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On the validity of the Aharonov-Bergmann-Lebowitz rule

It is argued that the proof of Cohen [Phys. Rev. A 51, 4373 (1995)] which shows that an application of the Aharonov-Bergmann-Lebowitz (ABL) rule leads to contradiction with predictions of quantum theory is erroneous. A generalization of the ABL rule for the case of an incomplete final measurement (which is needed for the analysis of Cohen's proof) is presented.

Cohen [1] examines a few surprising results that have been obtained for pre-and post-selected quantum system by applying the Aharonov, Bergmann and Lebowitz (ABL) rule [2]. Following Sharp and Shanks [3] he proves that the ABL rule is not valid in general by showing that in a particular situation it leads to a prediction contradicting quantum theory. He claimed, however, that the ABL rule is valid for a special class of situations which correspond to "consistent histories" [4]. This limitation, if true, reduces significantly the importance of the ABL rule. In this comment I will argue that there is a crucial error in Cohen's proof of the inconsistency of the ABL rule with quantum theory.
The proof of Cohen is a variation of the proof given earlier by Sharp and Shanks [3]. I have showed in details in Ref. [5] the flaw in these arguments, and here I will only present the key point and discuss details which are specific for Cohen's proof.
Cohen considers a modified Mach-Zehnder apparatus with a possible measurement performed by a "which way" detector D 3 (that lets detected particles pass through), see Fig. 1. Then he applies the ABL rule in a counterfactual sense, and arrives at a contradiction with quantum theory. This leads him to reject the ABL rule for counterfactual situations. I argue that the contradiction obtained by Cohen follows from a logical error in his equation and not from an inapplicability of the ABL formula in this case. The argument of Cohen is as follows. If detector D 3 is not placed, then the probabilities for getting the click in D 1 and in D 2 are equal, Prob(D 1 ) = Prob(D 2 ) = 1 2 . This is so because the beam-splitters are half-transparent and the Mach-Zehnder BS 2 -BS 3 is tuned in such a way that all particles moving towards BS 2 are detected by D 2 . If D 3 is in place, then the ABL formula yields (Cohen claims) a probability 1 4 for a click in D 3 given a click in D 1 , Prob(D 3 |D 1 ) = 1 4 , and a probability 1 2 for a click in D 3 given a click in D 2 , Prob(D 3 |D 2 ) = 1 2 . Combining these three statements Cohen obtains an unconditional probability for the click in D 3 : This result is in contradiction with quantum theory which yields Prob(D 3 ) = 1 4 . One difficulty with this derivation is that the versions of the ABL formula which were published so far are not applicable to Cohen's experiment. The original ABL formula is applicable to a situation in which there is a complete measurement at t 1 , a complete measurement at t 2 and a complete measurement at time t, t 1 < t < t 2 . "Complete" means that the outcome specifies the quantum state completely. In Ref. [6] a generalization of the ABL formula which is applicable for an arbitrary measurement at time t is given, but the measurements at t 1 and t 2 have to be complete. [7] The ABL formula of Ref. [6] yields the probability for the result C = c n at time t given that at time t 1 the system was prepared in the state |Ψ 1 and that at time t 2 the state |Ψ 2 was found (here, for simplicity, zero free Hamiltonian is assumed): In Cohen's example, however, the measurement at time t 2 is not complete: the click in D 1 does not distinguish between |a , the state of the particle arriving from the left and |b , the state of the particle arriving vertically from beam-splitter BS 3 . I will analyze the proper generalization of the ABL formula for this case below and I will reach a different result for the conditional probability Prob(D 3 |D 1 ) which, nevertheless, will not change Cohen's argument. However, I believe that for trying to show putative inconsistency of the ABL formula it is better to consider situations in which the present version of this formula (shown in Eq. 2) is applicable. Therefore, I will first modify Cohen's experiment in such a way that Eq. 2 is applicable while Cohen's argument still goes through.
In the simplest variation of the experiment which makes the final measurement complete, D 1 is modified in such a way that it distinguishes the particles in states |a and |b . This, however, is not suitable for our purpose since it will not lead to Cohen's type contradiction. Therefore, we will consider, instead, a detector D 1 which distinguishes between the states |+ ≡ 1 Again this differs from the prediction of quantum theory: Prob(D 3 ) = 1 4 . Equation (3) is indeed wrong, but not because the probabilities given by the ABL formula are incorrect. The probabilities Prob(D 1 , +), Prob(D 1 , −) and Prob(D 2 ) are obviously wrong. Indeed, these probabilities were calculated on the assumption that detector D 3 was not placed. But equation (3) yields the probability for click in D 3 . Therefore, it must be in place and the assumption on which the probabilities Prob(D 1 , +), Prob(D 1 , −) and Prob(D 2 ) were calculated is not fulfilled.
It is easy to correct the calculation of Prob(D 3 ). If D 3 is in place we obtain: Prob(D 1 , +) = Not surprisingly, it is the same number which can be immediately obtained using quantum rules without considering the results of the final measurement.
Let us now analyze the unmodified Cohen's experiment. To this end we have to first generalize the ABL formula (2) for the case of an incomplete final measurement. The proof of Eq. (2) is given on p. 2317 of Ref. [6] and it can be repeated line by line for the case of incomplete final measurement, say with the result B = b. Essentially, the only change required is the replacement of "|Ψ f = |Ψ 2 " by "B = b" and the final result will be: Now we can analyze Cohen's original example. The generalized ABL formula (5) Since Cohen's proof of inconsistency of the ABL rule is erroneous, his conclusions about the limitations on the applicability of the ABL rule are unfounded. The surprising consequences of the ABL rule in the examples which Cohen considered are still valid.
This research was supported in part by a grant No. 614/95 from the Israel Science Foundation.