Complexity in algebraic QFT
Stefan Hollands and Alessio Ranallo
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Submission date: 21. Feb. 2023
MSC-Numbers: 81P45, 81T05, 46L10
Keywords and phrases: complexity theory, Quantum Field Theory, Operator algebras
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We consider a notion of complexity of quantum channels in relativistic continuum quantum ﬁeld theory (QFT) deﬁned by the distance to the trivial (identity) channel. Our distance measure is based on a speciﬁc divergence between quantum channels derived from the Belavkin-Staszewski (BS) divergence. We prove in the prerequisite generality necessary for the algebras in QFT that the corresponding complexity has several reasonable properties: (i) the complexity of a composite channel is not larger than the sum of its parts, (ii) it is additive for channels localized in spacelike separated regions, (iii) it is convex, (iv) for an N-ary measurement channel it is log N, (v) for a conditional expectation associated with an inclusion of QFTs with ﬁnite Jones index it is given by log(Jones Index). The main technical tool in our work is a new variational principle for the BS divergence.