  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.
