Exact Calculations of Coherent Information for Toric Codes under Decoherence: Identifying the Fundamental Error Threshold

Abstract

The toric code is a canonical example of a topological error-correcting code. Two logical qubits stored within the toric code are robust against local decoherence, ensuring that these qubits can be faithfully retrieved as long as the error rate remains below a certain threshold. Recent studies have explored such a threshold behavior as an intrinsic information-theoretic transition, independent of the decoding protocol. These studies have shown that information-theoretic metrics, calculated using the Renyi (replica) approximation, demonstrate sharp transitions at a specific error rate. However, an exact analytic expression that avoids using the replica trick has not been shown, and the connection between the transition in information-theoretic capacity and the random bond Ising model (RBIM) has only been indirectly established. In this work, we present the first analytic expression for the coherent information of a decohered toric code, thereby establishing a rigorous connection between the fundamental error threshold and the criticality of the RBIM.

5 pages + 5 pages (supplementary material), 3 figures

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• Revisiting Nishimori multicriticality through the lens of information measures
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• Holographic duality between bulk topological order and boundary mixed-state order
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• Universal Decay of Mutual Information and Conditional Mutual Information in Gapped Pure- and Mixed-State Quantum Matter
• Non-Commutative weak measurements: Entanglement, Symmetry Breaking, and the Role of Readout
• Phases of Floquet code under local decoherence
• Spacetime Markov length: a diagnostic for fault tolerance via mixed-state phases
• Boundary Criticality at the Nishimori Multicritical Point
• Error Threshold of SYK Codes from Strong-to-Weak Parity Symmetry Breaking