Recovery With Incomplete Knowledge: Fundamental Bounds on Real-Time Quantum Memories

The recovery of fragile quantum states from decoherence is the basis of building a quantum memory. Many recovery techniques, such as quantum error correction, rely on the prior knowledge of the environment noise parameter to achieve their best performance. However, such parameters are likely to drift in time in the context of implementing long-time quantum memories. This necessitates the use of a ``spectator'' system, which makes an estimate of the noise parameter in real time, then feeds the outcome back to the recovery protocol as a classical side-information. In this article, I present information-theoretic bounds on the performance of such a spectator-based recovery. First, I show that there is a fundamental bound in the performance of any recovery operation, as a function of the entanglement fidelity of the overall dynamics. Then, I provide information-theoretic characterizations of the incomplete knowledge of the noise parameter to the lower bound. Finally, I provide fundamental bounds for multicycle recovery in the form of recurrence inequalities. The latter suggests that incomplete knowledge could be an advantage. These results are illustrated for the approximate [4,1] code of the amplitude-damping channel.