Quantum state reduction: Generalized bipartitions from algebras of observables
Abstract
Reduced density matrices are a powerful tool in the analysis of entanglement structure, approximate or coarsegrained dynamics, decoherence, and the emergence of classicality. It is straightforward to produce a reduced density matrix with the partialtrace 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 is reduced 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 statereduction 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 limitedresolution 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 onedimensional Ising model. Our results have relevance for quantum information, bulk reconstruction in holography, and quantum gravity.
 Publication:

Physical Review A
 Pub Date:
 March 2020
 DOI:
 10.1103/PhysRevA.101.032303
 arXiv:
 arXiv:1909.12851
 Bibcode:
 2020PhRvA.101c2303K
 Keywords:

 Quantum Physics;
 High Energy Physics  Theory
 EPrint:
 78 pages, 10 figures, comments welcome