Continuous measurements in probability representation of quantum mechanics

TL;DR

This paper explores continuous quantum measurements using probability representations, deriving classical propagators for non-selective measurements and illustrating with quantum oscillator and particle examples.

Contribution

It introduces a method to represent continuous quantum measurements via classical propagators within the probability (symplectic tomogram) framework, including the derivation of non-selective measurement propagators.

Findings

Derived classical propagator for non-selective measurements

Applied the approach to quantum oscillator and particle

Connected continuous quantum measurement with classical path integrals

Abstract

The continuous quantum measurement within the probability representation of quantum mechanics is discussed. The partial classical propagator of the symplectic tomogram associated to a particular measurement outcome is introduced, for which the representation of a continuous measurement through the restricted path integral is applied. The classical propagator for the system undergoing a non-selective measurement is derived by summing these partial propagators over the entire outcome set. The elaborated approach is illustrated by considering non-selective position measurement of a quantum oscillator and a particle.