Quantum State Reduction: Generalized Bipartitions from Algebras of Observables

Reduced density matrices are a powerful tool in the analysis of entanglement structure, coarse-grained dynamics, decoherence, and the emergence of classicality. While one often uses the partial trace map to produce a reduced density matrix, in many natural situations (such as limited resolution experiments) this reduction may not be achievable. We investigate the general problem of identifying how the quantum state reduces given a restriction on the observables where the appropriate state-reduction map can be defined via a generalized bipartition, which is associated with the structure of irreducible representations of the algebra generated by the restricted set of observables. One of our main technical results is a general algorithm for finding irreducible representations of matrix algebras. We demonstrate the viability of this approach with two examples of limited-resolution observables. The definition of quantum state reductions can also be extended beyond algebras of observables by a more flexible notion of bipartition, the partial bipartition, which describes coarse-grainings preserving information about a limited set (not necessarily algebra) of observables.