Coherent information as a mixed-state topological order parameter of fermions

TL;DR

This work links quantum error correction to topological phases by recasting the coherent information (CI) of a decohered toric code as a mixed-state topological order parameter for disordered Majorana fermions. Through exact mappings to the random-bond Ising model and its Majorana representation, CI becomes a boundary-sensitive measure that signals the decoding threshold via a zero-crossing tied to vortex fugacity and self-duality. The Majorana formulation reveals a bulk–vortex correspondence, where the presence of Majorana zero modes trapped in vortices corresponds to a robust code space, and the thermodynamic limit yields a quantized order parameter. Numerically, CI captures thresholds across multiple 2D stabilizer codes, underscoring the broad applicability of this approach and suggesting practical means to estimate error thresholds and phase diagrams in quantum memories.

Abstract

Quantum error correction protects quantum information against decoherence provided the noise strength remains below a critical threshold. This threshold marks the critical point for the decoding phase transition. Here we connect this transition in the toric code to a topological phase transition in disordered Majorana fermions at high temperatures. A quantum memory in the error correctable phase is captured by the presence of a Majorana zero mode, trapped in vortex defects associated with twisted boundary conditions. These results are established by expressing the coherent information, which measures the amount of recoverable quantum information in a given noisy code, in terms of a mixed-state topological order parameter of fermions. Our work hints at a broader connection of the robustness of quantum information in stabilizer codes and mixed-state topological phase transitions in symmetry protected fermion matter.

Figures (16)

• Figure 1: Overview of different representations of the coherent information (CI) for the toric code under both bit-flip and phase errors. Here, the CI measures the residual information that is recoverable under decoherence, with the zero point of the CI marking the decoding phase transition. This point acquires different physical interpretations in different representations: It corresponds to the critical error threshold in the decohered toric code (representation 1), and the low/high temperature self-duality point in the random-bond Ising model (RBIM, representation 2) fan2024prxQ. In this work, we create a link to symmetry-protected topological quantum matter in terms of disordered Majorana fermions (representation 3). In this representation, the coherent information is directly tied to a mixed-state topological order parameter of fermions under twisted boundary conditions. The decoding phase transition is described by a topological phase transition of fermions in a mixed quantum state. A non-trivial quantum memory in the error correctable phase is captured by the presence of a Majorana zero mode, induced by vortex defects associated to the boundary conditions. For a further summary of results, see Sec. \ref{['sec:TSC']} and Fig. \ref{['fig:braiding']}.
• Figure 2: Summary of the relation between the coherent information (CI) zero crossing point, the critical point, and the self-dual point. The CI zero point ($I_c(p,\ N)=0$) can depend on both the error rate $p$ and the system size $N$, as schematically illustrated by the blue line in panel (a). Mapping the decohered toric code to a disordered Majorana model reveals the physical meaning of $I_c=0$ as the zero vortex fugacity point. The zero vortex fugacity point rapidly converges to a renormalization group (RG) fixed point, leading to an emergent coincidence between the zero CI point and the critical point in the thermodynamic limit ($p=p_c$, red star in (a)). Additionally, we find an exact correspondence between the zero CI point and the self-duality point in the random-bond Ising model (RBIM) for any system size $N$ (blue frame in (a)), where the RBIM arises from a statistical mechanics mapping of the CI. Together, these results establish the relation between the zero CI point, the critical point, and the self-dual point, as shown in panel (b).
• Figure 3: Coherent information setup (a) and its dependence on the error rate $p$ for a single qubit under bit-flip and phase errors (b). In (a), $Q$ represents the quantum memory with density matrix $\rho_{Q}$, purified to $|\Psi_{RQ}\rangle$ by introducing a reference qubit $R$, i.e., $\rho_{Q}=\text{Tr}_{R}\left(|\Psi_{RQ}\rangle\langle\Psi_{RQ}|\right)$. Decoherence occurs due to coupling with the environment $E$, evolving the initial state $|\Psi_{RQ}\rangle\otimes|0_{E}\rangle$ to $|\Psi_{RQ^{\prime}E^{\prime}}\rangle=U_{QE}|\Psi_{RQ}\rangle\otimes|0_{E}\rangle$. In (b), $I_c$ and $I_c^{(2,\ 3)}$ represent the coherent information and its Rényi-$2, 3$ counterparts, respectively, both of which decrease monotonically with $p$.
• Figure 4: Illustration of the toric code: (a) shows the stabilizers $A_{s}$ and $B_{p}$ in the toric code Hamiltonian (Eq. \ref{['eq:Ham_toric']}), as well as the non-contractible loops in the $x$- and $y$- direction, $\mathcal{L}_{x_{1/2}}$ and $\mathcal{L}_{y_{1/2}}$. (b) depicts a string excitation with $A_{s}=-1$ at both ends. Applying $B_{p}$ (blue shaded area), changes the string's shape, while keeping the ends intact. (c) plots error chains, or string excitations created by chains of bit-flip ($\mathcal{S}_{X}$) or phase ($\mathcal{S}_{Z}$) errors. The error corrupted density matrix $\rho_{Q^\prime/RQ^\prime}$ (Eq. \ref{['eq:rho_RQ_p']}) represents a weighted ensemble of these chains. (d) illustrates that in the Rényi-$n$ entropy (used for the Rényi-$n$ CI in Eq. \ref{['eq:Renyi_n_CI_2qubit']}), an error chain (e.g., $\mathcal{S}_{X}$), contributes only when $\mathcal{S}_{X}^{\left(f\right)}$ and $\mathcal{S}_{X}^{\left(f+1\right)}$ in adjacent replicas form a closed loop.
• Figure 5: Illustration of (a) the equivalence of anti-periodic boundary conditions with a $\mathbb{Z}_2$ flux line insertion, and (b) the random-bond Ising model (RBIM) in the one-site limit. In (a), an anti-periodic boundary along the $y$-axis is equivalent to a $\mathbb{Z}_2$ flux line ($\eta=-1$) along the $x$-axis, resulting in $\prod_{\langle i,\ j\rangle\in \mathcal{L}_y}\eta_{ij}=-1$ for any non-contractible loop in the $y$ direction. In (b), the hollow circle represents the Ising spin, with the $\mathbb{Z}_2$ flux lines $\eta_{xx}$ and $\eta_{yy}$ changing the interaction sign (green double line).