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Complete Gradient Estimates of Quantum Markov Semigroups.

Melchior Wirth1, Haonan Zhang1

  • 1Institute of Science and Technology Austria (IST Austria), Am Campus 1, 3400 Klosterneuburg, Austria.

Communications in Mathematical Physics
|November 15, 2021
PubMed
Summary
This summary is machine-generated.

We developed a complete gradient estimate for quantum Markov semigroups, proving entropy semi-convexity with respect to the noncommutative 2-Wasserstein distance. This estimate is stable under products and leads to a modified logarithmic Sobolev inequality for free group factors.

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Area of Science:

  • Quantum probability
  • Noncommutative geometry
  • Functional analysis

Background:

  • Quantum Markov semigroups are fundamental in quantum probability.
  • Understanding their properties, like gradient estimates, is crucial for developing noncommutative analysis.
  • The noncommutative 2-Wasserstein distance offers a new framework for studying metric properties.

Purpose of the Study:

  • Introduce a complete gradient estimate for symmetric quantum Markov semigroups.
  • Establish implications for entropy semi-convexity and the noncommutative 2-Wasserstein distance.
  • Investigate the stability and applicability of this estimate.

Main Methods:

  • Development of a novel complete gradient estimate.
  • Analysis of stability under tensor and free products.
  • Application to derive inequalities for specific quantum structures.

Main Results:

  • A complete gradient estimate for quantum Markov semigroups is established.
  • This estimate implies semi-convexity of entropy with respect to the noncommutative 2-Wasserstein distance.
  • The estimate is shown to be stable under tensor and free products.

Conclusions:

  • The introduced gradient estimate provides a powerful tool in quantum analysis.
  • It offers new insights into the metric properties of quantum spaces.
  • Applications include proving a complete modified logarithmic Sobolev inequality for free group factors.