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Mixed states and module categories

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The theory of finite-dimensional (f.d.) pure state quantum mechanics is encapsulated by the dagger compact closed category Hilb of f.d. Hilbert spaces and linear maps. There are at least two good reasons why we might want to consider categories other than Hilb. Firstly, we are often interested in states/maps which are ‘covariant’ – that is, compatible with the action of some symmetry group G. The relevant generalisation here is the category Rep(G) of f.d. unitary representations of G. Secondly, in quantum communication, quantum error correction etc. we need to move from the pure-state theory to the more general mixed-state theory, which is usually formulated in terms of f.d. C*-algebras and completely positive linear maps. The relevant generalisation here is not a 1-category but a 2-category – namely 2Hilb, the 2-category of f.d. 2-Hilbert spaces. In general, it turns out that moving from a tensor category T to its 2-category of module categories Mod(T) is the right construction to obtain a mixed state theory from a pure one. In this talk I will explain why. If there is enough time, I will demonstrate how the graphical calculus of the 2-category Mod(Rep(G)) allows us to prove new results in covariant zero-error communication theory.

This talk is part of the Theoretical computer science seminar series.

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