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Oxford Quantum Talks Archive

Non-unital Frobenius algebras: first steps in infinite-dimensional categorical quantum mechanics

Samson Abramsky, University of Oxford
Flowin'Cat 2010, October 2010, University of Oxford

(Joint work with Chris Heunen) Frobenius algebras have emerged as important structures within categorical quantum mechanics, used e.g. to give an abstract notion of observables. When equipped with units, they induce compact structure, and hence can only have finite-dimensional realizations in Hilbert spaces. Since quantum observables in general live in infinite-dimensional Hilbert space, we are led to investigate to what extent the theory carries over when the assumption of units is dropped. In Hilbert spaces, we find a natural connection with beautiful work by Warren Ambrose from 1945 on H*-algebras, which admit perfect infinite-dimensional analogues of the classical Wedderburn structure theorems, which give the appropriate mathematical context for understanding the correspondences between Frobenius algebra structures and bases which have been observed in finite-dimensional Hilbert spaces. The notion of H*-algebra can be axiomatized in general monoidal dagger categories. It can be shown to imply the Frobenius axiom, while unital Frobenius algebras are H*. This leads to the question of whether Frobenius without units implies H* in general. We are able to show that it does in various cases, including quantale-valued relations and positive l2 matrices. The general question, and the case of Hilbert spaces in particular, remains open.

[video] [streaming video] [slides]