Orthogonal Evolution and Anticipation

TL;DR

This paper explores how quantum states evolving into orthogonal states can be anticipated at half-step times, revealing future and past states with significant probability, and introduces a framework for analyzing these anticipation effects.

Contribution

It introduces a novel analysis of anticipation amplitudes in quantum state evolution, characterizes the spectrum, and establishes analogs to classical relations, with applications in quantum information.

Findings

Anticipation amplitudes reveal information about future and past states.

Probability of measuring states N steps apart exceeds a constant lower bound.

Analysis includes spectrum characterization and a sampling scheme for anticipation probabilities.

Abstract

Quantum states evolving at equidistant steps into a set of mutually orthogonal states of finite or infinite cardinality p exhibit an interesting physical effect. The analysis of the amplitudes of the state at half the step time with the elements of this set (the anticipation amplitudes) shows, that for randomly chosen states measurements of the state at half-step time reveal information about the states at full step time, anticipating future states and reflecting past states with significant probability. For fixed N and p to infinity, the probability to measure a state which is N steps apart in future or past exceeds a constant lower bound. We characterize the spectrum, establish an analog to Plack's relation, define a random sampling scheme, analyze the resulting distribution of the anticipation probabilities and point out applications.