquette and vertex stabilizers of the $\mathbb{Z}_{N}$ Toric code.
• Figure 3: Coherent Information. We numerically compute the coherent information of the $\mathbb{Z}_{N}$ toric code for $N = 2, 4, 6,$ and $8$ under $X$-type dephasing. $L$ denotes the linear size of the Toric code. The decoherence parameters $p^{x}_{k}$ are fixed as in Eq. \ref{['eq:Fixed_Parameters']} with the decoherence temperature $T \equiv T^{x}$ acting as the sole control parameter. To enable a direct comparison across different values of $N$, the vertical axis is plotted in logarithmic units of base $N$. For $N \leq 4$, we observe a single phase transition in the disordered clock model, separating an ordered and a disordered phase. In the ordered phase, the coherent information saturates to $2 \log N$, indicating perfect decodability of the encoded logical information. In contrast, in the disordered phase the coherent information vanishes in the thermodynamic limit, signaling the complete loss of encoded quantum information. For $N > 4$, however, the corresponding clock model exhibits three distinct phases: an ordered phase, a disordered phase, and an intervening QLRO phase. Remarkably, within the QLRO phase the coherent information saturates to a nonzero fractional value of $2\log N$ as $L \to \infty$, indicating that a finite fraction of logical information is still protected. The extent of this fractionally decodable phase grows with increasing $N$, and we find that the second critical temperature $T^{(2)}_c \sim 0.38$ is approximately independent of the value of $N$. Interestingly, the locations of the critical points are found to agree to excellent precision with the self-dual identity given in Eq. \ref{['eq:self_duality']}.
• Figure 4: Wilson Loop Expectation. We plot the expectation value of the Wilson loop operator $\langle X_{\mathcal{C}}^{q} \rangle$ (shown here for $q=1$) with respect to the METTS wavefunction $\ket{\psi_{Q}(\textbf{1})}$ (Eq. \ref{['eq:METTS_Ansatz']}) as a function of the decoherence temperature for the $\mathbb{Z}_8$ Toric code. In the decodable phase, $\langle X_{\mathcal{C}}^{q} \rangle$ vanishes in the thermodynamic limit, signaling topological order, whereas in the non-decodable phase it saturates to 1 indicating topological triviality. In the information critical phase, $\langle X_{\mathcal{C}}^{q} \rangle$ approximately saturates to a constant that is roughly independent of system size and increases monotonically with decoherence strength. The inset shows the dependence of this saturation value on the loop charge $q \mod N$ within the information critical phase (at $T = 0.3$). We find that the saturation is sharply decreasing with increasing $q$ suggesting that the system can only tunnel within a restricted subset of topological sectors. Together with the power-law scaling of open Wilson strings in this phase, this behavior supports the interpretation in terms of Coulombic states with emergent gapless photons.
• Figure 5: Rényi-1 Correlator. We plot the Rényi-1 correlator in the Kramers Wannier dual of the decohered $\mathbb{Z}_{8}$ Toric code for a range of decoherence temperatures, where $r$ denotes the separation between the dual spins. We find that for $T<T^{(1)}_{c}$, corresponding to the decodable phase of the Toric code, $R_{1}(r) \sim e^{-r\alpha}$ indicating the persistence of strong $\mathbb{Z}_{N}$ symmetry. For $T>T^{(2)}_{c}$, corresponding to the non-decodable phase, $R_{1}(r) \sim \mathcal{O}(1)$ as $r \rightarrow \infty$ signaling SWSSB. For $T^{(1)}_{c}\leq T\leq T^{(2)}_{c}$, we find that $R_{1}(r)\sim r^{-\eta(T)}$ exhibits an approximate power-law scaling. This points to a superfluid phase with quasi-long-range order.