On the operational and algebraic quantum correlations

TL;DR

This paper explores the differences between algebraic and operational quantum correlations, establishing bounds related to measurement invasiveness and clarifying conditions for their equivalence, with applications to Leggett-Garg inequality violations.

Contribution

It provides a quantitative analysis of the relationship between algebraic and operational quantum correlations, introducing bounds and conditions for their equivalence.

Findings

Bound on differences between correlation definitions due to measurement invasiveness

Lower bound on discrepancy between operational and algebraic joint distributions

Condition identified for when operational and algebraic correlations coincide

Abstract

We investigate the intrinsic ambiguity in the definition of correlation functions arising from the inevitable invasiveness of quantum measurements. While algebraic correlations defined as expectation values of products of observables are widely used, their relationship to operational ones defined through actual measurement procedures remain unclear. We demonstrate that the differences among various definitions of correlation functions and those among their underlying (quasi-)joint probability distributions are bounded above by a quantitative measure of measurement invasiveness. We further obtain a lower bound on the discrepancy among operational and algebraic (quasi-)joint probability distributions, providing a new form of the uncertainty relation. In addition, we identify an equivalence condition under which operational and algebraic correlations coincide. As an application, we analyze the quantum violation of the Leggett-Garg inequality and clarify the structural origin of the equivalence among different approaches to observing the violation, including sequential projective measurements and weak-measurement. Our results provide an operational foundation for the commonly used algebraic concepts of quantum theory.