David Mermin’s essay is a welcome introduction to the quantum Bayesian, or QBist, approach to quantum theory. Unfortunately, it stops just as the story gets interesting. The QBist research program is illuminating not just because it makes old philosophic problems melt away but because it nourishes new ideas that take root in the solid ground of everyday physics.

For example, the field of quantum-state tomography is concerned with how to acquire information about unknown quantum states. But if one holds, as the QBists do, that quantum states are states of knowledge, information, or—most provocatively—belief, what can it mean for a state of knowledge to be unknown? A subjective Bayesian classical statistician is in the same quandary when confronted with the notion of an unknown probability. That problem was solved by Bruno de Finetti’s representation theorem, which establishes that “unknown probability” is sloppy shorthand for a rigorously defined experimental scenario. Informally speaking, the classical de Finetti theorem says that if one is willing to gamble that the order of the repeated trials in a multi-trial experiment does not matter, then he can act as though he is measuring an “unknown probability.” QBism resolves the conundrum of “unknown quantum states” with a quantum generalization of the de Finetti theorem. In turn, those developments, born from QBist philosophical inquiries, have led to practical advances in quantum tomography.1 

Furthermore the QBist drive to understand quantum states as states of belief has stimulated fascinating technical work on the old problem of how to reexpress quantum mechanics entirely in terms of the probabilities that the states catalog. That work has led to advances in areas ranging from quantum information theory to the pure mathematics of Lie algebras.2 

QBism treats measurement—the intervention of one piece of the natural world (the physical system) into the experience of another (the agent)—as a fundamental primitive process of quantum theory. Seeking a better definition of “measurement” from conventional pre-QBist quantum theory is rather like expecting the standard axioms of arithmetic to define the fundamental primitive terms “number,” “zero,” and “successor”: an exercise in missing the point.

Some theorists treat quantum theory as a noncommuting generalization of classical probability.3 Others think of it as overlapping with classical stochastic mechanics.4 QBism takes quantum theory to be a specialization of unadorned probability theory, in which the restriction in question is imposed by the character of the physical world. If this variety of options does not say that today is an exciting time to study quantum probability, what could?

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, “Network Theory,” http://math.ucr.edu/home/baez/networks.