M.C. Escher and Quantum Mechanics
(Contextuality and Paradoxes)

Last weekend I attended the New Direction conference in Washington DC and I have great new content from there. Today I will start presenting my favorite talk: “Contextuality: At the Borders of Paradox.” by Samson Abramsky from Oxford University. This talk was showcasing this preprint: http://arxiv.org/pdf/1502.03097v1.pdf

But what is contextuality? Here is how Abramsky puts it: "Contextuality can be understood as arising where we have a family of data which is locally consistent, but globally inconsistent." Look at the Escher picture above. On each side the stairs make sense, but not together.

Now how does Abramsky goes about quantifying contextuality? For the sake of a definite example, consider Hardy's paradox. Suppose Alice has two settings: \(a_1 \) and \(a_2\), and Bob has two settings as well: \(b_1\) and \(b_2\). Also during measurement they can obtain the outcomes 0 or 1.

Then in Hardy;s paradox case there are 4 logical formulas arising out of the experimental results:

\(a_1 \wedge b_1 , \lnot(a_1 \wedge b_1), \lnot (a_2 \wedge b_1), a_2 \vee b_2\)

Together those formulas lead to contradictions and we say that Hardy's paradox demands contextuality.

Now if we plot this as a fiber bundle we get the following picture:

and this kind of picture can be constructed for any quantum mechanics system. Now for the fireworks: one can investigate this using the tools of cohomology. The cohomology used is that of sheaf cohomology, but singular cohomology would work as well in this case. Basically cohomology gives you the topological obstruction to making global sense of the data. 

The amazing thing is that the philosophical concept of contextuality has a very precise mathematical representation in terms of algebraic topology tools. Even more surprising is the link with computer science in terms of relational databases. Relational databases consists of tables and those tables respect what is called the three normal forms. For performance reasons, programmers perform what is called a "denormalization", and the ultimate denormalization is to construct one single huge table containing the entire data. Now sometimes this is not possible, and the same cohomology theory used in analyzing quantum systems is used here as well to detect when this is impossible. Who knew quantum mechanics has a deep relationship with relational databases?

The ultimate motivation for Abramsky is however practical. Can we harness contextuality and use it to have better solutions in quantum information theory? The first step is to derive the mathematical tools needed to pose the question in a precise way. This is what Abramsky achieved with his amazing research.