Inversion of moments to retrieve joint probabilities in quantum sequential measurements

TL;DR

This paper demonstrates that quantum joint probabilities for sequential measurements cannot be reconstructed from moments, highlighting the fundamental non-existence of a consistent grand joint distribution, supported by theoretical analysis and NMR experiments.

Contribution

It provides a clear demonstration that moments do not suffice to retrieve quantum joint probabilities in sequential measurements, revealing a fundamental incompatibility.

Findings

Moments do not determine three-time quantum joint probabilities.

Experimental NMR results support the theoretical incompatibility.

Quantum sequential measurements lack a consistent grand joint distribution.

Abstract

A sequence of moments encode the corresponding probability distribution. Probing if quantum joint probability distribution can be retrieved from the associated set of moments -- realized in the sequential measurement of a dichotomic observable at different time intervals -- reveals a negative answer i.e., the joint probabilities of sequential measurements do not agree with the ones obtained by inverting the moments. This is indeed a reflection of the non-existence of a bonafide grand joint probability distribution, consistent with all the physical marginal probability distributions. Here we explicitly demonstrate that given the set of moments, it is not possible to retrieve the three-time quantum joint probability distribution resulting from quantum sequential measurement of a single qubit dichotomic observable at three different times. Experimental results using a nuclear magnetic resonance (NMR) system are reported here to corroborate these theoretical observations viz., the incompatibility of the three-time joint probabilties with those extracted from the moment sequence.