Quantum Mechanics and Decision Theory

By Sean Carroll | April 16, 2012 8:20 am

Several different things (all pleasant and work-related, no disasters) have been keeping me from being a good blogger as of late. Last week, for example, we hosted a visit by Andy Albrecht from UC Davis. Andy is one of the pioneers of inflation, and these days has been thinking about the foundations of cosmology, which brings you smack up against other foundational issues in fields like statistical mechanics and quantum mechanics. We spent a lot of time talking about the nature of probability in QM, sparked in part by a somewhat-recent paper by our erstwhile guest blogger Don Page.

But that’s not what I want to talk about right now. Rather, our conversations nudged me into investigating some work that I have long known about but never really looked into: David Deutsch’s argument that probability in quantum mechanics doesn’t arise as part of a separate ad hoc assumption, but can be justified using decision theory. (Which led me to this weekend’s provocative quote.) Deutsch’s work (and subsequent refinements by another former guest blogger, David Wallace) is known to everyone who thinks about the foundations of quantum mechanics, but for some reason I had never sat down and read his paper. Now I have, and I think the basic idea is simple enough to put in a blog post — at least, a blog post aimed at people who are already familiar with the basics of quantum mechanics. (I don’t have the energy in me for a true popularization at the moment.) I’m going to try to get to the essence of the argument rather than being completely careful, so please see the original paper for the details.

The origin of probability in QM is obviously a crucial issue, but becomes even more pressing for those of us who are swayed by the Everett or Many-Worlds Interpretation. The MWI holds that we have a Hilbert space, and a wave function, and a rule (Schrödinger’s equation) for how the wave function evolves with time, and that’s it. No extra assumptions about “measurements” are allowed. Your measuring device is a quantum object that is described by the wave function, as are you, and all you ever do is obey the Schrödinger equation. If MWI is to have some chance of being right, we must be able to derive the Born Rule — the statement that the probability of obtaining a certain result from a quantum measurement is the square of the amplitude — from the underlying dynamics, not just postulate it.

Deutsch doesn’t actually spend time talking about decoherence or specific interpretations of QM. He takes for granted that when we have some observable X with some eigenstates |xi>, and we have a system described by a state

$latex |psirangle = a |x_1rangle + b |x_2rangle , $

then a measurement of X is going to return either x1 or x2. But we don’t know which, and at this stage of the game we certainly don’t know that the probability of x1 is |a|2 or the probability of x2 is |b|2; that’s what we’d like to prove.

In fact let’s just focus on a simple special case, where

$latex a = b = frac{1}{sqrt{2}} . $

If we can prove that in this case, the probability of either outcome is 50%, we’ve done the hard part of the work — showing how probabilistic conclusions can arise at all from non-probabilistic assumptions. Then there’s a bit of mathematical lifting one must do to generalize to other possible amplitudes, but that part is conceptually straightforward. Deutsch refers to this crucial step as deriving “tends to from does,” in a mischievous parallel with attempts to derive ought from is. (Except I think in this case one has a chance of succeeding.)

The technique used will be decision theory, which is a way of formalizing how we make rational choices. In decision theory we think of everything we do as a “game,” and playing a game results in a “value” or “payoff” or “utility” — what we expect to gain by playing the game. If we have the choice between two different (mutually exclusive) actions, we always choose the one with higher value; if the values are equal, we are indifferent. We are also indifferent if we are given the choice between playing two games with values V1 and V2 or a single game with value V3 = V1 + V2; that is, games can be broken into sub-games, and the values just add. Note that these properties make “value” something more subtle than “money.” To a non-wealthy person, the value of two million dollars is not equal to twice the value of one million dollars. The first million is more valuable, because the second million has a smaller marginal value than the first — the lifestyle change that it brings about is much less. But in the world of abstract “value points” this is taken into consideration, and our value is strictly linear; the value of an individual dollar will therefore depend on how many dollars we already have.

