Weak values and the past of a quantum particle

We investigate four key issues with using a nonzero weak value of the spatial projection operator to infer the past path of an individual quantum particle. First, we note that weak measurements disturb a system, so any approach relying on such a perturbation to determine the location of a quantum particle describes the state of a disturbed system, not that of a hypothetical undisturbed system. Second, even assuming no disturbance, there is no reason to associate the nonzero weak value of an operator containing the spatial projection operator with the classical idea of “particle presence.” Third, weak values are only measurable over ensembles, and so to infer properties of individual particles from values of them is problematic. Finally, weak value approaches to the path of a particle do not provide information beyond standard quantum mechanics (and the classical modes supporting the experiment). We know of no experiment with testable consequences that demonstrates a connection between particle presence and weak values.

Figure 1

Nested-interferometer model, used to discuss cases where the two-state vector formalism contradicts “common-sense” continuous-path approaches [7].

Figure 2

Two-state vector formalism analysis of Salih et al.'s polarization-based single-outer-cycle protocol for counterfactual communication. Bob communicates with Alice by turning off or on his switchable mirrors to determine whether the photon goes to $D1$ or $D0$, respectively. We specify the polarization, given it determines direction of travel through the polarizing beam splitters (PBSs). The forward- and backward-traveling states do not overlap anywhere on the inner interferometer chain when there is a detection at $D0$, meaning, by weak value approaches, the particle detected at $D0$ was never at Bob. This means Bob's ability to communicate with Alice is not explained by weak value approaches any more than it is by standard quantum mechanics.