Eigenstate Thermalization and Representative States on Subsystems

TL;DR

This paper explores how, in certain quantum systems, expectation values within a subsystem can be accurately computed using a pure state on that subsystem alone, leveraging the eigenstate thermalization hypothesis.

Contribution

It introduces the concept of 'representative states' in subsystems, showing they can approximate expectation values without full knowledge of the environment, based on ETH insights.

Findings

Expectation values inside a subsystem can be approximated by pure states.

Representative states exist under conditions consistent with ETH.

Controlled errors are achievable in these approximations.

Abstract

We consider a quantum system A U B made up of degrees of freedom that can be partitioned into spatially disjoint regions A and B. When the full system is in a pure state in which regions A and B are entangled, the quantum mechanics of region A described without reference to its complement is traditionally assumed to require a reduced density matrix on A. While this is certainly true as an exact matter, we argue that under many interesting circumstances expectation values of typical operators anywhere inside A can be computed from a suitable pure state on A alone, with a controlled error. We use insights from quantum statistical mechanics - specifically the eigenstate thermalization hypothesis (ETH) - to argue for the existence of such "representative states".