There are various axioms assumed by decision theory, but for the purposes of this blog post I’ll treat them as largely intuitive. Let’s imagine that the game we’re playing takes the form of a quantum measurement, and we have a quantum operator X whose eigenvalues are equal to the value we obtain by measuring them. That is, the value of an eigenstate |x> of X is given by

$latex V[|xrangle] = x .$

The tricky thing we would like to prove amounts to the statement that the value of a superposition is given by the Born Rule probabilities. That is, for our one simple case of interest, we want to show that

$latex Vleft[frac{1}{sqrt{2}}(|x_1rangle + |x_2rangle)right] = frac{1}{2}(x_1 + x_2) . qquadqquad(1)$

After that it would just be a matter of grinding. If we can prove this result, maximizing our value in the game of quantum mechanics is precisely the same as maximizing our expected value in a probabilistic world governed by the Born Rule.

To get there we need two simple propositions that can be justified within the framework of decision theory. The first is:

Given a game with a certain set of possible payoffs, the value of playing a game with precisely minus that set of payoffs is minus the value of the original game.

Note that payoffs need not be positive! This principle explains what it’s like to play a two-person zero-sum game. Whatever one person wins, the other loses. In that case, the value of the game to the two participants are equal in magnitude and opposite in sign. In our quantum-mechanics language, we have:

$latex Vleft[frac{1}{sqrt{2}}(|-x_1rangle + |-x_2rangle)right] = – Vleft[frac{1}{sqrt{2}}(|x_1rangle + |x_2rangle)right] . qquadqquad (2)$

Keep that in mind. Here’s the other principle we need:

If we take a game and increase every possible payoff by a fixed amount k, the value is equivalent to playing the original game, then receiving value k.

If I want to change the value of a playing a game by k, it doesn’t matter whether I simply add k to each possible outcome, or just let you play the game and then give you k. I don’t think we can argue with that. In our quantum notation we would have

$latex Vleft[frac{1}{sqrt{2}}(|x_1+krangle + |x_2+krangle)right] = Vleft[frac{1}{sqrt{2}}(|x_1rangle + |x_2rangle)right] +k . qquadqquad (3)$

Okay, if we buy that, from now on it’s simple algebra. Let’s consider the specific choice

$latex k = -x_1 – x_2 $

and plug this into (3). We get

$latex Vleft[frac{1}{sqrt{2}}(|-x_2rangle + |-x_1rangle)right] = Vleft[frac{1}{sqrt{2}}(|x_1rangle + |x_2rangle)right] -x_1 – x_2. $

You can probably see where this is going (if you’ve managed to make it this far). Use our other rule (2) to make this

$latex -2 Vleft[frac{1}{sqrt{2}}(|x_1rangle + |x_2rangle)right] = -x_1 – x_2 , $

which simplifies straightaway to

$latex Vleft[frac{1}{sqrt{2}}(|x_1rangle + |x_2rangle)right] = frac{1}{2}(x_1 + x_2) , $

which is our sought-after result (1).

Now, notice this result by itself doesn’t contain the word “probability.” It’s simply a fairly formal manipulation, taking advantage of the additivity of values in decision theory and the linearity of quantum mechanics. But Deutsch argues — and on this I think he’s correct — that this result implies we should act as if the Born Rule is true if we are rational decision-makers. We’ve shown that the value of a game described by an equal quantum superposition of states |x1> and |x2> is equal to the value of a game where we have a 50% chance of gaining value x1 and a 50% chance of gaining x2. (In other words, if we acted as if the Born Rule were not true, someone else could make money off us by challenging us to such games, and that would be bad.) As someone who is sympathetic to pragmatism, I think that “we should always act as if A is true” is the same as “A is true.” So the Born Rule emerges from the MWI plus some seemingly-innocent axioms of decision theory.

While I certainly haven’t followed the considerable literature that has grown up around this proposal over the years, I’ll confess that it smells basically right to me. If anyone knows of any strong objections to the idea, I’d love to hear them. But reading about it has added a teensy bit to my confidence that the MWI is on the right track.


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About Sean Carroll

Sean Carroll is a Senior Research Associate in the Department of Physics at the California Institute of Technology. His research interests include theoretical aspects of cosmology, field theory, and gravitation. His most recent book is The Particle at the End of the Universe, about the Large Hadron Collider and the search for the Higgs boson. Here are some of his favorite blog posts, home page, and email: carroll [at] cosmicvariance.com .


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