What Are the Subjective Probabilities in Quantum Theories?

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1 What Are the Subjective Probabilities in Quantum Theories? June 1, 2018 Abstract Textbook orthodoxy maintains that some probabilities in quantum theories may be viewed as capturing ignorance of the true state of a system. It is natural to suppose that these probabilities are subjective in the sense familiar to philosophers that is, it is natural to suppose that they are Bayesian credences. This supposition leads to a paradox: while the probabilities in quantum theories do not depend on how systems are measured, agents may have coherent credences that do. I propose, instead, that subjective quantum probabilities are expectations for chances. I show that this approach resolves the paradox, and I argue that it is naturally motivated by Leibniz s principle. Contents 1 Introduction 1 2 Quantum gambles : a paradox The sense in which quantum theory is contextual How quantum gambles fail to explain noncontextuality Expectations for chance and Leibniz s principle 4 1 Introduction There are two sorts of probabilities that one encounters in standard quantum mechanics: those given by wavefunctions, and those given by mixtures of wavefunctions, or density operators. These latter probabilities are often taken to be subjective in some sense or another. Von Neumann, one of the folks who came up with density operators in the first place, used them to describe cases where we do not know what state is actually present (von Neumann 1955, p. 295). The modern version of this idea can be found in the field of quantum state tomography, wherein experimenters use density-operator probabilities to express their estimate of what quantum state is prepared in the lab (Gonçalves et al. 2017). What is the precise sense in which these probabilities are subjective? It is tempting to reach for an off-the-shelf solution. We have a rigorously developed notion of subjective probability from formal epistemology the Bayesians notion of degree of belief or credence. Could the subjective probabilities in quantum theories be credences? Perhaps. But at least one account of density-operator probabilities as credences the quantum gambles of Pitowsky (2003) leads to a nasty paradox. Density-operator probabilities do not depend on how systems are measured. But by the lights of quantum gambles, agents can have coherent credences that do. I don t think credences are the right solution. Instead, I think subjective quantum probabilities should be understood as expectations for chances. As we will see, this tack handily dissolves the paradox. 2 Quantum gambles : a paradox It is well-known that the all quantum probabilities both those yielded by wavefunctions and those yielded by density operators are noncontextual. That is, they do not depend on how a system is measured. I will 1

2 argue that, on Pitowsky s account, agents can have coherent credences that do depend on how systems are measured. But first, a clarification is in order. An oft-repeated mantra among physicists is that quantum theory is contextual, although the precise meaning of this mantra has mutated and splintered over the years. To show how credences fail to capture the noncontextuality of quantum probability, I will first briefly review the sense in which quantum theory is contextual. Then, I will show how this contextuality poses a problem for Pitowsky s quantum gambles. 2.1 The sense in which quantum theory is contextual As noted, it is an oft-repeated mantra that quantum theory is contextual. One of the more popular explications of this fact may be paraphrased as follows: so long as quantum theory gives the right labels to preparation procedures, measurement procedures, and outcomes, then there are quantum experiments for which no assignment of outcomes that is deterministic and noncontextual will be consistent with the support of any quantum probability (Spekkens 2005). This is an informal statement of what is known as the Kochen- Specker theorem (Kochen and Specker 1967). Even less formally, but more brusquely: any deterministic and noncontextual assignment of outcomes will posit the possibility of an event that we never see in the lab. There is an illuminating graphical way of viewing this result due to Cabello et al. (1996) that I include in Figure 1. Here, every node represents a possible outcome (some element k of a set of outcomes K), and each box represents a way of measuring (some element M of a set of measurements M) the outcomes it contains. No two mesaurements in the graph can be performed at the same time. All the events are mutually exclusive and exhaustive quantum probabilities (roughly, and taken objectively) predict that precisely one of the events in a box will occur given that the appropriate measurement is made. So let a coloring of black represent will occur (i.e. 