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Published October 10, 2014 | public
Journal Article

Hypercontractivity of quasi-free quantum semigroups

Abstract

Hypercontractivity of a quantum dynamical semigroup has strong implications for its convergence behavior and entropy decay rate. A logarithmic Sobolev inequality and the corresponding logarithmic Sobolev constant can be inferred from the semigroup's hypercontractive norm bound. We consider completely-positive quantum mechanical semigroups described by a Lindblad master equation. To prove the norm bound, we follow an approach which has its roots in the study of classical rate equations. We use interpolation theorems for non-commutative Lₚ spaces to obtain a general hypercontractive inequality from a particular p→q-norm bound. Then, we derive a bound on the 2→4-norm from an analysis of the block diagonal structure of the semigroup's spectrum. We show that the dynamics of an N-qubit graph state Hamiltonian weakly coupled to a thermal environment is hypercontractive. As a consequence this allows for the efficient preparation of graph states in time poly(log(N)) by coupling at sufficiently low temperature. Furthermore, we extend our results to gapped Liouvillians arising from a weak linear coupling of a free-fermion systems.

Additional Information

© 2014 IOP Publishing Ltd. We thank J Eisert, H Wilming for helpful discussions. We especially thank L Aolita for reminding us of the quasi-product property of graph state Hamiltonians. This work was supported by the Institute for Quantum Information and Matter, a NSF Physics Frontiers Center with support of the Gordon and Betty Moore Foundation (Grants No. PHY-0803371 and PHY-1125565). KT also acknowledges the support from the Erwin Schrödinger fellowship, Austrian Science Fund (FWF): J 3219-N16. MJK acknowledges support from the Alexander von Humboldt foundation and the EU (SIQS, RAQUEL).

Additional details

Created:
August 22, 2023
Modified:
October 18, 2023