Utility functions and quantum mechanics
Interpreting quantum mechanics throws an interesting wrench into utility calculation.
Utility functions, according to the interpretation typical in these parts, are a function of the state of the world, and an agent with consistent goals acts to maximize the expected value of their utility function. Within the many-worlds interpretation (MWI) of quantum mechanics (QM), things become interesting because “the state of the world” refers to a wavefunction which contains all possibilities, merely in differing amounts. With an inherently probabilistic interpretation of QM, flipping a quantum coin has to be treated linearly by our rational agent—that is, when calculating expected utility, they have to average the expected utilities from each half. But if flipping a quantum coin is just an operation on the state of the world, then you can use any function you want when calculating expected utility.
And all coins, when you get down to it, are quantum. At the extreme, this leads to the possible rationality of quantum suicide—since you’re alive in the quantum state somewhere, just claim that your utility function non-linearly focuses on the part where you’re alive.
As you may have heard, there have been several papers in the quantum mechanics literature that claim to recover ordinary rules for calculating expected utility in MWI—how does that work?
Well, when they’re not simply wrong (for example, by replacing a state labeled by the number a+b with the state |a> + |b>), they usually go about it with the Von Neumann-Morgenstern axioms, modified to refer to quantum mechanics:
Completeness: Every state can be compared to every other, preferencewise.
Transitivity: If you prefer |A> to |B> and |B> to |C>, you also prefer |A> to |C>.
Continuity: If you prefer |A> to |B> and |B> to |C>, there’s some quantum-mechanical measure (note that this is a change from “probability”) X such that you’re indifferent between (1-X)|A> + X|C> and |B>.
Independence: If you prefer |A> to |B>, then you also prefer (1-X)|A> + X|C> to (1-X)|B> + X|C>, where |C> can be anything and X isn’t 1.
In classical cases, these four axioms are easy to accept, and lead directly to utility functions with X as a probability. In quantum mechanical cases, the axioms are harder to accept, but the only measure available is indeed the ordinary amplitude-squared measure (this last fact features prominently in Everett’s original paper). This gives you back the traditional rule for calculating expected utilities.
For an example of why these axioms are weird in quantum mechanics, consider the case of light. Linearly polarized light is actually the same thing as an equal superposition of right-handed and left-handed circularly polarized light. This has the interesting consequence that even when light is linearly polarized, if you shine it on atoms, those atoms will change their spins—they’ll just change half right and half left. Or if you take circularly polarized light and shine it on a linear polarizer, half of it will go through. So anyhow, we can make axiom 4 read “If you are indifferent between left-polarized light and right-polarized light, then you must also be indifferent between linearly polarized light (i.e. left+right) and circularly polarized light (right+right).” But… can’t a guy just want circularly polarized light?
Under what sort of conditions does the independence axiom make intuitive sense? Ones where something more complicated than a photon is being considered. Something like you. If MWI is correct and you measure the polarization of linearly polarized light vs. circularly polarized light, this puts your brain in a superposition of linear vs. circular. But nobody says “boy, I really want a circularly polarized brain.”
A key factor, as is often the case when talking about recovering classical behavior from quantum mechanics, is decoherence. If you carefully prepare your brain in a circularly polarized state, and you interact with an enormous random system (like by breathing air, or emitting thermal radiation), your carefully prepared brain-state is going to get shredded. It’s a fascinating property of quantum mechanics that once you “leak” information to the outside, things are qualitatively different. If we have a pair of entangled particles and a classical phone line, I can send you an exact quantum state—it’s called quantum teleportation, and it’s sweet. But if one of our particles leaks even the tiniest bit, even if we just end up with three particles entangled instead of two, our ability to transmit quantum states is gone completely.
In essence, the states we started with were “close together” in the space where quantum mechanics lives (Hilbert space), and so they could interact via quantum mechanics. Interacting with the outside even a little scattered our entangled particles farther apart.