1) and let a coloring of white represent will not occur (i.e. 0). It s not hard to see that there is no coloring that leaves each box with precisely one black node; one failed coloring is shown. Figure 1: An assignment of 0 (white) and 1 (black) for Cabello s (1996) Kochen-Specker model. Note that the thick box contains no black nodes. This inconsistency is striking, perhaps in part because we never encounter it in classical physics. But at least the possibility of such inconsistency should not be terribly surprising, given that quite unlike in the classical case not all measurements of quantum systems can be performed simultaneously. And, moreover, there are two ways to posit hidden structure that easily avoid the inconsistency. One way out is to maintain that outcomes are determined in advance, but that the determined outcome crucially depends on how the system is measured. Another way out is to allow for quantum events to have objective probabilities or chances. We will return to this option later. For now, what is important to keep in mind is that any yes-no assignment of outcomes must depend on context. There are two sides to this result. We ve been discussing the first, which is that events cannot both be determined in advance and not depend on which measurement is made. The second is the following: what events end up happening must depend on which measurement is made. Here s another way of looking at this second side. The occurrence of events is often described by a classical truth valuation on an appropriate logical language. The above argument shows that there cannot be a classical assignment of truth values one that respects the mutually exclusive and exhaustive criterion stated above that does not depend on context. The second side to this result may seem rather tame. But it wrecks havoc for an understanding of density-operator probabilities as credences. Let s turn to one such understanding now: Pitowsky s quantum gambles. 2

3 2.2 How quantum gambles fail to explain noncontextuality Pitowsky s quantum gambles afford a tidy case study of an approach that treats quantum probabilities as credences. A quantum gamble proceeds in four stages: 1. A single physical system is prepared by the bookie. 2. A finite set M of incompatible measurements is announced by the bookie, and the agent is asked to place bets (i.e. credences) on possible outcomes (i.e. propositions about occurrences) of each one of them. 3. One of the measurements M M is chosen by the bookie and the money placed on all other measurements is promptly returned to the agent. 4. The chosen measurement is performed and the agent gains or loses in accordance with his bet on that measurement. Let B M be the subset of events in K that may occur given the measurement M is made; Pitowsky treats each B M as a Boolean algebra. 1 To recover the usual density-operator probability functions on these algebras, Pitowsky imposes two rules of gambling and a possibility criterion. The first rule states that credences must be classical probabilities over each B M (in Pitowsky s notation, p( M) must be a classical probability on B M ), and the second rule states that credences must be independent of context (so p(k M) = p(k M )). The possibility criterion simply establishes that the set K is the set of projections on some finite-dimensional Hilbert space. A result known as Gleason s theorem now ensures that credences will align with density-operator probabilities. 2 In defense of the first rule, Pitowsky cites de Finetti s (1972) usual Dutch book argument for probabilistic credence. But here s the catch: this argument assumes that possible outcomes are described by all and only the classical truth valuations on the appropriate logical language. In order for the usual argument to go through, Pitowsky must be assuming that the possible values over each B M yield all and only the classical truth valuations on this algebra. How can we get such assignments when M is given by the Kochen-Specker model in Figure 1? The most perspicuous way forward seems to be to embrace contextuality. That is: have one set of possible values Q M for each measurement M in Figure 1, and one expectation p( M) for each measurement. But note now that the usual Dutch book argument yields that any combination of classical probabilities p( M) is a coherent credence. We have no motivation for the ostensible noncontextuality of subjective quantum probabilities. That is: we have no motivation for Pitowsky s second rule. Pitowsky adopts two different attitudes towards this rule in the literature. There is an early view (Pitowsky 2003) that seems to treat it as a principle of rationality, and there is a later view (Hemmo and Pitowsky 2007) that treats it as a contingent empirical fact. I will now review both of these attitudes in turn. On the early view, Pitowsky insists that ignoring the second rule has