Mohrhoff proposes using the Aharonov–Bergmann–Lebowitz (ABL) rule for time-symmetric “objective” (meaning nonepistemic) probabilities corresponding to the possible outcomes of not-actually-performed measurements between specified pre- and post-selection measurement outcomes. It is emphasized that the ABL rule was formulated on the assumption that such intervening measurements are actually made and that it does not necessarily apply to counterfactual situations. The exact nature of the application of the ABL rule considered by Mohrhoff is made explicit and is shown to fall short of his stated counterfactual claim.

## REFERENCES

1.

U.

Mohrhoff

, “What Quantum Mechanics is Trying to Tell Us

,” Am. J. Phys.

68

, 728

–745

(2000

).2.

See, for example, R. I. G., Hughes,

*The Structure and Interpretation of Quantum Mechanics*(Harvard U.P., Cambridge, MA, 1989), p. 218;A. Shimony, “Search for a worldview which can accommodate our knowledge of microphysics,” in

*Philosophical Consequences of Quantum Theory*, edited by J. Cushing and E. McMullin (University of Notre Dame Press, Notre Dame, 1989), p. 27;*Search for a Naturatistic World View*(Cambridge U.P., New York, 1993), Vol. II, pp. 141–142. Mohrhoff defines subjective probabilities as applying only in cases in which measurements have been made and an observer is ignorant of the result of the measurement, which differs slightly from Hughes’ use of the term (see Ref. 3).

3.

Mohrhoff applies the term “fact” to measurement outcomes only (whether known or unknown), as opposed to possessed properties independent of measurement (which he denies).

4.

Y.

Aharonov

, P. G.

Bergmann

, and J. L.

Lebowitz

, “Time Symmetry in the Quantum Process of Measurement

,” Phys. Rev. B

134

, 1410

–1416

(1964

).5.

See, for instance,

J.

Bub

and H.

Brown

, “Curious Properties of Quantum Ensembles Which Have Been Both Preselected and Post-Selected

,” Phys. Rev. Lett.

56

, 2337

–2340

(1986

);W.

Sharp

and N.

Shanks

, “The Rise and Fall of Time-Symmetrized Quantum Mechanics

,” Philos. Sci.

60

, 488

–499

(1993

);O.

Cohen

, “Pre- and postselected quantum systems, counterfactual measurements, and consistent histories

,” Phys. Rev. A

51

, 4373

–4380

(1995

);O.

Cohen

, “Reply to Validity of the Aharonov–Bergmann–Lebowitz Rule

,” Phys. Rev. A

57

, 2254

–2255

(1998

);D. J.

Miller

, “Realism and Time Symmetry in Quantum Mechanics

,” Phys. Lett. A

222

, 31

–36

(1996

);R. E.

Kastner

, “Time-symmetrised Quantum Theory, Counterfactuals, and Advanced Action

,” Studies History Philos. Mod. Phys.

30

, 237

–259

(1999

);R. E.

Kastner

, “The Three-Box Paradox and Other Reasons to Reject the Counterfactual Usage of the ABL Rule

,” Found. Phys.

29

, 851

–863

(1999

);TSQT,

R. E.

Kastner

, “Elements of Possibility?

,” Studies History Philos. Mod. Phys.

30

, 399

–402

(1999

).6.

As it happens, Mohrhoff’s specific example of a counterfactual use of the ABL rule corresponds to a special case in which that use is valid (in the strong sense of statement 1). This is an example in which a particle is pre- and post-selected with outcomes

*a*and*b*corresponding to noncommuting observables A and B, and counterfactual measurements of either A or B are considered at time*t*. [The validity of a counterfactual usage of the ABL rule in cases like this has been shown in detail in Kastner (Ref. 5) and in Cohen (Ref. 5).] But this is a special case and, as has been discussed at length in the literature, “would”-type counterfactual uses of the ABL rule are generally invalid.7.

L.

Vaidman

, “Validity of the Aharonov–Bergmann–Lebowitz Rule

,” Phys. Rev. A

57

, 2251

–2253

(1998

).8.

