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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
Summary
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.
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.
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