Pusz–Woronowicz Functional Calculus and Extended Operator Convex Perspectives

Hiai, Fumio; Ueda, Yoshimichi; Wada, Shuhei

doi:10.1007/s00020-021-02676-y

Cited by 7 publications

(2 citation statements)

References 40 publications

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“…See [34,35,21,36] as general references. Central to the concept of operator mean and the divergences studied in this paper are the notions of operator monotoneand operator convex functions.…”

Section: Maximal F -Divergence For Bounded Operatorsmentioning

confidence: 99%

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Complexity in algebraic QFT

Hollands¹,

Ranallo²

2023

Preprint

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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.

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Section: Maximal F -Divergence For Bounded Operatorsmentioning

confidence: 99%

“…From this relation and ( 20) one can obtain the variational formula with ease, see also [36,Remark 9.2] for a closely related formula in the case of M = B(H ):…”

Section: Properties Of Maximal F -Divergencementioning

confidence: 99%

Complexity in algebraic QFT

Hollands¹,

Ranallo²

2023

Preprint

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Perspectives, Means and their Inequalities

Osaka

Wada

2022

Trends in Mathematics

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Channel Divergences and Complexity in Algebraic QFT

Hollands,

Ranallo

2023

Commun. Math. Phys.

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We consider a notion of divergence between quantum channels in relativistic continuum quantum field theory (QFT) that is derived from the Belavkin–Staszewski relative entropy and the concept of bimodules for general von Neumann algebras. Key concepts of the divergence that we shall prove based on a new variational formulation of that relative entropy are the subadditivity under composition and additivity under the tensor product between channels. Based on these properties, we propose to use the channel divergence relative to the trivial (identity-) channel as a novel measure of complexity. Using the properties of our channel 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 $$\log N$$ log N , (v) for a conditional expectation associated with an inclusion of QFTs with finite Jones index it is given by $$\log (\text {Jones Index})$$ log ( Jones Index ) .

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Pusz–Woronowicz Functional Calculus and Extended Operator Convex Perspectives

Cited by 7 publications

References 40 publications

Complexity in algebraic QFT

Complexity in algebraic QFT

Perspectives, Means and their Inequalities

Channel Divergences and Complexity in Algebraic QFT

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