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"Higher linear algebra in topology and quantum information theory", David Reutter, 2019

Reviewed September 12, 2023

Citation: Reutter, David. Higher linear algebra in topology and quantum information theory. Diss. University of Oxford, 2019.

Web: http://www.cs.ox.ac.uk/people/aleks.kissinger/theses/reutter-thesis.pdf

Tags: Hadamard-matrices, Monoidal-categories

This PhD thesis gives a very nice perspective on what the author calls "categorical quantum mechanics". A very convincing case is made for why higher categorical structures are natural tools for the study of quantum information. A very charming example is borrowed from

> Abramsky, Samson, and Bob Coecke. "A categorical semantics of quantum protocols." Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science, 2004.. IEEE, 2004.

In that paper, it is shown that a state in V\otimes W is maximally entangled if and only if it appears as the coevaluation map C\to V\otimes W in a duality between V and W.

An interesting section of this thesis to me is the one based off the paper

> Reutter, David J., and Jamie Vicary. "Biunitary constructions in quantum information." arXiv preprint arXiv:1609.07775 (2016).

Here, a very nice recent perspective on the use of Hadamard matrices (and Latin squares) for quantum error bases. Additionally, a construction of a new higher-category-inspired error basis is given which cannot be constructed from known bases. It is interesting to think about what topological phase these might correspond to.

Because quantum mechanics is based on Hilbert spaces, it is no surprise that the notation of 2-Hilbert space introduced in

> Baez, John C. "Higher-dimensional algebra II. 2-Hilbert spaces." Advances in Mathematics 127.2 (1997): 125-189.

plays a major role. According to Week 137 of John Baez's blog: A modular tensor category is a braided 2-H*-algebra whose Mueger center is trivial.