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We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint
1/2023

Complexity in algebraic QFT

Stefan Hollands and Alessio Ranallo

Abstract

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 $\log (\text{Jones Index})$.

The main technical tool in our work is a new variational principle for the BS divergence.

Received:
Feb 21, 2023
Published:
Feb 24, 2023
MSC Codes:
81P45, 81T05, 46L10
Keywords:
complexity theory, Quantum Field Theory, Operator algebras

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2023 Repository Open Access
Stefan Hollands and Alessio Ranallo

Complexity in algebraic QFT