Quasi-probabilities in Conditioned Quantum Measurement and a Geometric/Statistical Interpretation of Aharonov's Weak Value

摘要

We show that the joint behaviour of an arbitrary pair of quantum observablescan be described by quasi-probabilities, which are extensions of the standardprobabilities used for describing the behaviour of a single observable. Thephysical situations that require these quasi-probabilities arise when oneconsiders quantum measurement of an observable conditioned by some othervariable, with the notable example being the weak measurement employed toobtain Aharonov's weak value. Specifically, we present a general prescriptionfor the construction of quasi-joint-probability (QJP) distributions associatedwith a given pair of observables. These QJP distributions are introduced in twocomplementary approaches: one from a bottom-up, strictly operationalconstruction realised by examining the mathematical framework of theconditioned measurement scheme, and the other from a top-down viewpointrealised by applying the results of spectral theorem for normal operators andits Fourier transforms. It is then revealed that, for a pair of simultaneouslymeasurable observables, the QJP distribution reduces to their unique standardjoint-probability distribution, whereas for a non-commuting pair there existsan inherent indefiniteness in the choice, admitting a multitude of candidatesthat may equally be used for describing their joint behaviour. In the course ofour argument, we find that the QJP distributions furnish the space of operatorswith their characteristic geometric structures such that the orthogonalprojections and inner products of observables can, respectively, be givenstatistical interpretations as `conditionings' and `correlations'. The weakvalue $A_{w}$ for an observable $A$ is then given a geometric/statisticalinterpretation as either the orthogonal projection of $A$ onto the subspacegenerated by another observable $B$, or equivalently, as the conditioning of$A$ given $B$.