  Reduced density matrices are a powerful tool in the analysis of entanglement
structure, approximate or coarse-grained dynamics, decoherence, and the
emergence of classicality. It is straightforward to produce a reduced density
matrix with the partial-trace map by ``tracing out'' part of the quantum state,
but in many natural situations this reduction may not be achievable. We
investigate the general problem of identifying how the quantum state reduces
given a restriction on the observables. For example, in an experimental
setting, the set of observables that can actually be measured is usually modest
(compared to the set of all possible observables) and their resolution is
limited. In such situations, 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, not inherently
numeric, 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. To accomplish this task we
introduce a more flexible notion of bipartition, the partial bipartition, which
describes coarse-grainings preserving information about a limited set (not
necessarily algebra) of observables. We describe a variational method to choose
the coarse-grainings most compatible with a specified Hamiltonian, which
exhibit emergent classicality in the reduced state space. We apply this
construction to the concrete example of the 1-D Ising model. Our results have
relevance for quantum information, bulk reconstruction in holography, and
quantum gravity.

Quantum Physics;High Energy Physics - Theory