Quantum State Reduction: Generalized Bipartitions from Algebras of Observables

TL;DR

This paper develops a generalized framework for quantum state reduction based on restricted observable sets, introducing algorithms for irreducible representations and applications to models like the 1-D Ising, with implications for quantum information and holography.

Contribution

It introduces a generalized bipartition framework for quantum state reduction based on observable restrictions and provides algorithms for irreducible representations of matrix algebras.

Findings

Demonstrates a general algorithm for finding irreducible representations of matrix algebras.

Applies the framework to limited-resolution observables in quantum models.

Shows how to select coarse-grainings compatible with Hamiltonians to observe emergent classicality.

Abstract

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.