Session R07: The Author in Dialogue: Jeffery Bub, Bananaworld: Quantum Mechanics for Primates

Pragmatism: A Natural Home for Information-Theoretic Interpretations of Quantum Theory?

Interpretations of quantum theory are often divided into realist and instrumentalist camps. Dominant realist interpretations include the Everett (many worlds) interpretation, the de Broglie-Bohm interpretation and the GRW theory. The tradition of classical American pragmatism, with origins in the work of Peirce, Dewey and James, provides the philosophical basis for a `third way' which is neither instrumentalist nor realist in the traditional sense. Some recent interpretations of quantum theory may be seen as attempts to occupy the kind of philosophical position that the pragmatists opened up. These include the information theoretic interpretation presented by Jeff Bub in \textit{Bananaworld} and the QBist interpretation due to Chris Fuchs, R\"{u}diger Schack and others. In the case of QBism, there are clear lines of historical influence from the pragmatist philosophers. Both interpretations are also closely connected to ongoing developments in physics. They draw on the resources of the burgeoning field of quantum information theory and on ongoing efforts in the foundations of quantum theory to reaxiomatise the theory in operationalist or information-theoretic terms.   

Elliptopes and Polyhedra: Quantum Correlations and Their Classical Simulations

I use correlation arrays, the workhorse of Bub's Bananaworld, to analyze the correlations found in an experimental setup due to Mermin for measurements on the singlet state of a pair of spin-1/2 particles. Adopting an approach pioneered by Pitowsky and promoted in Bananaworld, I show that the class of correlations allowed by quantum mechanics in this setup can be represented geometrically as an elliptope in a non-signaling cube. I then introduce special raffles to determine which of these quantum correlations are allowed by local hidden-variable theories. The subclass of the quantum correlations that can be simulated in this way can be represented geometrically by a tetrahedron inscribed within the elliptope. I extend this analysis to the singlet state of two particles with higher spin. The class of correlations allowed by quantum mechanics in this case is still represented by the elliptope; the subclass of those whose main features I can simulate with my raffles can be represented by polyhedra that, with increasing spin, take up more and more of the volume of the elliptope. The elliptope is thus a general constraint on correlations of the kind studied in this Mermin setup, a result which predates quantum mechanics and was already recognized by the statistician Yule in 1896.   

The Measurement Problem, "Big" and "Small"

Bub's Bananaworld can be seen as the sequel to Bub and Pitowsky's "Two dogmas about quantum mechanics." Bub and Pitowsky introduced the notion of a "truthmaker" to characterize the difference between phase space in classical mechanics and Hilbert space in quantum mechanics. Points in phase space are truthmakers in the sense that they fix the values of all observables, represented by functions on phase space. State vectors in Hilbert space fail to do so in two ways. First, the state vector does not tell us which observables, represented by operators in Hilbert space, will be assigned definite values. Second, even when these observables are specified, the state vector only gives us probabilities for finding particular values. This then raises two questions. First, how is one (set of) observable(s) rather than another selected to be assigned definite values? Second, why does an observable, once selected, take on one value rather than another? These two questions correspond to what Bub and Pitowsky, with irony, call the "small" and the "big" measurement problem, respectively. They claim that abandoning the two dogmas they identify in their paper solves both. I will argue that they are right about the "big" but wrong about the "small" measurement problem.   

If relativity is about space and time, what is quantum mechanics about?

h $-abstract-$\backslash $pardThe theory of relativity is about the structure of space and time: we were wrong in thinking that events occur in a flat Euclidean 3$+$1-dimensional manifold. Similarly, quantum mechanics is fundamentally about probability: we were wrong in thinking that probability is just a measure of ignorance. The transition from classical to quantum mechanics involves going from a commutative to a non-commutative algebra of observables, equivalently from a Boolean to a non-Boolean algebra of 2-valued observables, which represent properties or propositions. The non-Boolean algebra of quantum mechanics is not embeddable into a Boolean algebra, which is to say there are no `hidden variables' whose values would `complete' the quantum state description to a consistent assignment of truth or falsity to all propositions (technically, a 2-valued homomorphism on the algebra). Non-Booleanity allows new sorts of nonlocal probabilistic correlations with no causal explanation, associated with `entangled' quantum states, that are not possible in a Boolean or classical theory. I will expand on these ideas with reference to the Bohr-Einstein debate about the completeness of quantum mechanics and recent arguments applying quantum mechanics to complex systems that include agents who are themselves using quantum theory.$\backslash $pard-/abstract-$\backslash $\tex