Abstract
The application of postselection to a weak quantum measurement leads to the phenomenon of weak values. Expressed in units of the measurement strength, the displacement of a quantum coherent measuring device is ordinarily bounded by the eigenspectrum of the measured observable. Postselection can enable an interference effect that moves the average displacement far outside this range, bringing practical benefits in certain situations. Employing the Fisher information metric, we argue that the amplified displacement offers no fundamental metrological advantage, due to the necessarily reduced probability of success. Our understanding of metrological advantage is the possibility of a lower uncertainty in the estimate of an unknown parameter with a large number of trials. We analyze a situation in which the detector is pixelated with a finite resolution, and in which the detector is afflicted by random displacements: imperfections which degrade the fundamental limits of parameter estimation. Surprisingly, weak-value amplification is no more robust to them than a technique making no use of the amplification effect brought about by a final, postselected measurement.
Highlights
In recent years, the Aharonov-Albert-Vaidman (AAV) effect [1,2,3] has received much attention
The phenomenon arises through a combination of both (i) a weak quantum measurement of an initial state of a “system” by a quantum coherent “meter” and (ii) judicious postselection of the system into an unlikely final state
The pre- and postselected ensemble is characteristic of the “two-state-vector approach,” which attempts to restore time symmetry to quantum mechanics [4]
Summary
The Aharonov-Albert-Vaidman (AAV) effect [1,2,3] has received much attention. A particle’s spatial wave function is weakly coupled to an internal observable A, causing it to undergo a small lateral shift conditional on the internal state. It is strongly postselected into the eigenstate of another observable B and allowed to impinge on a array of pixels with width r. The analyses in this paper apply if kx and xare interchanged, so that the interaction induces shifts in momentum space Under this dynamics, each of the eigenstates of the control observable becomes correlated to a separate wave function, which, up to the proportionality constant g, is peaked around the corresponding eigenvalue.
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