We study the outcomes in a general measurement with postselection, and derive upper bounds for the pointer readings in weak measurement. Using the idea of weak measurement, we study Hardy's gedanken experiment and show how the negative probabilities emerge in weak measurement. By calculating the information gain of the measuring device about which path the particles pass through, we show that the negative probabilities only emerge for cases when the information gain is little due to very weak coupling between the measuring device and the particles. When the coupling strength increases, we can unambiguously determine whether a particle passes through a given path every time, hence the average shifts always represent true probabilities, and the strange negatives probabilities disappear.
Weak values are the fundamental values for observables in a pre- and postselected system. Weak values are typically measured by weak measurement, in which weak values appear in the change of not the pre- and postselected system but the probe system. This indirect characteristic of weak measurement obscures the meaning of weak values for the pre- and postselected system, in contrast to conventional physical quantities, which have a clear operational meaning. In this study, we operationally formulate weak values as the sensitivity of postselection probability amplitude to small transformation in a pre- and postselected system. This formulation of weak values, which is free from the concept of probe shift assumed in weak measurement, gives a direct interpretation of strange weak values for the pre- and postselected system. Furthermore, this formulation can simplify weak-value measurement experiments because the associated measurements appearing in the modulus change of the postselection probability amplitude do not require external degrees of freedom.
The usual way to quantify measurement incompatibility is noise robustness: how much noise must be mixed in before some joint measurement becomes possible. That is an extremal, worst-case quantity, computed by optimization, and it says little about what a real experiment would actually record if it just kept running. We ask a plainer question. A qubit is measured, over and over, in two projective bases separated by a Bloch angle θ. Once the resulting chain settles into steady state, how far does the state typically move between one measurement and the next? For sharp measurements this rate, r(θ), turns out to have the closed form 21sinθ1+sinθ. It agrees with the standard robustness threshold ν(θ)=1−1/1+sinθ to first order near θ=0, then departs from it everywhere else. For unsharp (POVM) measurements in the weak-measurement limit λ→0, full-rank Kraus operators preserve purity exactly, and this collapses the problem onto a one-dimensional diffusion on the Bloch circle. The resulting rate is c(θ)=sin(θ/2)/[2E(m)] with m=1−tan2(θ/2) and E the complete elliptic integral of the second kind, giving the clean value c(π/2)=1/π at the orthogonal point. A blind spot shows up along the way: at θ=π/2, r(θ) stops responding to noise entirely, for any Pauli-type or amplitude-damping channel, simply because sin45∘=cos45∘. That is unfortunate news for anyone hoping to use r(θ) as a calibration check at the angle experimentalists tend to prefer. Purification under repeated or simultaneous incompatible measurement has by now a sizable literature, but in every case we examined the angle θ itself is held fixed and something else, measurement strength or the number of axes, is varied instead. Treating θ as the free parameter and solving for it in closed form is the gap this paper tries to close. Past the qubit, the same construction becomes a random walk on CPd−1 rather than on a circle, and its closed form we leave open.