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

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.

arXiv comment23 pages, including 5 figures, improved explanation of the fundamental relation between energy and time and the state-independence of B(t;E) in section II and III

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Cited by in corpus (3)

• Quantum enhancement of sensitivity achieved by photon-number-resolving detection in the dark port of a two-path interferometer operating at high intensities
• Control of particle propagation beyond the uncertainty limit by interference between position and momentum
• An exploration of the limits of control using quantum superpositions