Sandwiched Rényi Convergence for Quantum Evolutions

Alexander Müller-Hermes; Daniel Stilck Franca

doi:10.22331/q-2018-02-27-55

Sandwiched Rényi Convergence for Quantum Evolutions

Alexander Müller-Hermes¹ and Daniel Stilck Franca²

¹Department of Mathematical Sciences, University of Copenhagen, 2100 Copenhagen, Denmark
²Department of Mathematics, Technische Universität München, 85748 Garching, Germany

Published:	2018-02-27, volume 2, page 55
Eprint:	arXiv:1607.00041v3
Doi:	https://doi.org/10.22331/q-2018-02-27-55
Citation:	Quantum 2, 55 (2018).

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Abstract

We study the speed of convergence of a primitive quantum time evolution towards its fixed point in the distance of sandwiched Rényi divergences. For each of these distance measures the convergence is typically exponentially fast and the best exponent is given by a constant (similar to a logarithmic Sobolev constant) depending only on the generator of the time evolution. We establish relations between these constants and the logarithmic Sobolev constants as well as the spectral gap. An important consequence of these relations is the derivation of mixing time bounds for time evolutions directly from logarithmic Sobolev inequalities without relying on notions like lp-regularity. We also derive strong converse bounds for the classical capacity of a quantum time evolution and apply these to obtain bounds on the classical capacity of some examples, including stabilizer Hamiltonians under thermal noise.

► BibTeX data

@article{MullerHermes2018sandwichedrenyi,
  doi = {10.22331/q-2018-02-27-55},
  url = {https://doi.org/10.22331/q-2018-02-27-55},
  title = {Sandwiched {R}{\'{e}}nyi {C}onvergence for {Q}uantum {E}volutions},
  author = {M{\"{u}}ller-Hermes, Alexander and Franca, Daniel Stilck},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {2},
  pages = {55},
  month = feb,
  year = {2018}
}

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