the consequence of destroying the identities of the events and the logical relations between them, because we are no longer assuming that a node is really the same event regardless of which box containing it denotes the measurement actually performed (2003, pp. 400, 409). But if we take the perspicuous approach above, then each set of possible values Q M contains truth valuations describing the occurrence and non-occurrence of events indexed to the context M. The actual value of the occurrence of any event in K is given by one of these truth valuations, so long as M is performed. And if the actual value for the occurrence depends on context, why isn t my expectation for this value allowed to depend on context? Indeed: if we take quite seriously Pitowsky s claim that a node really represents the same event regardless of context, we seem to be forced into viewing it as a random variable that is assigned just one actual value. But this only-one-actual-value approach requires us to let the possible values be given by noncontextual yes-no assignments. And this, of course, will result in a failed coloring of the sort seen in Figure 1. The devotee of this approach could insist that such evaluations reflect real possibilities but this devotee now 1 The reader familiar with the original paper of Kochen and Specker (1967) may view K as a partial algebra in exactly these authors sense. 2 For Hilbert spaces of finite dimension d 3. 3

4 seems to be in the tenuous position of insisting that some events both are and are not possible. And more to the point: the possible values of variables over a given context will no longer be all and only the classical truth valuations over that context. So it is unclear how the usual Dutch book argument will restrict an agent s expectations even just to those functions that are classical probabilities over contexts! Motivating the noncontextuality of subjective quantum states, then, seems to be a persistent problem for this subjective approach to density-operator probabilities as credences at least if we hope to get by with a set of possible values and constraints on rational credence alone. On the later view, Pitowksy seems to concede this point. He now stresses that the noncontextuality rule cannot be a principle of rationality, and that we must rely on repeated trials and Bayesian updating to see the extent to which it is justified (Hemmo and Pitowsky 2007). This later view suggests that we baptize Pitowsky s second rule as a contingent empirical principle added to the two base ingredients of possible occurrences and a coherence condition. Roughly: it has just always turned out to be the case that agents who follow the noncontextuality rule do a heck of a lot better in quantum gambles than agents who do not, and we should learn from this experience. I think that this assessment is essentially correct but it is important to note what is lost when we embrace it. There is no longer any a priori reason for agents to match their credences to quantum probabilities; they should do so merely as a matter of contingent empirical fact. And moreover: different experimenters will have different initial credences and different experiences. It is not at all obvious that all experimenters will eventually converge on noncontextual credences via updating (Bayesian or otherwise). And, as noted, contextual credences are perfectly reasonable by the lights of the usual Dutch book argument! Pitowsky s program is not the only game in town when in comes to treating density-operator probabilities as credences. Of the other extant programs, the most famous is the Quantum Bayesianism of Caves, Fuchs, and Schack (Caves et al. 2002). I cannot pretend to be able to give the QBists a fair shake here. Rather, I will simply hope that my assessment of Pitowsky s program provides some light motivation to be skeptical of a credence-based approach. After all, there is a different approach available that rather naturally explains the noncontextuality of subjective quantum probability. As I noted earlier, an easy way out of the Kochen-Specker no-go is to allow quantum measurement events to occur with objective probability, or chance. On my view, subjective quantum probabilities should be viewed as expectations for such chances. 