# 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>](http://en.wikipedia.org/wiki/Bra-ket_notation)), they usually go about it with the [Von Neumann-Morgenstern axioms](http://en.wikipedia.org/wiki/Von_Neumann%E2%80%93Morgenstern_utility_theorem), modified to refer to quantum mechanics:

1.  Completeness: Every state can be compared to every other, preferencewise.
2.  Transitivity: If you prefer |A> to |B> and |B> to |C>, you also prefer |A> to |C>.
3.  Continuity: If you prefer |A> to |B> and |B> to |C>, there's some quantum-mechanical [measure](http://en.wikipedia.org/wiki/Probability_measure) (note that this is a change from "probability") X such that you're indifferent between (1-X)|A> + X|C> and |B>.
4.  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](http://en.wikipedia.org/wiki/Quantum_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](http://en.wikipedia.org/wiki/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.