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PreprintCC BY 4.0arXiv

The Differential Structure of Generators of KMS-Symmetric Quantum Markov Semigroups

  • Matthijs Vernooij
  • Melchior Wirth
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Abstract

A KMS-symmetric quantum Markov semigroup is completely determined by the quadratic form in induces on the GNS Hilbert space. The quadratic forms that arise in this way are called quantum Dirichlet forms. In this article we give a complete description of bounded quantum Dirichlet forms in terms of derivations. In particular, we exhibit an explicit characterization of the bounded quantum Dirichlet forms on type I factors. We also show that any unbounded quantum Dirichlet form can be represented in terms of derivations. The key technical insight is that the modular group always restricts to a strongly continuous group on the domain of a quantum Dirichlet form such that the square root of the analytic generator is accretive with respect to the form norm.

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Cite

BibTeXRIS
BibTeX
@misc{vernooij2026differential,
  title = {{The Differential Structure of Generators of KMS-Symmetric Quantum Markov Semigroups}},
  author = {Matthijs Vernooij and Melchior Wirth},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.09678},
  eprint = {2610.09678},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.09678},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - The Differential Structure of Generators of KMS-Symmetric Quantum Markov Semigroups
AU  - Matthijs Vernooij
AU  - Melchior Wirth
PY  - 2026
DA  - 2026/10/07
PB  - arXiv
DO  - 10.48550/arxiv.2610.09678
UR  - https://arxiv.org/abs/2610.09678
AB  - A KMS-symmetric quantum Markov semigroup is completely determined by the quadratic form in induces on the GNS Hilbert space. The quadratic forms that arise in this way are called quantum Dirichlet forms. In this article we give a complete description of bounded quantum Dirichlet forms in terms of derivations. In particular, we exhibit an explicit characterization of the bounded quantum Dirichlet forms on type I factors. We also show that any unbounded quantum Dirichlet form can be represented in terms of derivations. The key technical insight is that the modular group always restricts to a strongly continuous group on the domain of a quantum Dirichlet form such that the square root of the analytic generator is accretive with respect to the form norm.
ER  -