3 Expectations for chance and Leibniz s principle What is an expectation for a chance? Nothing more than an agent s best estimate of an objective probability value. Note that this construal of expectation for a chance leaves wide-open the question of what you take an objective probability value to be. I, for one, have something like Sober s (2010) no-theory theory in mind. An objective probability yields predictions for frequencies, it is evidenced by them, and that s all there is to it. This tack requires no additional commitments regarding objective probability s relationship to credence or frequency; in particular, it accommodates agnosticism regarding Lewis s (1980) principal principle. Why should expectations for chances be noncontextual? Spekkens provides a compelling and principled reason: Leibniz s principle of the identity of indiscernibles, or PII. Of course, there are multifarious well-explored ways of construing and using PII. I will focus only on Spekkens s approach to to PII as a methodological principle. On this tack, the principle amounts to the following advice: if a physical model posits that two scenarios are empirically indistinguishable but nonetheless represents them as ontologically distinct, this model should be rejected and replaced with one that makes them ontologically the same (Spekkens 2017). That is: we should strive for the ontological identity of empirical indiscernibles. As Spekkens notes, contextual chances clearly violate this principle. Here s why: suppose you ve accepted that there are chances, at the very least in the thin sense that Sober sketches. If you do, then the stable frequencies for quantum measurement events that you observe in the lab evidence chances that do not depend on how an event is measured. Now suppose you posit {0, 1}-valued, contextual chances underlying these empirical chances. This posit introduces hidden {0, 1}-valued chances that do depend on how an event is measured. They introduce an ontological difference where there is no empirical difference. Such, it would seem, is the very definition of otiose metaphysics. 4

5 Given that the empirical chances do not depend on context, it is metaphysically perspicuous to posit that the hidden chances fixed by the state of the quantum system do not depend on context. So expectations for their values should not depend on context, either. Last but not least: the usual Dutch book argument may be lightly modified to assess agents expectations for chance. It turns out that if either wavefunctions or density operators are taken to yield possible chances, then this modified argument yields that density-operator probabilities are all and only the coherent expectations for these chances. For the details of this argument, see Steeger (2017). In sum: there is a principled realism on which the noncontextuality of expectations for chance has a straightforward a priori explanation. This realism requires a commitment to only the thinnest notion of objective probability. And the usual Dutch book argument may be lightly modified to vindicate densityoperator probabilities as all and only the reasonable expectations for chances. I think this explanatory success of expectations for chance is as good a reason as any for philosophers of science to make room in their hearts for this new notion of subjective probability. Bayesian credences are extremely powerful tools; I do not dispute their impeccable track record. But for some tasks, we need expectations for chance to get the job done. References Cabello, Adán, José M. Estebaranz, and Guillermo García-Alcaine (1996), Bell-Kochen-Specker Theorem: A Proof with 18 Vectors, Physics Letters A, 212, 4, pp Caves, Carlton M., Christopher A. Fuchs, and Rüdiger Schack (2002), Unknown quantum states: the quantum de Finetti representation, Journal of Mathematical Physics, 43, 9, pp De Finetti, Bruno (1972), Probability, Induction, and Statistics: The Art of Guessing, John Wiley & Sons, London. Gonçalves, D. S., C. L. N. Azevedo, C. Lavor, and M. A. Gomes-Ruggiero (2017), Bayesian inference for quantum state tomography, Journal of Applied Statistics, pp Hemmo, Meir and Itamar Pitowsky (2007), Quantum probability and many worlds, Studies in History and Philosophy of Modern Physics, 38, pp Kochen, Simon and Ernst P. Specker (1967), The Problem of Hidden Variables in Quantum Mechanics, Journal of Mathematics and Mechanics, 17, pp Lewis, David (1980), A Subjectivist s Guide to Objective Chance, in Studies in Inductive Logic and Probability, ed. by Richard C Jeffrey, University of California Press, Berkeley, vol. 2, pp Pitowsky, Itamar (2003), Betting on the Outcomes of Measurements: a Bayesian Theory of Quantum Probability, Studies in History and Philosophy of Modern Physics, 34, pp Sober, Elliott (2010), Evolutionary theory and the reality of macro-probabilities, in The Place of Probability in Science, ed. by Ellery Eells and James H. Fetzer, Springer, Dordrecht, pp Spekkens, Robert W. (2005), Contextuality for Preparations, Transformations, and Unsharp Measurements, Physical Review A, 71, 5, p (2017), Noncontextuality: how we should define it, why it is natural, and what to do about its failure, Contextuality: Conceptual Issues, Operational Signatures, and Applications, ayflash.php?id= Steeger, Jeremy (2017), Betting on Quantum Objects, version 3, ArXiv e-prints (July 2017), arxiv: [hist-ph]. Von Neumann, John (1955), Mathematical Foundations of Quantum Mechanics, Princeton University Press, Princeton. 5