Significance of the imaginary part of the weak value

摘要

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