Efficient decoders for symmetric monitored quantum dynamics

We consider learnability transitions in monitored quantum systems with a global symmetry. We propose a systematic way of constructing ``decoders'' for such systems, which allow the observer to distinguish states with distinct global quantum numbers from local measurements. Such decoders are obtained from marginalizing over the unitary part of the dynamics, corresponding to quantum channels that can be efficiently simulated using tensor network methods. We test numerically the accuracy of such decoders for various symmetry groups, and compare them to optimal (but non-scalable) decoders.