I have built on the work of Peres, Mermin and GHZ (Greenberger, Horne and Zeilinger) to give a proof of Bell's theorem without inequalities that involves only two distant observers, rather than the three involved in the classic proof of GHZ. My proof has the interesting feature that the observers first establish the BKS theorem independently and then find, on getting together and comparing their observations, that they have also established Bell's theorem. Similar work was done independently, and at about the same time, by Adan Cabello. One area of activity that has been impacted by this work is Quantum Pseudo-Telepathy.

I pointed out that the three-particle GHZ state is similar to the Borromean rings in that making a measurement on one of the particles disentangles the other two, just as cutting one of the Borromean rings allows the other two to be pulled apart. However I encountered difficulties in pushing this analogy further because of the subtle nature of measurement in quantum mechanics. The knot theorist Lou Kauffman became interested in the problem and took up the challenge of making a connection between knots and quantum entanglement, although he proceeded along lines rather different from mine. You can learn about some of his work from this talk or paper . However the most fruitful connection between knots (or braids) and the quantum theory is probably through anyons and topological quantum computing , which holds the promise of leading to robust (i.e., error-free) quantum computers.

Vaidman, Aharonov and Albert introduced a quantum retrodiction problem they termed the "mean king's problem" and showed how to solve it for a qubit or two-state system. Essentially, Bob gives Alice a qubit and tells her to measure its spin along the x-, y- or z- direction; Alice then tells Bob the direction along which she measured but not what she got, and Bob has to guess her result, which he always does correctly. But the quantum theory says that a qubit cannot have definite spin components along all three axes at the same time, so Bob's (or VAA's) feat seems nothing short of magical. Aharonov and Englert showed how to extend the trick to a particle whose state space has a prime dimension and I then showed how to generalize their solution to all prime power dimensions by exploiting the known MUBs (mutually unbiased bases) in these dimensions. This problem illustrates nicely how the availability of quantum resources allows a party to perform a feat that would be impossible with any amount of conventional (i.e. non-quantum) resources alone.

Together with my ex-graduate student and good friend Mordecai Waegell, I have been involved in an expansive project to identify quantum contextual sets in symmetrical objects of all kinds in dimension 4 and up. Such sets are of interest in connection with applications such as quantum computing, random number generation and parity oblivious transfer. We have found numerous examples of quantum contextual sets in structures as diverse as the exceptional four-dimensional polytopes, Gosset's polytope in eight dimensions, a variety of complex polytopes associated with the N-qubit Pauli group, the Witting polytope and the binary and ternary Golay codes. The sets come in a variety of guises, with "parity proofs" and "Diophantine proofs" being the easiest to recognize, but with others being neither easy to recognize or characterize. The sets also possess many interesting geometrical and combinatorial properties that we have uncovered only partially. Our recent discovery of quantum contextual sets in states derived from the Golay codes leads us to suspect that the Leech lattice and the sporadic groups might harbor such sets as well, although it is likely to take a heroic amount of work (and much more knowledge than we possess) to settle this question. The current focus of effort in contextuality studies (see Budroni et al for an authoritative overview) seeks to broaden our understanding of the concept by seeing, for example, how it can be given an operational meaning that allows it to be established more decisively through experimental tests or by demonstrating that it is the source of the quantum advantage in a particular application. It is our hope that the rich lode of raw material we have unearthed on contextuality in a variety of symmetrical structures in progressively higher dimensions can be used in conjunction with more fundamental studies to give us a better understanding of this still elusive concept and also assist with the search for new applications.