Open-System Quantum Error Correction Conditions

In this work, we present QEC conditions for a system undergoing open-system dynamics. Here, we describe the noise on the system as originating from a joint completely-positive, trace-preserving map on the system-bath composite, after which we trace out bath degrees of freedom. Our noise model can be viewed as an intermediate picture between the standard system-only quantum channel model and a system-bath Hamiltonian noise model: It goes beyond a Markovian description for the system dynamics, and yet retains a quantum dynamical semigroup structure for the problem. Our general noise model fits naturally into many physical scenarios where one has a relatively strong coupling between the system and an "intermediate" bath, which also couples weakly to a large dissipative bath. Despite the physical motivation from an intermediate bath, our noise model is mathematically general however, and in turn contains both system-only quantum channel model and system-bath Hamiltonian noise model as extreme cases.

We derive and study the QEC conditions for our general noise model, with the emphasis that the recovery operation acts on the system only. When the noise is only approximately correctable, we obtain a lower bound for the performance of the QEC, as quantified by worst-case fidelity.