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Topological quantum memory can protect information against local errors up to finite error thresholds. Such thresholds are usually determined based on the success of decoding algorithms rather than the intrinsic properties of the mixed states describing corrupted memories. Here we provide an intrinsic characterization of the breakdown of topological quantum memory, which both gives a bound on the performance of decoding algorithms and provides examples of topologically distinct mixed states. We employ three information-theoretical quantities that can be regarded as generalizations of the diagnostics of ground-state topological order, and serve as a definition for topological order in error-corrupted mixed states. We consider the topological contribution to entanglement negativity and two other metrics based on quantum relative entropy and coherent information. In the concrete example of the two-dimensional (2D) Toric code with local bit-flip and phase errors, we map three quantities to observables in 2D classical spin models and analytically show they all undergo a transition at the same error threshold. This threshold is an upper bound on that achieved in any decoding algorithm and is indeed saturated by that in the optimal decoding algorithm for the Toric code. Published by the American Physical Society 2024

How long does it take to entangle two distant qubits in a quantum circuit evolved by generic unitary dynamics? We show that if the time evolution is followed by measurements of all but two infinitely separated test qubits, then the entanglement between them can undergo a phase transition and become nonzero at a finite critical time t_{c}. The fidelity of teleporting a quantum state from an input qubit to an infinitely distant output qubit shows the same critical onset. Specifically, these finite-time transitions occur in short-range interacting two-dimensional random unitary circuits and in sufficiently long-range interacting one-dimensional circuits. The phase transition is understood by mapping the random continuous-time evolution to a finite-temperature thermal state of an effective spin Hamiltonian, where the inverse temperature equals the evolution time in the circuit. In this framework, the entanglement between two distant qubits at times t>t_{c} corresponds to the emergence of long-range ferromagnetic spin correlations below the critical temperature. We verify these predictions using numerical simulation of Clifford circuits and propose potential realizations in existing platforms for quantum simulation.

How long does it take to entangle two distant qubits in a quantum circuitevolved by generic unitary dynamics? We show that if the time evolution isfollowed by measurements of all but two infinitely separated test qubits, thenthe entanglement between them can undergo a phase transition and become nonzeroat a finite critical time $t_c$. The fidelity of teleporting a quantum statefrom an input qubit to an infinitely distant output qubit shows the samecritical onset. Specifically, these finite-time transitions occur inshort-range interacting two-dimensional random unitary circuits and insufficiently long-range interacting one-dimensional circuits. The phasetransition is understood by mapping the random continuous-time evolution to afinite-temperature thermal state of an effective spin Hamiltonian, where theinverse temperature equals the evolution time in the circuit. In thisframework, the entanglement between two distant qubits at times $t>t_c$corresponds to the emergence of long-range ferromagnetic spin correlationsbelow the critical temperature. We verify these predictions using numericalsimulation of Clifford circuits and propose potential realizations in existingplatforms for quantum simulation.