Weak Value and Weak Measurements

The weak value of a variable O is a description of an effective interaction with that variable in the limit of weak coupling. For a pre-and post-selected system described at time t by the two-state vector Φ| |Ψ (see entry Two-State Vector Formalism), the weak value is [1]: O w ≡ Φ|O|Ψ Φ|Ψ. (1) Contrary to classical physics, variables in quantum mechanics might not have definite values at a given time. In the complete description of a usual (pre-selected) quantum system, the state |Ψ yields probabilities p i for various outcomes o i of (an ideal) measurement of the variable O. Numerous measurements on an ensemble of identical systems yield an average – expectation value of O: p i o i. Since p i = |O = o i |Ψ| 2 , the expectation value can be expressed as Ψ|O|Ψ. If the coupling to the measuring device is very small, this expression is related directly to the response of the measuring device, and the measurement does not reveal the eigenvalues o i and their probabilities p i. Specifically, Ψ|O|Ψ is the shift of the quantum state of the pointer variable of the measuring device, which, otherwise, is not distorted significantly due to the measurement interaction. For pre-and post-selected quantum system, the response of the measuring device or any other system coupled weakly to the variable O, is the shift of the quantum state by the weak value (1). The coupling can be modeled by the von Neumann measurement interaction H = g(t)P O,