University of Oxford
Talks by Bob Coecke
For well over a decade, we developed an entirely pictorial (and formally rigorous!) presentation of quantum theory [*]. At the present, experiments are being setup aimed at establishing the age at which children could effectively learn quantum theory in this manner. Meanwhile, the pictorial language has also been successful in the study of natural language, and very recently we have started to apply it to model cognition, where we employ GPT-alike models. We present the key ingredients of the pictorial language language as well as their interpretation across disciplines. [*] B.
We establish a tight relationship between two key quantum theoretical notions: non-locality and complementarity. In particular, we establish a direct connection between Mermin-type non-locality scenarios, which we generalise to an arbitrary number of parties, using systems of arbitrary dimension, and performing arbitrary measurements, and a new stronger notion of complementarity which we introduce here. Our derivation of the fact that strong complementarity is a necessary condition for a Mermin scenario provides a crisp operational interpretation for strong complementarity.
It is now exactly 75 years ago that John von Neumann denounced his own Hilbert space formalism: ``I would like to make a confession which may seem immoral: I do not believe absolutely in Hilbert space no more.'' (sic)  His reason was that Hilbert space does not elucidate in any direct manner the key quantum behaviors. One year later, together with Birkhoff, they published "The logic of quantum mechanics". However, it is fair to say that this program was never successful nor does it have anything to do with logic. So what is logic?
Our starting point is a particular `canvas' aimed to `draw' theories of physics, which has symmetric monoidal categories as its mathematical backbone. With very little structural effort (i.e. in very abstract terms) and in a very short time this categorical quantum mechanics research program has reproduced a surprisingly large fragment of quantum theory.
Multipartite quantum states constitute a (if not the) key resource for quantum computations and protocols. However obtaining a generic, structural understanding of entanglement in N-qubit systems is still largely an open problem. Here we show that multipartite quantum entanglement admits a compositional structure. The two SLOCC-classes of genuinely entangled 3-qubit states, the GHZ-class and the W-class, exactly correspond with the two kinds of commutative Frobenius algebras on C^2, namely `special' ones and `anti-special' ones.
In our approach, rather than aiming to recover the 'Hilbert space model' which underpins the orthodox quantum mechanical formalism, we start from a general `pre-operational' framework, and verify how much additional structure we need to be able to describe a range of quantum phenomena. This also enables us to investigate which mathematical models, including more abstract categorical ones, enable one to model quantum theory. Till now, all of our axioms only refer to the particular nature of how compound quantum systems interact, rather that to the particular structure of state-spaces.
Researchers in quantum foundations claim (D'Ariano, Fuchs, ...): Quantum = probability theory + x and hence: x = Quantum - probability theory Guided by the metaphorical analogy: probability theory / x = flesh / bones we introduce a notion of quantum measurement within x, which, when flesing it with Hilbert spaces, provides orthodox quantum mechanical probability calculus.
Yes, that's indeed where it happens. These pictures are not ordinary pictures but come with category-theoretic algebraic semantics, support automated reasoning and design of protocols, and match perfectly the developments in important areas of mathematics such as representation theory, proof theory, TQFT & GR, knot theory etc. More concretely, we report on the progress in a research program that aims to capture logical structures within quantum phenomena and quantum informatic tasks in purely diagrammatic terms.