The state of a quantum system is not a complete description for retrodiction

TL;DR

This paper explores how different interpretations of mixed quantum states affect retrodiction, revealing that beliefs about past states can differ despite identical future predictions, highlighting a unique quantum aspect of Bayesian inference.

Contribution

It introduces a framework for quantum retrodiction and establishes conditions for belief equivalence, emphasizing the operational impact of interpretational differences.

Findings

Different interpretations lead to distinct retrodictive beliefs.

Identical density matrices can imply different past state inferences.

Operational consequences in quantum state recovery are demonstrated.

Abstract

A mixed quantum state can be taken as capturing an unspecified form of ignorance; or as describing the lack of knowledge about the true pure state of the system ("proper mixture"); or as arising from entanglement with another system that has been disregarded ("improper mixture"). These different views yield identical density matrices and therefore identical predictions for future measurements. But when used as prior beliefs for inferring the past state from later observations ("retrodiction"), they lead to different updated beliefs. This is a purely quantum feature of Bayesian agency. Based on this observation, we establish a framework for retrodicting on any quantum belief and we prove a necessary and sufficient condition for the equivalence of beliefs. We also illustrate how these differences have operational consequences in quantum state recovery.