Complex Probability Measure and Aharonov's Weak Value

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.