System-time entanglement in a discrete time model

TL;DR

This paper introduces a quantum model using system-clock entanglement to measure evolution distinguishability, providing a quantum circuit implementation and analyzing properties for various states and Hamiltonians.

Contribution

It proposes a novel measure of quantum evolution based on system-time entanglement and offers a quantum circuit model for efficient representation of discrete time evolution.

Findings

System-time entanglement vanishes for stationary states.

Maximum entanglement occurs when the system jumps to orthogonal states.

The measure relates to energy eigenstate spread and satisfies an energy-time uncertainty relation.

Abstract

We present a model of discrete quantum evolution based on quantum correlations between the evolving system and a reference quantum clock system. A quantum circuit for the model is provided, which in the case of a constant Hamiltonian is able to represent the evolution over $2^n$ time steps in terms of just $n$ time qubits and $n$ control gates. We then introduce the concept of system-time entanglement as a measure of distinguishable quantum evolution, based on the entanglement between the system and the reference clock. This quantity vanishes for stationary states and is maximum for systems jumping onto a new orthogonal state at each time step. In the case of a constant Hamiltonian leading to a cyclic evolution it is a measure of the spread over distinct energy eigenstates, and satisfies an entropic energy-time uncertainty relation. The evolution of mixed states is also examined. Analytical expressions for the basic case of a qubit clock, as well as for the continuous limit in the evolution between two states, are provided.