History states of systems and operators

TL;DR

This paper explores the properties of discrete system-time history states in quantum systems, analyzing their entanglement, representations, and how they relate to unitary evolution and energy spread, with implications for quantum measurement.

Contribution

It introduces a detailed analysis of system-time history states, including their entanglement properties, representations, and the role of the unitary operator as an operator history state.

Findings

History states are maximally entangled when evolution operators form an orthogonal set.

Quadratic system-time entanglement entropy is analytically evaluated and bounded.

Simple measurements on the clock can determine overlaps and evolution operators efficiently.

Abstract

We discuss some fundamental properties of discrete system-time history states. Such states arise for a quantum reference clock of finite dimension and lead to a unitary evolution of system states when satisfying a static discrete Wheeler-DeWitt-type equation. We consider the general case where system-clock pairs can interact, analyzing first their different representations and showing there is always a special clock basis for which the evolution for a given initial state can be described by a constant Hamiltonian $H$. It is also shown, however, that when the evolution operators form a complete orthogonal set, the history state is maximally entangled for any initial state, as opposed to the case of a constant $H$, and can be generated with a simple two-clock setting. We then examine the quadratic system-time entanglement entropy, providing an analytic evaluation and showing it satisfies strict upper and lower bounds determined by the energy spread and the geodesic evolution connecting the initial and final states. We finally show that the unitary operator that generates the history state can itself be considered as an operator history state, whose quadratic entanglement entropy determines its entangling power. Simple measurements on the clock enable to efficiently determine overlaps between system states and also evolution operators at any two times.