Conditional probabilities of measurements, quantum time and the Wigner's friend case

TL;DR

This paper derives unambiguous conditional probabilities for quantum measurements within a timeless formalism, clarifying measurement roles and resolving paradoxes like Wigner's friend without contradictions.

Contribution

It introduces a minimal assumption framework using the Gleason-Bush theorem to derive conditional probabilities, including in complex scenarios like Wigner's friend.

Findings

Conditional probabilities are uniquely derived for quantum measurements.

The formalism clarifies the roles of Wigner and his friend as being in superposition.

No paradoxes arise in the considered measurement scenarios.

Abstract

Considering a minimal number of assumptions and in the context of the timeless formalism, conditional probabilities are derived for subsequent measurements in the non-relativistic regime. Only unitary transformations are considered with detection processes described by generalized measurements (POVM). One-time conditional probabilities are unambiguously derived via the Gleason-Bush theorem, including for puzzling cases like the Wigner's friend scenario where their form underlines the relativity aspect of measurements. No paradoxical situations emerge and the roles of Wigner and Wigner can be seen by his friend as being in a superposition.