Abstract
We study the monoidal dagger category of Hilbert C*-modules over a commutative C*-algebra from the perspective of categorical quantum mechanics. The dual objects are the finitely presented projective Hilbert C*-modules. Special dagger Frobenius structures correspond to bundles of uniformly finite-dimensional C*-algebras. A monoid is dagger Frobenius over the base if and only if it is dagger Frobenius over its centre and the centre is dagger Frobenius over the base. We characterise the commutative dagger Frobenius structures as finite coverings, and give nontrivial examples of both commutative and central dagger Frobenius structures. Subobjects of the tensor unit correspond to clopen subsets of the Gelfand spectrum of the C*-algebra, and we discuss dagger kernels.
Highlights
Categorical quantum mechanics [36] provides a powerful graphical calculus for quantum theory
We study the monoidal category of Hilbert modules over a commutative C*-algebra
– Just like commutative C*-algebras are dual to locally compact Hausdorff spaces, we prove that Hilbert modules are equivalent to bundles of Hilbert spaces over locally compact Hausdorff spaces
Summary
Categorical quantum mechanics [36] provides a powerful graphical calculus for quantum theory. – We prove that dagger Frobenius structures correspond to finite-dimensional C*algebras that vary continuously over the base space Commutative Frobenius structures are equivalent to finite coverings of the base space The proof of this fact uses that Frobenius structures have dual objects, otherwise finite branched coverings might be allowed [43]; we leave open a characterisation of commutative H*-algebras [2]. Frobenius structures in a category like that of Hilbert modules need not copy classical information elementwise as previously thought: there may be no copyable states at all.
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