Resolving the Three-Box Paradox

An apparent paradox proposed by Aharonov and Vaidman in which a single particle can be found with certainty in two (or more) boxes is analyzed. It is found that the paradox arises from a counterfactual usage of the Aharonov-Bergmann-Lebowitz (ABL) rule. The paradox is resolved by observing that the counterfactual usage of the rule is, in general, not valid. 1. Background. The Aharonov-Bergmann-Lebowitz (ABL) rule is a well-known formula for calculating the probabilities of the possible outcomes of observables measured at an intermediate time t between preand post-selection measurements at times t1 and t2, respectively [Aharonov, Bergmann, and Lebowitz, 1964]. If an intermediate measurement of nondegenerate observable C with eigenvalues {ci} is performed at time t, the ABL rule states that the probability of outcome cj in the case of a preselection for the state |ψ1〉 and a post-selection for the state |ψ2〉 is given by: PABL(cj|ψ1, ψ2) = |〈ψ2|cj〉||〈cj||ψ1〉| ∑ i |〈ψ2|ci〉|2|〈ci||ψ1〉|2 (1) It has recently been shown that the ABL rule cannot be used in a counterfactual sense: i.e., in general, it cannot be used to calculate the probabilities of possible outcomes of observables that have not actually been measured at time t. 2. The three-box example. In his ‘Weak-Measurement Elements of Reality’ (1996), Vaidman discusses an example of what he terms an “ideal-measurement element of reality of the preand post-selected [email protected] In this case, the Hamiltonian H = 0; the ABL rule also applies to the more general case of nonzero Hamiltonian, with a time-dependence of the preand post-selection states in the usual way. Kastner 1998a, 1998b; see also Sharp and Shanks (1993), Cohen (1995), Miller (1996).