A loose and separate certainty: Caves, Fuchs and Schack on quantum probability one

https://doi.org/10.1016/j.shpsb.2011.02.001Get rights and content

Abstract

Carlton Caves, Fuchs, and Schack (2002) have recently appealed to an argument of mine (Stairs, 1983) to address a problem for their subjective Bayesian account of quantum probability. The difficulty is that on the face of it, quantum mechanical probabilities of one appear to be objective, but in that case, the Born Rule would yield a continuum of probabilities between zero and one. If so, we end up with objective probabilities strictly between zero and one. The authors claim that objective probabilities of one leads to a dilemma: give up locality or fall into contradiction. I argue that this conclusion depends on an overly strong interpretation of objectivism about quantum probabilities.

Introduction

Carlton Caves, Christopher Fuchs and Rüdiger Schack (henceforth CFS) have long defended a subjective Bayesian account of quantum mechanical probability. This may seem implausible for probabilities of one, but that case is important for their view. Assigning probability one to a quantum proposition typically generates a continuum of probabilities between zero and one via the Born rule. If probability one is objective, it would presumably follow that these other probabilities are as well. Consequently the viability of CFS's program requires them to deny that quantum probability is objective even for probability one. They write:

The statement that the measurement outcome is 1 with certainty is… not a proposition that is true or false of the system, but an agent's belief – and another agent might make a different prediction. (Caves et al., 2002, p. 267)

In order to make their case, CFS appeal to a paper of mine from some years ago (Stairs, 1983). Though I am flattered by the attention to my work, I do not think their argument goes through. The appearance that it does rests on an overly strong reading of what objectivism calls for.

What follows is not intended as a full defense of objectivism about quantum probability (henceforth we will just say “objectivism.”) CFS try to show that if probability one is objective, we face a dilemma: embrace non-locality or fall into contradiction. The main goal of this paper is to show that there is no such dilemma. As for quantum probabilities strictly between 0 and 1, the argument would not be that they must be treated objectively, but rather that nothing CFS say rules this out. I will sketch what I take to be a promising strategy for objectivism about quantum probabilities, but working out that strategy – or any other – goes beyond this paper.

CFS's case breaks into three parts: general arguments on behalf of subjective Bayesianism, a brief against an objective view of state preparation, and an argument that if we treat quantum certainty as objective but accept locality, we wind up in contradiction. I will urge that the general considerations are not compelling, that the case against the objective view of state preparation does not succeed, and that the argument about quantum certainty can be turned aside by some careful reflection on the connection between probabilities, properties and counterfactuals.

Section snippets

General considerations

According to CFS, propositions and probabilities lie on opposite sides of a category divide. Probability has an objective component: events or facts, which agents can settle unambiguously, and the rules of probability, including the Born rule. However, probabilities themselves are degrees of belief, and are neither true nor false. Probabilities do not follow from facts, and unlike physical parameters, they cannot be determined unambiguously—not even approximately. And though David Lewis's

Objective preparations

On the face of it, arguing that quantum probability one cannot be objective seems like a hard row to hoe. After all, we seem to be able to prepare states; indeed, we seem to do it all the time. And many of the states we prepare seem to provide us with unit probabilities. Just think of preparing a beam of photons all polarized in some particular direction. CFS dub the view that this can be done the “objective preparations” view. If it is correct, we can give objective and, indeed, classical

Probability one and correlations

At this point, CFS appeal to my 1983 paper. I argued there that if we combine Kochen and Specker’s well-known result (1967) with appropriate correlations and a locality requirement, we can show that locally well-defined classical values are impossible. CFS streamline the argument and take it in a different direction; we’ll streamline what they say in turn.

Off to see the wizard

Let us slow down. In fact, let us leave quantum theory aside for the time being. Here is a tale that does not happen to be true, but is a perfectly good story in spite of that. The example at its core is based on (and isomorphic to) one developed by Liang, Spekkens, and Wiseman (in preparation), who in turn were inspired by a paper by Ernst Specker (1960).

A certain wizard makes wondrous pairs of boxes. Each box has three drawers in a row, labeled 1–3. If you open a drawer, a brilliant light

Quantum states

Our imaginary case was theory-free. It assumed that there is no special set of terms or concepts required to tell someone how to make the boxes, and so we have the analog of objective state preparation. The total “theory” of the boxes consists of three claims: that the strict correlations are as we have described them, that the correlations are non-signaling, and that the boxes are causally local. For all this says, there could be a more detailed theory, and the boxes could be a special case of

Concluding remarks

Here is the main point in brief: supposing quantum theory is correct, there are counterfactual-supporting lawful generalizations such as if a pair of qubits prepared in the singlet state are both subjected to spin measurements in direction d, then the results will sum to zero. Generalizations like this ground conditional probabilities of one. But they can hold without instruction sets that fix individual outcomes, and without causal signals passing amidst the pair. This means that when Alice

Acknowledgments

Thanks to Jeffrey Bub for ongoing conversations about the issues discussed here. Thanks also to Chris Fuchs for generously hosting me as a visitor at the Perimeter Institute and for helping me to appreciate the Bayesian perspective better than I once did, even if I’m not yet converted. I also acknowledge the support of the National Science Foundation under NSF Grant no. 0822545 during portions of the research for this paper.

References (17)

  • T Maudlin

    What could be objective about probabilities?

    Studies in History and Philosophy of Modern Physics

    (2007)
  • Barrett, J., Linden, N., Massar, S., Pironio, S., Popescu, S., & Roberts, D. (2005). Nonlocal correlations as an...
  • C.M. Caves et al.

    Subjective probability and quantum certainty

    Studies in History and Philosophy of Modern Physics

    (2002)
  • Frigg, R., & Hoefer, C. (2010). Determinism and chance from a Humean perspective. In Dieks, D., Gonzalez, W., Hartmann,...
  • Fine, Arthur (1980). Correlations and physical locality. PSA column two: symposia and invited papers. Philosophy of...
  • Fuchs, C. (2002). Quantum mechanics as quantum information (and only a little more). Available from:...
  • Hegedüs, R., Åkesson, S., Wehner, R., Horváth, G. (2007). Procedings of the Royal Society A 463,...
  • C. Hoefer

    The third way on objective probability: A skeptic’s guide to objective chance

    Mind

    (2007)
There are more references available in the full text version of this article.

Cited by (0)

View full text