Quantum probability from subjective likelihood: Improving on Deutsch's proof of the probability rule

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Abstract

I present a proof of the quantum probability rule from decision-theoretic assumptions, in the context of the Everett interpretation. The basic ideas behind the proof are those presented in Deutsch's recent proof of the probability rule, but the proof is simpler and proceeds from weaker decision-theoretic assumptions. This makes it easier to discuss the conceptual ideas involved in the proof, and to show that they are defensible.

Introduction

The mathematical formalism of the quantum theory is capable of yielding its own interpretation (DeWitt, 1970).

If I were to pick one theme as central to the tangled development of the Everett interpretation of quantum mechanics, it would probably be: the formalism is to be left alone. What distinguished Everett's original paper both from the Dirac-von Neumann collapse-of-the-wavefunction orthodoxy and from contemporary rivals such as the de Broglie–Bohm theory was its insistence that unitary quantum mechanics need not be supplemented in any way (whether by hidden variables, by new dynamical processes, or whatever).

Many commentators on the Everett interpretation—even some, like David Deutsch, who are sympathetic to it (Deutsch, 1985)—have at various points and for various reasons retreated from this claim.1 The “preferred basis problem”, for instance, has induced many to suppose that quantum mechanics must be supplemented by some explicit rule that picks out one basis as physically special. Many suggestions were made for such a rule (Barrett, 1999 discusses several); all seem to undermine the elegance (and perhaps more crucially, the relativistic covariance) of Everett's original proposal. Now the rise of decoherence theory has produced a broad consensus among supporters of Everett (in physics, if perhaps not yet in philosophy) that the supplementation was after all not necessary. (For more details of this story, see Wallace, 2002, Wallace, 2003a and references therein.)

Similarly, various commentators (notably Albert and Loewer, 1988, Bell, 1981, Bell, 1987, Butterfield, 1996) have suggested that the Everett interpretation has a problem with persistence of objects (particles, cats, people, etc.) over time, and some have been motivated to add explicit structure to quantum mechanics in order to account for this persistence (in particular, this is a prime motivation for Albert and Loewer's original Many-Minds theory). Such moves again undermine the rationale for an Everett-type interpretation. And again, a response to such criticism which does not require changes to the formalism was eventually forthcoming. It was perhaps implicit in Everett's original discussion of observer memory states, and in Gell-Mann and Hartle's later notion of IGUSs (Gell-Mann & Hartle, 1990); it has been made explicit by Simon Saunders’ work (Saunders, 1998) on the analogy between Everettian branching and Parfittian fission.

In both these cases, my point is not that the critics were foolish or mistaken: Everettians were indeed obliged to come up with solutions both to the preferred-basis problem and to the problem of identity over time. However, in both cases the obvious temptation—to modify the formalism so as to solve the problem by fiat—has proved to be unnecessary: it has been possible to find solutions within the existing theory, and thus to preserve those features of the Everett interpretation which made it attractive in the first place.

Something similar may be going on with the other major problem of the Everett interpretation: that of understanding quantitative probability. One of the most telling criticisms levelled at Everettians by their critics has always been their inability to explain why, when all outcomes objectively occur, we should regard one as more likely than the other. The problem is not merely how such talk of probability can be meaningful; it is also how the specific probability rule used in quantum mechanics is to be justified in an Everettian context. Here too, the temptation is strong to modify the formalism so as to include the probability rule as an explicit extra postulate.

Here too, it may not be necessary. David Deutsch has, I believe, transformed the debate by attempting (Deutsch, 1999) to derive the probability rule within unitary quantum mechanics, via considerations of rationality (formalised in decision-theoretic terms). His work has not so far met with wide acceptance, perhaps in part because it does not make it at all obvious that the Everett interpretation is central (and his proof manifestly fails without that assumption).

In Wallace (2003b), I have presented an exegesis of Deutsch's proof in which the Everettian assumptions are made explicit. The present paper may be seen as complementary to my previous paper: it presents an argument in the spirit of Deutsch's, but rather different in detail. My reasons for this are two-fold: firstly, I hope to show that the mathematics of Deutsch's proof can be substantially simplified, and his decision-theoretic axioms greatly weakened; secondly and perhaps more importantly, by simplifying the mathematical structure of the proof, I hope to be able to give as clear as possible a discussion of the conceptual assumptions and processes involved in the proof. In this way I hope that the reader may be in a better position to judge whether or not the probability problem, like other problems before it, can indeed be solved without modifications to the quantum formalism.

