Quantum theory from information inference principles

Tuesday, Nov 11th
Philipp Hoehn, Perimeter Institute 
Title: Quantum theory from information inference principles 
PDF of the talk (800k)
Audio [.wav 40MB]

by Matteo Smerlak, Perimeter Institute

When a new theory enters the scene of physics, a succession of events normally takes place: at first, nobody cares; then a minority starts playing with the maths while the majority insists that the theory is obviously wrong; farther down the road, we find the majority using the maths on a daily basis and all arguing that the theory is so beautiful, it can only be right; along the way, thanks to many years of practice, a new kind of intuition grows out of the formalism, and our entire picture of reality changes accordingly. This is the process of science.

For some reason, though, the eventual shift from formalism to intuition never happened for quantum mechanics (QM). Ninety years after its discovery, specialists still call QM “weird”, teachers still quote Feynman claiming that “nobody really understands QM”, and philosophers still discuss whether QM requires us to be “antirealist”, “neo-Kantian”, “Bayesian”… you name it. Niels Bohr wanted new theories to be “crazy enough”, but it seems this one is just too crazy. And yet it works!

In the face of this puzzle, a school of thought initiated by Birkhoff and von Neumann in the thirties has declared it its mission to reconstruct QM. The idea is simple: if you don’t get how the machine works, then roll up your sleeves, take the machine apart, and build it again—from scratch. Indeed this is how Einstein delt with the symmetry group of Maxwell’s equations (and its mysterious action on lengths and durations): he found intuitive two physical principles—the relativity principles—and derived the Lorentz group (the set of symmetries of Maxwell's equations) from them. Thus special relativity was “really understood”.

Much recent work towards a reconstruction of QM has taken place within a framework called “generalized probability theories” (GPT). This approach elaborates on basic notions such as preparations, transformations and measurements. The main achievement of GPT has been to locate QM within a more general landscape of possible modifications of classical probability theory. It has showed for instance that QM is not the most non-local theory consistent with what is known as no-signaling property: stronger correlations than quantum entanglement are in principle possible, though they are not realized in nature. To understand what is, we must know what else could have been—thus speak GPT proponents. 

Philipp uses a different language for his reconstruction of QM: instead of measurements and states, he talks about questions and answers. The semantic shift is not innocent: while a “measurement” uncovers the intrinsic state of a system, a “question” only brings information to whoever asks it—that is, a question relates to two entities (the system and the observer/interrogator) rather than just one (the system). Because there isn’t anybody out there to ask questions about everything, there is no such thing as the “state of the universe”, Philipp says!

This so-called “relational” questions/answers approach to QM was advocated twenty years ago by Rovelli, who emphasized its similarity with the structure of gravitation (time is relative, remember?). He also proposed two basic informational principles: one states that the total information that an observer O can gather about a system S is limited; the second specifies that, even when O has obtained the maximum amount of information about S, she can still learn something about S by asking other, “complementary” questions. Thence non commuting operators! Similar ideas where discussed independently by Zeilinger and Brukner—and Philipp embraces them wholeheartedly.

But he also takes a big step further. Adding four more postulates to Rovelli’s (which he calls completeness, preservation, time evolution and locality), Philipp shows how to reconstruct the set Σ of all possible states of S relative to O (together with its isometry group, representing possible time evolutions). For a quantum system allowing only one independent question—a qubit—Σ is a three-dimensional ball, the Bloch sphere. (Note that a 3-ball is a much bigger space than a 1-ball, the state space of a classical bit—enter quantum computing…) For systems with more independent questions, i.e. N qubits, Σ is the mathematical structure known as the convex cone over some complex projective space—not quite what is known as a Calabi-Yau manifold, but still a challenge for the mind to picture.

N=2 turns out to be the most difficult case: once this one is solved—Philipp says this took him a full year, with inputs from his collaborator Chris Wever—, higher N’s follow rather straightforwardly. This is a reflection of a crucial aspect of QM: quantum systems are “monogamous”, meaning that they can establish strong correlations (aka “entanglement”) with just one partner at a time. Philipp’s questions/answers formulation provides a new and detailed understanding of this peculiar correlation structure, which he represents as a spherical tiling. “QM is beautiful!”, says Philipp.

One limitation of Philipp’s current approach—also pointed out by the audience—is the restriction to binary (or yes/no) questions. A spin-1 particle, for instance, falls outside this framework, for it can give three different answers to the question “what is your spin in the z direction?”, namely “up”, “down” or “zero”. Can Philipp deal with such ternary question, and reconstruct the 8 dimensional state space of a quantum “trit”? We wish him to find the answer within… less than a year!