On the role of complex phases in the quantum statistics of weak measurements

TL;DR

This paper explores how complex phases in weak measurements reveal the dynamic response of quantum systems to transformations, linking statistical overlaps with quantum correlations and paradoxes.

Contribution

It demonstrates that the phases of weak conditional probabilities directly relate to the action of unitary transformations, unifying statistical and dynamical aspects of quantum mechanics.

Findings

Complex phases describe system response to unitary transformations

Phases identify the action maximizing state overlap

Quantum correlations can diverge from classical expectations

Abstract

Weak measurements performed between quantum state preparation and post-selection result in complex values for self-adjoint operators, corresponding to complex conditional probabilities for the projections on specific eigenstates. In this paper, it is shown that the complex phases of these weak conditional probabilities describe the dynamic response of the system to unitary transformations. Quantum mechanics thus unifies the statistical overlap of different states with the dynamical structure of transformations between these states. Specifically, it is possible to identify the phase of weak conditional probabilities directly with the action of a unitary transform that maximizes the overlap of initial and final states. This action provides a quantitative measure of how much quantum correlations can diverge from the deterministic relations between physical properties expected from classical physics or hidden variable theories. In terms of quantum information, the phases of weak conditional probabilities thus represent the logical tension between sets of three quantum states that is at the heart of quantum paradoxes.