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

Ze-Min Huang1, Luis Colmenarez2,3, Markus Müller2,3, and Sebastian Diehl1

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Phys. Rev. Research 7, 043009 – Published 1 October, 2025

DOI: https://doi.org/10.1103/fx56-8nvy

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.

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Phys. Rev. Research 7, 043009– Published 1 October, 2025
  • Received 21 December 2024
  • Revised 21 March 2025
  • Accepted 5 August 2025
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