The question is how much the game changes if, as part of their strategizing, Alice and Bob prepare some quantum state on which they can later perform measurements. We say the graphs G and H are quantum isomorphic if there is a way for Alice and Bob to fool the referee with this additional resource.
Ada Chan and I showed that any two Hadamard graphs on the same number of vertices are quantum isomorphic. This was the first instance where more than two graphs were shown to be pairwise quantum isomorphic. The result for Hadamard graphs follows from a more general recipe for showing quantum isomorphism of graphs arising from certain association schemes. The main result is built from three tools. A remarkable recent result of Mančinska and Roberson shows that graphs G and H are quantum isomorphic if and only if, for any planar graph F, the number of graph homomorphisms from F to G is equal to the number of graph homomorphisms from F to H. A generalization of partition functions called "scaffolds" affords us some basic reduction rules such as series-parallel reduction and can be applied to counting homomorphisms. The final tool is the classical theorem of Epifanov showing that any plane graph can be reduced to a single vertex and no edges by extended series-parallel reductions and Delta-Wye transformations. This last sort of transformation is available to us in the case of exactly triply regular association schemes.
The goal of the talk is to walk through these ideas without making things unnecessarily technical. No knowledge of physics is assumed and, for this narrative, we will avoid the the quantum commuting framework and work in a finite-dimensional quantum tensor framework.
instrumentalmethod that he claims as his own invention. (He also includes this in Book VIII in the context of a discussion of mechanics!) In this talk, to set the stage, we will discuss the reduction of cube doubling to the problem asking for the construction of two mean proportionals in continued proportion between two given lines. We will present Pappus's method in full (and discuss what it has to do with mechanics and why it does not contradict the fact that some of us learned in Modern Algebra that this problem cannot be solved with straightedge and compass). Then we will turn to a fascinating and historically rich passage at the start of Book III where Pappus gives what can be described as something almost like a
referee's reporton a different proposed solution of the problem. As we'll see, Pappus's judgment is very negative, although with the eyes of modern mathematicians, we can see that the proposed method does have some merit. Please click HERE for the slides of the talk.