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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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Stefan Hollands and Alessio Ranallo

Complexity in algebraic QFT