Significance of the imaginary part of the weak value

Abstract

Unlike the real part of the generalized weak value of an observable, which can in a restricted sense be operationally interpreted as an idealized conditioned average of that observable in the limit of zero measurement disturbance, the imaginary part of the generalized weak value does not provide information pertaining to the observable being measured. What it does provide is direct information about how the initial state would be unitarily disturbed by the observable operator. Specifically, we provide an operational interpretation for the imaginary part of the generalized weak value as the logarithmic directional derivative of the postselection probability along the unitary flow generated by the action of the observable operator. To obtain this interpretation, we revisit the standard von Neumann measurement protocol for obtaining the real and imaginary parts of the weak value and solve it exactly for arbitrary initial states and postselections using the quantum operations formalism, which allows us to understand in detail how each part of the generalized weak value arises in the linear response regime. We also provide exact treatments of qubit measurements and Gaussian detectors as illustrative special cases, and show that the measurement disturbance from a Gaussian detector is purely decohering in the Lindblad sense, which allows the shifts for a Gaussian detector to be completely understood for any coupling strength in terms of a single complex weak value that involves the decohered initial state.

On the relation between measurement outcomes and physical properties

“…(12) indicates that any projective measurement {| m } can have a measurement error of ε 2 (A) = 0 if the corresponding set of weak values of m are chosen for the estimatesà m . However, weak values are generally complex, and the imaginary part is usually obtained in a dynamical response of the system that is not directly connected to the quantity represented by [34,35]. Sinceà m is an estimate of the quantityÂ, it does not have an imaginary part and Eq.…”

Disturbance in weak measurements and the difference between quantum and classical weak values

“…On one hand it is clear from (C3) that we must have R =Ô, on the other handÔ is the only operator on H in the game, so we should also have A I ∝Ô. Indeed, an explicit calculation shows that [25,37] A =Ô − i {X,P } s σ auxÔ .…”

Measurement of a spin-1 system

“…Here, we shall derive the main result following the approach of [25,26] with a slight improvement that shows the connection with the classical and the quantum characteristic function of the detector [58], rather than the partial Fourier transform of its Wigner function. We remark that this approach works only ifQ, the write-in variable of the detector, has the continuous unbounded spectrum (−∞, +∞), and is thus akin to a position operator.…”

“…We remark that this approach works only ifQ, the write-in variable of the detector, has the continuous unbounded spectrum (−∞, +∞), and is thus akin to a position operator. Notice that our variableQ corresponds top in [25,26], while the readout variableP corresponds tox.…”