Generalized Probability Measures and the Framework of Effects

Abstract

This paper is dedicated to the memory of Itamar Pitowsky. It develops the idea that the generalized probability measures of quantum mechanics are the probabilities of “effects.” As explained below, effects are included among measurement outcomes but are not exhausted by them. They also differ in key respects from propositions which attribute dynamical properties to the systems that are probed by measurements. These differences are elaborated, and an interpretation of the implicit probability theory of quantum mechanics in terms of effects is outlined. A central feature of this interpretation is that it supports a form of realism that accommodates the no hidden variable theorem of Kochen and Specker, and it does so without appealing to any notion of contextuality.

This paper was intended for presentation to the December 2008 Jerusalem conference in Itamar’s honor; unfortunately, circumstances prevented me from attending the conference. The paper is a sequel to “Effects and propositions” [1], to which the reader is referred for a more complete list of references and the historical context of some of the views presented here. A very early version of this paper was given as a talk to Chris Fuchs’s Perimeter Institute seminar on the foundations of quantum mechanics in May of 2008. Many thanks to Chris for numerous discussions and comments on subsequent drafts. I am indebted to the Social Sciences and Humanities Research Council of Canada and to the Killam Foundation for support of my research.

Appendix: Itamar on Locality as a Special Case of Non-contextuality

This paper was close to its final form when I sent it to Itamar for his reaction to it. I had intended to incorporate a suggestion of his into the paper before submitting the final version for publication. With his death, it occurred to me to simply quote from an e-mail in which he commented at some length on the paper. I think the e-mail conveys some of the remarkable mix of warmth and generosity—greatness of soul—that were characteristic of him and that were so highly valued by his friends.

The context of Itamar’s letter is my discussion of contextuality and locality which begins on p. 207. Toward the conclusion of this discussion (beginning on p. 208) I give an argument, based on Kochen and Specker’s proof of their principal theorem, for why effects rather than propositions should be understood as the proper subject of the probability assignments of quantum mechanics. Itamar’s letter supplements this argument with another which he traces to a thought experiment of Vaidman. Our two arguments are linked by the connection Itamar draws between what I call “propositions belonging to particles” and EPR’s elements of reality. Itamar and I argue that both notions should be rejected in favor of effects.

I am very much indebted to Jeffrey Bub for an extended e-mail correspondence which led to the clarification of Itamar’s argument and to the suggestion that Vaidman [12] is a plausible choice for the paper of Vaidman’s that Itamar had in mind. What we think is the relevant passage from Vaidman is quoted after Itamar’s letter. In quoting from Itamar’s letter, I have made some very minor stylistic changes which I’ve left invisible. More significant changes which correct or slightly elaborate Itamar’s remarks are enclosed in square brackets.

From Itamar, September 28, 2009:

…Back to your paper and to the discussion of locality as a special case of [non-]contextuality, I think that’s exactly right and it brings into focus the question about the relation between EPR’s “elements of reality” and the concepts of proposition and effect. As you recall, something is an element of reality if its existence can be predicted with certainty. The EPR argument is built upon assigning elements of reality, by this criterion, to properties whose existence is never actually measured. They never leave a trace, and that’s where they fail. I think that EPR’s criterion is at best necessary but insufficient, and proper elements of reality should also be required to leave a trace that can in principle be retrodicted, at least for a short time after the effect (in fact according to quantum mechanics if there is an effect the wave function changes, while unmeasured “elements” don’t change the wave-function). So EPR’s mistake is like what you describe for the Kochen-Specker case of assigning truth values to propositions, in particular the value “true” to a contradiction. GHZ’s (or Mermin’s) version of EPR shows that the two arguments, Kochen-Specker and Bell’s, are the same in this respect. The lesson of EPR is not about locality but about how their criterion of reality is insufficient.

There is a paper by Vaidman who shows the shortcomings in the EPR criterion (although Vaidman does not see it that way and does not use the result for this purpose). He creates a simple thought experiment with a system that has no locality issues (no tensor products), and considers a measurement on the system at t ₁ and a subsequent measurement (of some other observable) at t ₂. Now he considers a hypothetical measurement of property A at a t between t ₁ and t ₂ and asks: If we measured A at t, would we have discovered that the system had property A? By construction the results of the [measurements at t ₁ and t ₂] force the answer YES with certainty. However if we use [the measurement at t ₁ and] the later measurement at t ₂ and ask the same question about a hypothetical measurement [of B at the earlier time t, the answer is also YES with certainty. But A and B are contrary propositions, and therefore cannot be true together. Hence the answer to whether A is true] is NO with probability one. So the property A cannot be assigned a truth value that will be consistent across time.

From (Vaidman [12], pp. 134–135):

A peculiar example of time symmetric counterfactuals is the three box paradox […]. Consider a single particle prepared at time t ₁ in a superposition of being in three separate boxes:

$$ |{\Psi_1}\rangle = {1}/\surd {3 }(|{\hbox{A}}\rangle + |{\hbox{B}}\rangle + |{\hbox{C}}\rangle ). $$

At a later time t ₂ the particle is found in another superposition:

$$ |{\Psi_2}\rangle = {1}/\surd {2 }(|{\hbox{A}}\rangle + |{\hbox{B}}\rangle - |{\hbox{C}}\rangle ). $$

For this particle, a set of counterfactual statements, which are elements of reality according to the … definition,

[If we can infer with certainty that the result of measuring at time t of an observable O is o, then, at time t, there exists an element of reality O = o,]

is:

$$ {{\hbox{P}}_{\rm{A}}} = {1}, $$

$$ {{\hbox{P}}_{\rm{B}}} = {1}. $$

Or, in words: if we open box A, we find the particle there for sure; if we open box B (instead), we also find the particle there for sure.⁶