On the operational and algebraic quantum correlations

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.

Figures (2)

• Figure 1: The duality of quasi-classicalization and quantization as an adjoint operations.
• Figure 2: The application of our inequality to the qubit system for the case of $\theta=\frac{\pi}{3}$ in the states satisfies $\langle \sigma_y \rangle_\rho =0$. The red surface describes the difference between the two operational probabilities. The blue and green ones shows the upper and the lower bounds derived from our inequality, respectively. Here, red and green ones coincide everywhere.

Theorems & Definitions (32)

• Definition 1: Operational quantum correlations
• Definition 2: Algebraic quantum correlations
• Definition 3: Invasiveness measure
• Definition 4: disturbance operator Ozawa2003
• Definition 5: maximum disturbance
• Proposition 1
• proof
• Proposition 2
• proof
• Corollary 3