Where Quantum Probability Comes From
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In A Philosophical Essay on Probabilities, published in 1814, Pierre-Simon Laplace introduced a notorious hypothetical creature: a “vast intelligence” that knew the complete physical state of the present universe. For such an entity, dubbed “Laplace’s demon” by subsequent commentators, there would be no mystery about what had happened in the past or what would happen at any time in the future. According to the clockwork universe described by Isaac Newton, the past and future are exactly determined by the present.
Laplace’s demon was never supposed to be a practical thought experiment; the imagined intelligence would have to be essentially as vast as the universe itself. And in practice, chaotic dynamics can amplify tiny imperfections in the initial knowledge of the system into complete uncertainty later on. But in principle, Newtonian mechanics is deterministic.
A century later, quantum mechanics changed everything. Ordinary physical theories tell you what a system is and how it evolves over time. Quantum mechanics does this as well, but it also comes with an entirely new set of rules, governing what happens when systems are observed or measured. Most notably, measurement outcomes cannot be predicted with perfect confidence, even in principle. The best we can do is to calculate the probability of obtaining each possible outcome, according to what’s called the Born rule: The wave function assigns an “amplitude” to each measurement outcome, and the probability of getting that result is equal to the amplitude squared. This feature is what led Albert Einstein to complain about God playing dice with the universe.
Researchers continue to argue over the best way to think about quantum mechanics. There are competing schools of thought, which are sometimes referred to as “interpretations” of quantum theory but are better thought of as distinct physical theories that give the same predictions in the regimes we have tested so far. All of them share the feature that they lean on the idea of probability in a fundamental way. Which raises the question: What is “probability,” really?
Like many subtle concepts, probability starts out with a seemingly straightforward, commonsensical meaning, which becomes trickier the closer we look at it. You flip a fair coin many times; whether it comes up heads or tails on any particular trial is completely unknown, but if we perform many trials we expect to get heads 50% of the time and tails 50% of the time. therefore say that the probability of obtaining heads is 50%, and likewise for tails.
We know how to handle the mathematics of probability, thanks to the work of the Russian mathematician Andrey Kolmogorov and others. Probabilities are real numbers between zero and one, inclusive; the probabilities of all independent events add up to one; and so on. But that’s not the same as deciding what probability actually is.
There are numerous approaches to defining probability, but we can distinguish between two broad classes. The “objective” or “physical” view treats probability as a fundamental feature of a system, the best way we have to characterize physical behavior. An example of an objective approach to probability is frequentism, which defines probability as the frequency with which things happen over many trials, as in our coin-tossing example.