Weak Values Beyond Post-selection

Cosmin Andrei, Vlatko Vedral

Weak values are conventionally extracted in the limit of vanishing measurement strength. At finite strength, the corresponding post-selected conditional averages generally acquire nonlinear corrections and need not retain their weak value interpretation. Here we identify a class of measurement protocols for which this expectation fails: the conditional value reconstructed from the measuring probe is exactly independent of the probe strength and therefore coincides with the real part of the weak value. We first analyze this phenomenon in the measurement--disturbance protocol introduced by Lund and Wiseman. We then establish conditions under which this invariance persists, showing that it is determined by the interplay between the probe interaction, a subsequent measurement interaction with an ancillary apparatus, and the final post-selection. These results demonstrate that weak coupling, although generally sufficient for accessing weak values, is not always necessary.

Oded Zilberberg, Alessandro Romito, Yuval Gefen

Stefan Hillmich, Robert Wille

Riuji Mochizuki

The physical meaning of weak values and measurements can be completely understood with Born rule and the general probability theory. It is known that the weak value of an observable $\hat {A}$ with post-selection 〈F| may be out of the eigenvalue range of $\hat {A}$ . This is because the weak value of $\hat {A}$ with the post-selection is, in general, not the expectation value of $\hat {A}$ , but the expectation value of $\hat {A}| F\rangle \langle F|$ boosted by the post-selection.

Ron Cohen, Avshalom C. Elitzur, Eliahu Cohen

Violations of Bell inequalities are a hallmark of entanglement, with only entangled states capable of exceeding classical bounds in standard Bell tests. Here we analyze anomalous weak values of the CHSH-Bell operator in pre- and post-selected (PPS) quantum ensembles, using them to define separability-constrained bounds on Bell-type nonlocal correlations in the presence of post-selection. Fixing the overlap between the pre- and post-selected states, we compare three scenarios: unrestricted boundary states, one separable boundary state, and both boundary states separable. For each case, we derive both the maximal weak value for a fixed Bell operator and the maximal bound obtained by further optimizing over all CHSH operators. Our results show that post-selection and entanglement are distinct operational resources: post-selection alone can enhance correlations, but entanglement is necessary to exceed the corresponding separable PPS bounds, and their combination yields the strongest attainable correlations. Thus the separable PPS bound plays the role of a post-selected separability benchmark, distinct from the standard Bell bound. We further show that the enhancement beyond the separable bound closely tracks the concurrence of the states that optimize the bounds, identifying entanglement as the source of the additional correlation strength. Finally, we show that nonlocal weak values provide post-selected entanglement witnesses, and we give a constructive protocol that detects every pure two-qubit source state with nonzero concurrence in the ideal state-adapted setting, even in regimes where the corresponding standard CHSH entanglement test is inconclusive. More broadly, our results motivate hybrid protocols that combine post-selection and entanglement, with possible applications to improved quantum sensing, weak-value amplification, and quantum information processing.

Alessandro Romito, Yuval Gefen