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arXiv:0908.0967 · Communications in Mathematical Physics

Generators of KMS Symmetric Markov Semigroups on $${\mathcal{B}({\rm h})}$$ Symmetry and Quantum Detailed Balance

Franco Fagnola, Veronica Umanità

Published Feb 23, 2010• 65 citations
arXivHTMLPDF
DOI
10.1007/s00220-010-1011-1

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Generators of KMS Symmetric Markov Semigroups on B(h){\mathcal{B}({\rm h})}B(h) Symmetry and Quantum Detailed Balance
THE DECOHERENCE-FREE SUBALGEBRA OF A QUANTUM MARKOV SEMIGROUP ON B(h)\mathcal{B}({\rm{h}})B(h)
We present a systematic investigation of bimodule quantum Markov semigroups within the framework of quantum Fourier analysis. We introduce the concepts of bimodule detailed balance conditions and bimodule KMS symmetry, which not only generalize the classical notions of detailed balance but also expose interesting structures of quantum channels. We demonstrate that the evolution of densities governed by the bimodule quantum Markov semigroup with bimodule detailed balance is the gradient flow for the relative entropy with respect to the hidden density. Consequently, we obtain a modified logarithmic Sobolev inequality and a Talagrand inequality with respect to a hidden density from higher dimensional structure. Furthermore, we establish a Poincaré inequality for irreducible inclusions and relative ergodic bimodule quantum semigroups.