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Derivations and KMS-Symmetric Quantum Markov Semigroups.

Matthijs Vernooij1, Melchior Wirth2

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Researchers proved that the generator of KMS-symmetric quantum Markov semigroups can be the square of a derivation in a Hilbert bimodule. This extends prior work and introduces a new map crucial for the Hilbert bimodule inner product.

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Area of Science:

  • Quantum probability theory
  • Operator algebras
  • Noncommutative geometry

Background:

  • Quantum Markov semigroups are fundamental in quantum probability.
  • Previous work established connections between generators and derivations for specific symmetry types (GNS, tracially symmetric).
  • A gap existed for KMS-symmetric semigroups.

Purpose of the Study:

  • To extend the representation of quantum Markov semigroup generators as squares of derivations to KMS-symmetric cases.
  • To introduce a novel mathematical tool for analyzing these semigroups.
  • To deepen the understanding of the structure of KMS-symmetric quantum Markov semigroups.

Main Methods:

  • Introduction of a new completely positive map acting on bounded operators.
  • Utilizing the GNS (Gelfand-Naimark-Segal) Hilbert space framework.
  • Developing a novel inner product structure on a Hilbert bimodule.

Main Results:

  • The generator of a KMS-symmetric quantum Markov semigroup is proven to be the square of a derivation with values in a Hilbert bimodule.
  • The newly introduced completely positive map transforms symmetric Markov operators into symmetric Markov operators.
  • This map is essential for constructing the Hilbert bimodule and its inner product.

Conclusions:

  • The study successfully extends the square of a derivation representation to KMS-symmetric quantum Markov semigroups.
  • The introduced completely positive map is a key element in this extension.
  • This work provides a significant advancement in the spectral theory of quantum Markov semigroups.