Any virus, dust speck, or human being is constantly interacting with the outside world. States that are far enough apart to be perceptibly different to us aren’t just “one parallel world away,” like would make a good story—they are cracked wide open, spread out in the atmosphere as soon as you breathe it, spread by the Earth as soon as you push on it with your weight. If we were photons, one could easily connect with their “other selves”—if you try to change your polarization, whether you succeed or fail will depend on the orientation of your oppositely-polarized “other self”! But once you’ve interacted with the Earth, this quantum interference becomes negligible—so negligible that we seem to neglect it. When we make a plan, we don’t worry that our nega-self might plan the opposite and we’ll cancel each other out.
Does this sort of separation explain an approximate independence axiom, which is necessary for the usual rules for expected utility? Yes.
Because of decoherence, non-classical interactions are totally invisible to unaided primates, so it’s expected that our morality neglects them. And if the states we are comparing are noticeably different, they’re never going to interact, so independence is much more intuitive than in the case of a single photon. Taken together with the other axioms, which still make a lot of sense, this defines expected utility maximization with the Born rule.
So this is my take on utility functions in quantum mechanics—any living thing big enough to have a goal system will also be big enough to neglect interaction between noticeably different states, and thus make decisions as if the amplitude squared was a probability. With the help of technology, we can create systems where the independence axiom breaks down, but these systems are things like photons or small loops of superconducting wire, not humans.
Thanks for the article, which clarified some stuff for me that I should have been able to put together on my own, but hadn’t!
I’d like to disagree with your conclusion, though, which seems to be that my utility function should necessarily be additive if interference is negligible. (If you’re just saying that this is what evolution is predicted to produce on most branches, I have no quibbles with that.)
Suppose that you wake up in a white room, and Omega appears to you and tells you that it’s simulating you, for one year, after which it will shut down the simulation. You have three options for how this time will be spent. In option (A), Omega will reveal to you the true physical Theory of Everything, and help you understand how it all really works. In option (B), Omega will explain to you in detail the workings of the human mind, and help you understand how subjective experience arises from it. In option (C), Omega will flip two fair quantum coins; if they land (heads,heads), it will run simulation (A), if they land (heads,tails), it will run (B), and if the first coin lands tails, Omega will shut down the simulation immediately.
If Omega also explains that Copenhagen is definitely true, picking (C) would make little sense, since with 50% probability you’d just die, which surely isn’t a reasonable price for not having to choose what simulation you’re going to be in. But if Omega explains that MWI is definitely true, option (C) would mean that you get to learn about both subjects, on distinct Everett branches. Sure, these two branches of you won’t ever get to compare notes, but does that really mean that on abstract grounds I’m not allowed to be willing to trade half of my amplitude for having the more diverse experience in the other half?
[ETA: By the way, the two simulations are deterministic—your further experiences will be exactly the same on all branches running a given simulation, except in the very small minority of branches where enough cosmic rays hit Omega’s circuits etc. that something goes seriously awry in its innards.]
Actually, after a little more thought, there’s a non-physically-impossible version of this that’s a bit more counterintuitive.
Suppose you’re about to decide what you’ll do with the rest of your life—you have a choice to study the true physical theory of everything, or you have a choice to study the workings of the human mind. You have similar confidence for both that you’ll find success. You calculate that you’ll have slightly higher utility if you do (pick your favorite).
So the question is, do you flip a qubit in order to choose what to do, on the premise that you’d rather “someone out there” study the other thing too?
Doing things this way fixes the fact that Omega is non-quantum-mechanical, so it gives you impossible certainty of the future and current wavefunctions. The fact that this is more counterintuitive suggests a few other approximations we might be making.
I’m pretty sure we agree that since this example is non-observable, evolution isn’t going to select for something like it :P
Yet here we are as a by-product of selecting for other things, apparently. So that’s a reasonable point. You appear to have come up with an intuitive “circularly polarized brain.”
