Past exam of the quantum information course of the University of Cambridge / 2016 / Paper 1 / 3 / a

A weak measurement couples an observable only weakly to a meter, so the pointer distributions associated with its different eigenvalues overlap strongly. One trial supplies little information and a small disturbance; an ensemble is needed to estimate a small mean shift. It is not an exact determination of a previously hidden eigenvalue with no disturbance.

For pointer position $q$ and conjugate momentum $p$, a short von Neumann measurement interaction has unitary $U=\exp (-igA\otimes p/\hslash )$. An eigenvalue $a$ translates the pointer by $ga$. The weak regime requires those translations to be small relative to the pointer width. Without postselection the first-order mean displacement is $g⟨A⟩$ for an initially uncorrelated, suitably centered meter.

If the system is prepared in $|i⟩$ and postselected in $|f⟩$, the unnormalized meter state isFor a real centered Gaussian pointer with no position–momentum correlation, the conditional position shift is $g\mathrm{Re}{A}_w$; the imaginary part affects the conjugate momentum distribution. The weak value can be complex or have a real part outside the observable's eigenvalue interval, because it is a conditional transition-amplitude ratio, not an eigenvalue.

An example apparatus uses photon polarization as the two-level system, a weak polarization-dependent transverse displacement as the interaction, and an output polarizer for postselection. Repeated position detections estimate the selected beam's displacement. Nearly orthogonal selection can make the conditional shift large, but also reduces the accepted sample and eventually invalidates the first-order expansion. The original weak-value proposal emphasized preselection, a weak coupling and postselection, rather than an anomalous result from an ordinary sharp spin measurement.^[ref]

Weak coupling limits quantum measurement backaction per trial; it does not abolish the information–disturbance tradeoff. Repeating weak observations can accumulate substantial backaction, and discarding rejected trials cannot automatically improve the information available per prepared system. Conditional pointer amplification and a statistically improved parameter estimate are distinct claims.