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 $L_p$ spaces to obtain a general hypercontractive inequality from a particular $p\to q$-norm bound. Then, we derive a bound on the $2\to 4$-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 $\mathrm{poly}(\log (N))$ 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.

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• Rapid quantum ground state preparation via dissipative dynamics
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• Potential output purity of completely positive maps
• Simple and efficient end-to-end quantum thermal and ground state preparation
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• Nonlocal games with noisy maximally entangled states are decidable
• Energy exchange and entropy for quasi-free fermionic semigroups