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-Hermes, Daniel Stilck Franca

Department of Mathematical Sciences

8 Citations (Scopus)

51 Downloads (Pure)

Abstract

We study the speed of convergence of a primitive quantum time evolution towards its fixed point in the distance of sandwiched Renyi 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.

Original language	English
Article number	55
Journal	Quantum
Volume	2
Pages (from-to)	1-35
ISSN	2521-327X
DOIs	https://doi.org/10.22331/q-2018-02-27-55
Publication status	Published - 27 Feb 2018

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title = "Sandwiched R{\'e}nyi Convergence for Quantum Evolutions",

abstract = "We study the speed of convergence of a primitive quantum time evolution towards its fixed point in the distance of sandwiched Renyi 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.",

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T1 - Sandwiched Rényi Convergence for Quantum Evolutions

AU - Müller-Hermes, Alexander

AU - Franca, Daniel Stilck

PY - 2018/2/27

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N2 - We study the speed of convergence of a primitive quantum time evolution towards its fixed point in the distance of sandwiched Renyi 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.

AB - We study the speed of convergence of a primitive quantum time evolution towards its fixed point in the distance of sandwiched Renyi 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.

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DO - 10.22331/q-2018-02-27-55

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SN - 2521-327X

VL - 2

SP - 1

EP - 35

JO - Quantum

JF - Quantum

M1 - 55

ER -

Sandwiched Rényi Convergence for Quantum Evolutions

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