The structure of the paper is as follows. In Section 2 I review the account of branching that Everettians must give, and distinguish two rather different viewpoints that are available to them; in Section 3 I consider how probability might fit into such an account. Sections 4 and 5 are the mathematical core of the paper: they present an extremely minimal set of decision-theoretic assumptions and show how, in combination with an assumption which I call equivalence, they are sufficient to derive the quantum probability rule. The next three sections are a detailed examination of this postulate of equivalence. I argue that it is unacceptable as a principle of rationality for single-universe interpretations (Section 6), but is fully defensible for Everettians—either via the sort of arguments used by Deutsch (Section 7) or directly (Sections 8 and 9). Section 10 is the conclusion.

Section snippets

Thinking about branching

The conceptual problem posed by branching is essentially one of transtemporal identity: given branching events in my future, how can it even make sense for me to say things like “I will experience such-and-such”? Sure, the theory predicts that people who look like me will have these experiences, but what experiences will I have? Absent some rule to specify which of these people is me, the only options seem to be (1) that I am all of them, in which case I will presumably have all of their

Weight and probability

The paradigm of a quantum measurement is something like this: prepare a system (represented by a Hilbert space H) in some state (represented by a normalised vector ψ in H). Carry out some measurement process (represented by a discrete-spectrum self-adjoint operator X1 over H) on the system, and look to see what result (represented by some element of the spectrum of X1) is obtained.

Suppose some such measurement process is denoted by M, and suppose that associated with M is some set SM of

A rudimentary decision theory

In this section I will develop some of the formal details of the subjectivist program, in a context which will allow ready application to quantum theory. Our starting point is the following: define a likelihood ordering as some two-place relation holding between ordered pairs E,M, where M is a quantum measurement and E is an event in EM (that is, E is a subset of the possible outcomes of the measurement). We write the relation as :E|MF|Nis then to be read as “It's at least as likely that

A quantum representation theorem

It goes without saying that this set of axioms alone is insufficient to derive the quantum probability rule: absolutely no connection has yet been made between the decision-theoretic axioms and quantum theory. We can make this connection, however, via two further posits. Firstly, we need to assume that we have a fairly rich set of quantum measurements available to us: rich, in fact, in the sense of the following definition.

Weight richness: A set M of quantum measurements is rich provided that,

Equivalence and the single universe

So far, so good. The “decision-theoretic turn” suggested by Deutsch not only allows us to make sense of probability in an Everettian universe, but it also allows us to derive the quantitative form of the probability rule from assumptions—equivalence and the richness of the set of measurements—which prima facie are substantially weaker than the rule itself.

Nonetheless the situation remains unsatisfactory. Equivalence has the form of a principle of pure rationality: it dictates that any agent who

Equivalence via measurement neutrality

If we have shown that equivalence is implausible except in the Everett interpretation, still we have not shown that it is plausible for Everettians; that is our next task. An obvious starting point is Deutsch's original work; but Deutsch makes no direct use of equivalence. Instead, he uses—implicitly, but extensively—a principle which I have elsewhere (Wallace, 2003b) called measurement neutrality: the principle that once we have specified which system is being measured, which state that system

Equivalence, directly

Perhaps the reader is not prepared to accept the SU viewpoint; or perhaps s/he is unconvinced by the SU-dependent defence of measurement neutrality which I offered above. Either way, it seems worth looking directly at equivalence, to see if it can be justified without recourse to measurement neutrality. This is also of interest because it allows a direct reply to those critics of Deutsch (such as Lewis, 2003) who argue that there can be no decision-theoretic reason to be indifferent between

Branching indifference

There are two closely related lacunae in the erasure proof of equivalence. Firstly, erasure may lead to branching: realistic erasure processes will usually lead not to a single ‘erased(i)’, reward but to a superposition of them. Secondly, equivalence must hold not just when we have two equally weighted branches, but when we have one branch whose weight equals the combined weights of several other branches.

Both of these lacunae would be resolved if we could establish

Branching indifference: An

Conclusion

In Wallace (2003b), I identified four assumptions which I claimed (and still claim!) are required for Deutsch's proof of the probability rule to go through: the Everett interpretation, the SU viewpoint on that interpretation, measurement neutrality, and a “fairly strong set of decision-theoretic axioms”. I also argued that measurement neutrality was at least plausibly a consequence of the SU viewpoint (using roughly the argument of (the current paper's) Section 7).

The present paper may be seen

Acknowledgements

For useful conversations and detailed feedback at various points over the evolution of this paper, I would like to thank Harvey Brown, Adam Elga, Hilary Greaves, Peter Lewis, David Papineau, and especially Jeremy Butterfield and Simon Saunders.

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