This post is an explanation of a recent paper coauthored by Sean Carroll and Charles Sebens, where they propose a derivation of the Born rule in the context of the Many World approach to quantum mechanics. While the attempt itself is not fully successful, it contains interesting ideas and it is thus worthwhile to know.
A note to the reader: here I will try to enlighten the preconditions and give only a very general view of their method, and for this reason you won’t find any equation. It is my hope that if after having read this you’re still curious about the real math, you will point your browser to the preceding link and read the paper for yourself.
If you are not totally new to LessWrong, you should know by now that the preferred interpretation of quantum mechanics (QM) around here is the Many World Interpretation (MWI), which negates the collapse of the wave-function and postulates a distinct reality (that is, a branch) for every base state composing a quantum superposition.
MWI historically suffered from three problems: the absence of macroscopic superpositions, the preferred basis problem, the Born rule derivation. The development of decoherence famously solved the first and, to a lesser degree, the second problem, but the role of the third still remains one of the most poorly understood side of the theory.
Quantum mechanics assigns an amplitude, a complex number, to each branch of a superposition, and postulates that the probability of an observer to find the system in that branch is the (squared) norm of the amplitude. This, very briefly, is the content of the Born rule (for pure states).
Quantum mechanics remains agnostic about the ontological status of both amplitudes and probabilities, but MWI, assigning a reality status to every branch, demotes ontological uncertainty (which branch will become real after observation) to indexical uncertainty (which branch the observer will find itself correlated to after observation).
Simple indexical uncertainty, though, cannot reproduce the exact predictions of QM: by the Indifference principle, if you have no information privileging any member in a set of hypothesis, you should assign equal probability to each one. This leads to forming a probability distribution by counting the branches, which only in special circumstances coincides with amplitude-derived probabilities. This discrepancy, and how to account for it, constitutes the Born rule problem in MWI.
There have been of course many attempts at solving it, for a recollection I quote directly the article:
One approach is to show that, in the limit of many observations, branches that do not obey the Born Rule have vanishing measure. A more recent twist is to use decision theory to argue that a rational agent should act as if the Born Rule is true. Another approach is to argue that the Born Rule is the only well-deﬁned probability measure consistent with the symmetries of quantum mechanics.
These proposals have failed to uniformly convince physicists that the Born rule problem is solved, and the paper by Carroll and Sebens is another attempt to reach a solution.
Before describing their approach, there are some assumptions that have to be clarified.
The first, and this is good news, is that they are treating probabilities as rational degrees of belief about a state of the world. They are thus using a Bayesian approach, although they never call it that way.
The second is that they’re using self-locating indifference, again from a Bayesian perspective.
Self-locating indifference is the principle that you should assign equal probabilities to find yourself in different places in the universe, if you have no information that distinguishes the alternatives. For a Bayesian, this is almost trivial: self-locating propositions are propositions like any other, so the principle of indifference should be used on them as it should on any other prior information. This is valid for quantum branches too.
The third assumption is where they start to deviate from pure Bayesianism: it’s what they call Epistemic Separability Principle, or ESP. In their words:
the outcome of experiments performed by an observer on a speciﬁc system shouldn’t depend on the physical state of other parts of the universe.
This is a kind of a Markov condition: the request that the system is such that it screens the interaction between the observer and the system observed from every possible influence of the environment.
It is obviously false for many partitions of a system into an experiment and an environment, but rather than taking it as a Principle, we can make it an assumption: an experiment is such only if it obeys the condition.
In the context of QM, this condition amounts to splitting the universal wave-function into two components, the experiment and the environment, so that there’s no entanglement between the two, and to consider only interactions that can factors as a product of an evolution for the environment and an evolution for the experiment. In this case, environment evolution act as the identity operator on the experiment, and does not affect the behavior of the experiment wave-function.
Thus, their formulation requires that the probability that an observer finds itself in a certain branch after a measurement is independent on the operations performed on the environment.
Note though, an unspoken but very important point: probabilities of this kind depends uniquely on the superposition structure of the experiment.
A probability, being an abstract degree of belief, can depend on all sorts of prior information. With their quantum version of ESP, Carroll and Sebens are declaring that, in a factored environment, probabilities of a subsystem does not depend on the information one has about the environment. Indeed, in this treatment, they are equating factorization and lack of logical connection.
This is of course true in quantum mechanics, but is a significant burden in a pure Bayesian treatment.
That said, let’s turn to their setup.
They imagine a system in a superposition of base states, which first interacts and decoheres with an environment, then gets perceived by an observer. This sequence is crucial: the Carroll-Sebens move can only be applied when the system already has decohered with a sufficiently large environment.
I say “sufficiently large” because the next step is to consider a unitary transformation on the “system+environment” block. This transformation needs to be of this kind:
- it respects ESP, in that it has to factor as an identity transformation on the “observer+system” block;
- it needs to equally distribute the probability of each branch in the original superposition on a different branch in the decohered block, according to their original relative measure.
Then, by a simple method of rearranging labels of the decohered base, one can show that the correct probabilities comes out by the indifference principle, in the very same way that the principle is used to derive the uniform probability distribution in the second chapter of Jaynes’ Probability Theory.
As an example, consider a superposition of a quantum bit, and say that one branch has a higher measure with respect to the other by a factor of square root of 2. The environment needs in this case to have at least 8 different base states to be relabeled in such a way to make the indifference principle work.
In theory, in this way you can only show that the Born rule is valid for amplitudes which differ one another by the square root of a rational number. Again I quote the paper for their conclusion:
however, since this is a dense set, it seems reasonable to conclude that the Born Rule is established.
Evidently, this approach suffers from a number of limits: the first and the most evident is that it works only in a situation where the system to be observed has already decohered with an environment. It is not applicable to, say, a situation where a detector reads a quantum superposition directly, e.g. in a Stern-Gerlach experiment.
The second limit, although less serious, is that it can work only when the system to be observed decoheres with an environment which has sufficiently base states to distribute the relative measure in different branches. This number, for a transcendental amplitude, is bound to be infinite.
The third limit is that it can only work if we are allowed to interact with the environment in such a way as to leave the amplitudes of the interaction between the system and the observer untouched.
All of these, which are understood as limits, can naturally be reversed and considered as defining conditions, saying: the Born rule is valid only within those limits.
I’ll leave it to you to determine if this constitutes a sufficient answers to the Born rule problem in MWI.