Approximate recovery with locality and symmetry constraints

Numerous quantum many-body systems are characterized by either fundamental or emergent constraints---such as gauge symmetries or parity superselection for fermions---which effectively limit the accessible observables and realizable operations. Moreover, these constraints combine non-trivially with the potential requirement that operations be performed locally. The combination of symmetry and locality constraints influence our ability to perform quantum error correction in two counterposing ways. On the one hand, they constrain the effect of noise, limiting its possible action over the quantum system. On the other hand, these constraints also limit our ability to perform quantum error correction, or generally to reverse the effect of a noisy quantum channel. We analyze the conditions that local channels should satisfy in the constrained setting, and characterize the resulting optimal decoding fidelity. In order to achieve this result, we introduce a concept of local complementary channel, and prove a new local information-disturbance tradeoff.