Complex Probability Measure and Aharonov's Weak Value

TL;DR

This paper introduces a complex probability measure for double quantum states, linking it to Aharonov's weak value, and interprets it as representing mixed quantum processes with classical probabilistic superpositions.

Contribution

It extends the standard probability measure to double states in quantum mechanics, providing a new interpretation of weak values as expectations over this measure.

Findings

A complex probability measure for double states is formulated.

Aharonov's weak value is derived as an expectation value of this measure.

The measure can be physically interpreted as a superposition of classical probabilistic processes.

Abstract

We present a complex probability measure relevant for double (pairs of) states in quantum mechanics, as an extension of the standard probability measure for single states that underlies Born's statistical rule. When the double states are treated as the initial and final states of a quantum process, we find that Aharonov's weak value, which has acquired a renewed interest as a novel observable quantity inherent in the process, arises as an expectation value associated with the probability measure. Despite being complex, our measure admits the physical interpretation as mixed processes, i.e., an ensemble of processes superposed with classical probabilities.