On the Quasi-probability associated with the Weak Value

Aharonov’s weak value has gained much attention in recent years, partly due to its intriguing capabiity for understating quantum puzzles such as Danan's implausible photons, and partly due to its potential capacity for the use of precision measurement. Despite this success, some people still have deep reservations for the weak value because of its complex nature which hampers our direct recognition as physical quality. In this talk, I argue that behind the apparently baffling nature of the weak value lies quasi-probability, which is defined for two non-commuting observables in quantum mechanics. This will be seen in three ways. One is to look at the double (or multiple) slit experiments and see how interference takes place quantum mechanically. The second is to examine the possible form of probability measure for post-selected measurement relevant to the weak value. The third is to contemplate the role of the weak value in the context of the general hidden variable theory. We shall see that all of these lead to the notion of quasi-probability, with which the weak value arises simply as its expectation value. I also mention that the quasi-probability we arrive at forms a family in which the familiar Wigner function appears along with other less-known but statistically interesting functions. [slides]

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