It feels like a good time to enumerate the science things I’ve written or co-written over this past year. In reverse chronological order:
1. C. A. Fuchs and BCS, “QBism: Quantum Theory as a Hero’s Handbook.”
This paper represents an elaboration of the lectures delivered by one of us (CAF) during “Course 197 — Foundations of Quantum Physics” at the International School of Physics “Enrico Fermi” in Varenna, Italy, July 2016. Much of the material for it is drawn from arXiv:1003.5209, arXiv:1401.7254, and arXiv:1405.2390. However there are substantial additions of original material in Sections 4, 7, 8 and 9, along with clarifications and expansions of the older content throughout. Topics include the meaning of subjective probability; no-cloning, teleportation, and quantum tomography from the subjectivist Bayesian perspective; the message QBism receives from Bell inequality violations (namely, that nature is creative); the import of symmetric informationally complete (SIC) quantum measurements for the technical side of QBism; quantum cosmology QBist-style; and a potential meaning for the holographic principle within QBism.
2. M. Appleby, C. A. Fuchs, BCS and H. Zhu, “Introducing the Qplex: A Novel Arena for Quantum Theory.”
We reconstruct quantum theory starting from the premise that, as Asher Peres remarked, “Unperformed experiments have no results.” The tools of modern quantum information theory, and in particular the symmetric informationally complete (SIC) measurements, provide a concise expression of how exactly Peres’s dictum holds true. That expression is a constraint on how the probability distributions for outcomes of different, mutually exclusive experiments mesh together, a type of constraint not foreseen in classical thinking. Taking this as our foundational principle, we show how to reconstruct the formalism of quantum theory in finite-dimensional Hilbert spaces. Along the way, we derive a condition for the existence of a d-dimensional SIC.
We take a tour of a set of equiangular lines in eight-dimensional Hilbert space. This structure defines an informationally complete measurement, that is, a way to represent all quantum states of three-qubit systems as probability distributions. Investigating the shape of this representation of state space yields a pattern of connections among a remarkable spread of mathematical constructions. In particular, studying the Shannon entropy of probabilistic representations of quantum states leads to an intriguing link between the questions of real and of complex equiangular lines. Furthermore, we will find relations between quantum information theory and mathematical topics like octonionic integers and the 28 bitangents to a quartic curve.
Recently, Zhu classified all the SIC-POVMs whose symmetry groups act doubly transitively. Lattices of integers in the complex numbers, the quaternions and the octonions yield the key parts of these symmetry groups.
Also, some things I’d written earlier found official homes in 2016. This paper—which I worked on because I’d never taken material all the way from a lecture recording to written form, and it sounded like a fun thing to try—became a book chapter.
5. C. A. Fuchs and BCS, “Some Negative Remarks on Operational Approaches to Quantum Theory.”
Over the last 10 years there has been an explosion of “operational reconstructions” of quantum theory. This is great stuff: For, through it, we come to see the myriad ways in which the quantum formalism can be chopped into primitives and, through clever toil, brought back together to form a smooth whole. An image of an IQ-Block puzzle comes to mind. There is no doubt that this is invaluable work, particularly for our understanding of the intricate connections between so many quantum information protocols. But to me, it seems to miss the mark for an ultimate understanding of quantum theory; I am left hungry. I still want to know what strange property of matter forces this formalism upon our information accounting. To play on something Einstein once wrote to Max Born, “The quantum reconstructions are certainly imposing. But an inner voice tells me that they are not yet the real thing. The reconstructions say a lot, but do not really bring us any closer to the secret of the `old one’.” In this talk, I hope to expand on these points and convey some sense of why I am fascinated with the problem of the symmetric informationally complete POVMs to an extent greater than axiomatic reconstructions.
This is now a chapter in Quantum Theory: Informational Foundations and Foils, G. Chiribella and R. W. Spekkens, eds. (Springer, 2016.)
An unexpected connection exists between compatibility criteria for quantum states and symmetric informationally complete POVMs. Beginning with Caves, Fuchs and Schack’s “Conditions for compatibility of quantum state assignments” [Phys. Rev. A 66 (2002), 062111], I show that a qutrit SIC-POVM studied in other contexts enjoys additional interesting properties. Compatibility criteria provide a new way to understand the relationship between SIC-POVMs and mutually unbiased bases, as calculations in the SIC representation of quantum states make clear. This, in turn, illuminates the resources necessary for magic-state quantum computation, and why hidden-variable models fail to capture the vitality of quantum mechanics.
This was published as Mathematics 4, 2 (2016): 36.
7. BCS, “Von Neumann Was Not a Quantum Bayesian.”
Wikipedia has claimed for over three years now that John von Neumann was the “first quantum Bayesian.” In context, this reads as stating that von Neumann inaugurated QBism, the approach to quantum theory promoted by Fuchs, Mermin and Schack. This essay explores how such a claim is, historically speaking, unsupported.
This began with me getting irritated by one sentence on Wikipedia and writing 25 pages in response. (I fully endorse spite as a motivator for science!) It was published as Phil. Trans. Roy. Soc. A, 374, 2068 (2016), 20150235.