# Derivation of the statistics of quantum measurements from the action of unitary dynamics

## TL;DR

This paper explores how quantum measurement statistics and interference effects can be derived from the unitary dynamics of quantum systems, linking energy-time relations to observable measurement outcomes and quantum phases.

## Contribution

It establishes a detailed connection between quantum dynamics, measurement statistics, and Hilbert space products, revealing the fundamental role of interactions in quantum randomness.

## Key findings

- Arrival times encode Hilbert space products between energy and observable eigenstates
- Quantum phases appear as an action relating energy to time in measurement dynamics
- Quantum interference arises from coherent control of energy, explaining measurement randomness

## Abstract

Quantum statistics is defined by Hilbert space products between the eigenstates associated with state preparation and measurement. The same Hilbert space products also describe the dynamics generated by a Hamiltonian when one of the states is an eigenstate of energy E and the other represents an observable B. In this paper, we investigate this relation between the observable time evolution of quantum systems and the coherence of Hilbert space products in detail. It is shown that the times of arrival for a specific value of B observed with states that have finite energy uncertainties can be used to derive the Hilbert space product between eigenstates of energy E and eigenstates of the dynamical variable B. In these Hilbert space products, quantum phases and interference effects appear in the form of an action that relates energy to time in the experimentally observable dynamics of localized states. Quantum effects emerge in the measurement statistics when the precise control of energy in quantum state preparation results in a coherent randomization of the dynamics, such that two different arrival times contribute to the quantum statistics of the same measurement outcome B. The non-classical features associated with quantum interference can thus be explained as a consequence of quantum dynamics and its role in state preparation and measurement, indicating that the apparent randomness of control described by the energy-time uncertainties is not merely a technical problem but rather originates from the fundamental nature of interactions between physical systems.