Though that’s not to say I’d take (C) - getting shot half the time is a rather awful trade for someone out there like me knowing something cool.
What if Omega is inside a simulation of a Copenhagen universe and was lied to by meta-Omega about it being a MWI metaverse?
Then Omega would have been wrong—and Omega is never wrong!
(Recall that this is a thought experiment...)
Where did you get these axioms? They don’t seem to make much sense, and they are in fact even mathematically inconsistent with QM (you have to at least replace (1-X) and X with sqrt(1-X) and sqrt(X) or you get non-unit state vectors).
I think it makes more sense to consider Von Neumann-Morgenstern axioms over density operators, which can be interpreted as (equivalence classes of) probability distributions on pure states. You can always consider these states as parts of the pure states of a larger system with unobservable degrees of freedom ( quantum state purification ).
Why? I understand that the people here tend to think that’s a good idea, but it doesn’t allow for Dutch book betting like violating Bayes’ Theorem does.
Not exactly. If we were to make a bet on the millionth digit of pi, or the accuracy of string theory, or whether or not there is a particle with a mass in a certain range, the result would be the same in every Everett branch.
The thing I am asserting without bothering to back it up is called the expected utility hypothesis.
Fair enough. I believe you get the point though.
It’s equivalent to having different priors and doing it linearly (although you might have to involve infinitesimal probabilities to get it to work exactly right). That does raise the question of whether those things really can be considered separate.
Yes, though the result of the bet (i.e. your observations) would be different in an infinitesimal fraction of them.
/useless nitpick that doesn’t actually change your point
You mean, half of it will go through.
Right, fix’d.
So? In such a case, regular suicide would also be rational. (That is, instead of surviving all of the time with utility 1, you would prefer to survive p of the time with utility U>1/p.)
The standard formulation is that you would only prefer to commit quantum suicide if, if you happen to survive, you’ll be better off by some standard. Overdosing but waking up in the hospital doesn’t count, but gambling your life in interesting ways does.
The reason quantum suicide is attractive when regular suicide isn’t is because MWI gives but there’s still a chance extra emotional weight.
The “only states in which I’m alive matter” utility function will tell you to play Russian Roulette at any odds, so long as the payoff is positive, regardless of whether you believe MWI or not.
And even if you call death utility 0, then it can still be rational to play Russian Roulette, so long as the payoff in utilons is high enough to justify the chance of 0 utility, again whether you believe MWI or not.
The results you quote are very interesting and answer questions I’ve been worrying about for some time. Apologies for bringing up two purely technical inquiries:
Could you provide a reference for the result you quote? You referred to Eq. (34) in Everett’s original paper in another comment, but this doesn’t seem to make the link to the VNM axioms and decision theory.
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That seems wrong to me. There has to be a formulation of the form if the two initially perfectly entangled particles get only slightly entangled with other particles, then quantum teleportation still works with high fidelity / a high probability of success—otherwise quantum teleportation wouldn’t be feasible in reality.
Manfred, what about quantum computers? They can be big enough to have a goal system, and will work only when their states are coherent… Will they generate a new kind of morality?
If you can care about a quantum computer, then the quantum computer will generate no new kind of morality by caring about itself.
Sure, it puts is in an unusual position—but the goal of this post was to show that this position isn’t qualitatively different, jsut something we haven’t had much reason to care about yet.
Can you explain why “the only measure available is indeed the ordinary amplitude-squared measure”?
Also, I’m confused about this:
According to the Wikipedia entry you linked to, a probability measure is a real-valued function, but X here is apparently just a number? What’s the significance of your parenthetical note here?
It’s kind of an abstract mathematical fact. If you are fine with that, I recommend reading Everett’s original paper, which includes this fact as equation 34.
The significane is that now we’re talking about some “fundamental” property of the universe, rather than probability, which is more about our ignorance of what’s going to happen. Um, so if that description of probability didn’t make sense, the distinction won’t make much sense.