# Quantum Physics and the Law of Large Numbers

On page 1 of this recent paper by Bousso and Susskind we read.

This question is not about philosophy. Without a precise form of decoherence, one cannot claim that anything really "happened", including the specific outcomes of experiments. And without the ability to causally access an infinite number of precisely decohered outcomes, one cannot reliably verify the probabilistic predictions of a quantum-mechanical theory.

According to the Law of Large Numbers there is no requirement that I need to perform an infinite number of experiments in order to verify that results will asymptotically approach an expected value. Why would Quantum Mechanics be any different?

UPDATE: I wanted to find an relatively simple proof of the strong law of large numbers, best I can come up with is the combination of the following two wikipedia entries:

Kolmogorov's Three series Theorem

Kronecker's lemma

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Of course that you're right. In reality, we always perform a finite number of finite-accuracy experiments, and they're not even quite well-defined because the quantities never perfectly decohere. For example, we can't say that the Moon is over there and flying over there because its position and velocity don't even commute. Still, if the probability of some unpleasant option is smaller than e.g. $10^{-200}$, we can surely neglect such an option - or slight inconsistency - for all practical and almost all impractical purposes. It's silly to require that the accuracy etc. is infinite. –  Luboš Motl May 21 '11 at 16:04
In my opinion, the authors of that paper are utterly wrong when they say This question is not about philosophy. –  Peter Shor May 22 '11 at 2:11
The law of large numbers is also valid in quantum mechanics. –  Raskolnikov May 22 '11 at 9:25
Is the key word used here 'reliably'? Limiting our accuracy as above to 10^-200 I actually quite fancy our chances of reliability across the length of the universe! –  Nic May 22 '11 at 14:34
I wish that physicists would stop misusing the word 'philosophy' –  Jerry Schirmer May 22 '11 at 18:14

The article by Bousso and Susskind is a very eccentric interpretation of quantum mechanics based upon a total misunderstanding of decoherence and collapse motivated by a desperate need to get rid of macroscopic superpositions in our world.

Decoherence is not about diagonalizing the reduced density matrix. As they mentioned, this is very subjective and sensitive to what is traced over. Their causal diamond proposal picks out Planck scale quantum fluctuations at the boundary of the causal diamond, not what they were expecting. This is by no means objective.

They reject the many worlds interpretation but then try to resurrect it by an infinite ensemble of identical quantum systems in the same universe, all accessible to the same observer at the "Census Bureau". For that to work, they need to assume a Copenhagen-style collapse at the "Census Bureau" to definite outcomes. Their idea is to reject the many-worlds interpretation but then resurrect it using the Copenhagen interpretation applied to an infinite ensemble to get back objectively measurable probabilities! Probabilities must be exactly objectively measurable! If this sounds insane, that is because it is!

This is followed by some mumbo-jumbo invocations of horizon complementarity ending up with the conclusion that the hat receives infinitely many copies of quantum information about the causal patches of the other multiverses ending up with a big crunch. This violates the no-cloning principle!

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The Law of Large Numbers does not say as much as you seem to think it says. There is a great deal of misunderstanding about this. Suppose that the probability of a coin's toss being heads is one half. The Law of Large Numbers says nothing at all about what the asymptotic results will be. It only says what the probability is that the asymptotic results will be $X$. That the asymptotic result of the ratio of heads will be unity has a probability of zero, for example. But probability zero does not mean physically impossible, and hence the Law does not say that the asymptotic result will not be unity. The great English analyst Littlewood explained this clearly in a famous maths club talk, later published in his collection of essays, A Mathematician's Miscellany, entitled "The Dilemma of Probability". Kolmogorov himself said essentially the same thing, in print. For more references and a discussion (oh, and other people have also referred to the play Rosencrantz and Guildenstern Are Dead, by Tom Stoppard), see my "Logic of Physical Probability Assertions", http://arxiv.org/abs/quant-ph/0508059 , and prof. Jan von Plato's important article cited there.

The excerpt from Littlewood is as follows:

Mathematics \dots has no grip on the real world; if probability is to deal with the real world it must contain elements outside mathematics, the \it meaning\rm\ of «probability» must relate to the real world; and there must be one or more «primitive» propositions about the real world, from which we can then proceed deductively (i.e., mathematically).
We will suppose (as we may by lumping several primitive propositions together) that there is just one primitive proposition, the «probability axiom», and we will call it «$A$» for short.

\dots the «real» probability problem; what are the axiom $A$ and the meaning of «probability» to be, and how can we justify $A$? It will be instructive to consider the attempt called the «frequency theory». It is natural to believe that if (with the natural reservations) an act like throwing a die is repeated $n$ times the proportion of 6's will, with certainty, tend to a limit, $p$ say, as $n \rightarrow \infty$. (Attempts are made to sublimate the limit into some Pickwickian sense---«limit» in inverted commas.
But either you mean the ordinary limit, or else you have the problem of explaing how «limit» behaves, and you are no further. You do not make an illegitimate conception legitimate by putting it into inverted commas.) If we take this proposition as $A$ we can at least settle off-hand the other problem, of the meaning of probability, we can define its measure for the event in question to be the number $p$. But for the rest this $A$ takes us nowhere. Suppose we throw 1000 times and wish to know what to expect. Is 1000 large enough for the convergence to have got under way, and how far? $A$ does not say. We have, then, to add to it something about the rate of convergence.

Now an $A$ cannot assert a certainty about a particular number $n$ of throws, such as «the proportion of 6's will certainly be within $p\pm\epsilon$ for large enough $n$ (the largeness depending on $\epsilon$)».
It can only say « the proportion will lie between $p\pm\epsilon$ with at least such and such probability (depending on $\epsilon$ and $n_o$) whenever $n>n_o$.» The vicious circle is apparent. We have not merely failed to justify a workable $A$; we have failed even to state one which would work if its truth were granted.

http://www.library.uu.nl/digiarchief/dip/diss/1957294/c4.pdf

and

http://philsci-archive.pitt.edu/archive/00000367/00/ergodic.ps

are two reviews of prof. von Plato's ergodic theory of probability, which itself is not on-line.

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Verification is a separate issue from meaning. (Except for the, fortuately now rare, logical positivists who say the meaning of a statement is the procedure you would use to verify it, or at least falsify its denial.) The long answer is about the meaning of probability. This answer addresses the part of the question about verification.

Even in classical physics, verification is never 100% certain. For all practical purposes, you can verify a probability statement by looking at lots and lots of trials, and for all practical purposes, you can do this without understanding the logical basis of probability or its physical meaning or the proof of Kolmogoroff's Strong Law of Large Numbers.

There is no easy proof of Kolmogoroff's Strong Law of Large Numbers, which is the one about asymptotic tendencies (i.e., probabilities of limits of ratios). The Weak Law of Large Numbers (due to Bernoulli) is much easier, and is the one about limits of probabilities as discussed by Littlewood above. (Probabilities of limits is not at all the same as limits of probabilities.) If you knew what characteristic functions were, I think there could be given an easy proof...but the theory of characteristic functions relies on Bochner's theorem which is not easy....

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By the way, probabilities of $10^{-200}$ are not something that can be ignored. Most probabilities of observations of complex events are extremely tiny! Suppose that we throw a die 1000 times and ask for the probability that we obtained precisely the sequence of numbers that we actually obtained. The probability for it is $6^{-1000}\ll 10^{-200}$ although we just experienced an event with this tiny probability.