In his reply to this Comment, Mohrhoff rejects my defense of the Sharp and Shanks proof on the basis that I allegedly assume the reality of ensembles in situations concerning only one particle. This remark misunderstands my use of ensembles. The term “ensemble” as I am using it denotes not a real collection of particles, but rather a

*conceptual*ensemble in the sense of statistical mechanics [cf. R. K. Pathria,*Statistical Mechanics*(Pergamon, Oxford, 1972), p. 4].9.

In his reply to this Comment, Mohrhoff essentially repeats the objection to the Sharp and Shanks proof previously given by Vaidman (Ref. 7). In this attempt to refute the proofs of Sharp and Shanks and others (which all have essentially the same structure), Vaidman and Mohrhoff simply assume that the “counterfactual” measurement

*is performed*in all expressions employed in the proof. Then, of course, there can be no inconsistency with the predictions of quantum mechanics. But this is no refutation, for it explicitly*assumes as true that which is manifestly false*: namely, that the “counterfactual” measurement is actually made. This procedure is then justified by claiming that the computation applies not to the actual world but to a specially chosen possible world. If such a procedure were to be allowed, then one could argue for something manifestly false in the actual world merely by finding a specially chosen possible world in which it is true.10.

“Objective probabilities, quantum counterfactuals, and the ABL rule: Apropos of Kastner’s comment,” quant-ph/0006116 2000 (a preprint version of Mohrhoff’s reply to the present Comment).

11.

“If we think of the measurement of

*Q*as taking place in a possible world, we consider a world in which all the relevant facts are exactly as they are in the actual world, except that in this possible world there is one additional relevant fact indicating the value possessed by*Q*at a time between $ta$ and $tb.$”12.

Sharp and Shanks (1993) consider an ensemble of spin-$12$ particles prepared at time $t1$ in the state $|a1\u3009$ (read as “spin up along direction

**a**”). They then assume that this ensemble is subjected to a final post-selection spin measurement at time $t2$ along direction**b**(i.e., the observable $\sigma b$ is measured). This measurement yields two subensembles $Ei,$ $i=1,2$ corresponding to results spin up or spin down along direction**b**. The weight of each subensemble $Ei$ is given by $|\u3008bi|a1\u3009|2.$ Now they consider each subensemble individually, asking the counterfactual question: If we had measured the spin of these particles along direction**c**(i.e., observable $\sigma c$) at a time*t*between $t1$ and $t2,$ what would have been the probability for outcome $c1?$ They use the ABL rule to calculate the probability of outcome $c1$ for each subensemble $Ei$ for such a counterfactual measurement. They then show that the total probability of outcome $c1$ derived from the above calculation, taking into account the weights of the two subensembles $Ei,$ in general disagrees with the quantum mechanical probability, which is given simply by $|\u3008ci|a1\u3009|2.$13.

One can, of course, deny this statement if one assumes fatalism (i.e., everything that happens

*must*happen). But then it must also be assumed that there is no possibility of a “counterfactual” measurement at time*t*, since it is a recordable matter of fact that no such measurement occurred, and according to fatalism, that documented absence of a measurement is also a fact that*must*happen.14.

Mohrhoff has confirmed in a private correspondence that the diagram discussed herein correctly illustrates his proposed possible world structure.

15.

However, this assumption can be disputed, since under the given construct (which assumes that outcome

*b*definitely occurs at $t2$), the conditional probabilities $P(b|qjk)$ (where $qjk$ is an eigenvalue of the associated observable $Qj$) are unity, rather than the standard quantum mechanical conditional probabilities as assumed in the ABL rule.16.

ABL (1964), abstract.

17.

ABL (1964) do say “We shall now consider an ensemble of systems whose initial and final states are fixed to correspond to the particular eigenvalues

*a*and*b*, respectively;we ask for the probability that the outcome of the intervening measurements are $dj,\u2026dn,$ respectively.” (p. B1412) But those outcomes correspond to

*actually performed*measurements.
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