A Meta Logarithmic-Sobolev Inequality for Phase-Covariant Gaussian
Channels
- URL: http://arxiv.org/abs/2311.09572v1
- Date: Thu, 16 Nov 2023 05:14:17 GMT
- Title: A Meta Logarithmic-Sobolev Inequality for Phase-Covariant Gaussian
Channels
- Authors: Salman Beigi, Saleh Rahimi-Keshari
- Abstract summary: We show that our inequality provides a general framework to derive information theoretic results regarding phase-covariant Gaussian channels.
Specifically, we explicitly compute the optimal constant $alpha_p$, for $1leq pleq 2$, of the $p$-log-Sobolev inequality associated to the quantum Ornstein-Uhlenbeck semigroup.
- Score: 3.6985338895569204
- License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
- Abstract: We introduce a meta logarithmic-Sobolev (log-Sobolev) inequality for the
Lindbladian of all single-mode phase-covariant Gaussian channels of bosonic
quantum systems, and prove that this inequality is saturated by thermal states.
We show that our inequality provides a general framework to derive information
theoretic results regarding phase-covariant Gaussian channels. Specifically, by
using the optimality of thermal states, we explicitly compute the optimal
constant $\alpha_p$, for $1\leq p\leq 2$, of the $p$-log-Sobolev inequality
associated to the quantum Ornstein-Uhlenbeck semigroup. These constants were
previously known for $p=1$ only. Our meta log-Sobolev inequality also enables
us to provide an alternative proof for the constrained minimum output entropy
conjecture in the single-mode case. Specifically, we show that for any
single-mode phase-covariant Gaussian channel $\Phi$, the minimum of the von
Neumann entropy $S\big(\Phi(\rho)\big)$ over all single-mode states $\rho$ with
a given lower bound on $S(\rho)$, is achieved at a thermal state.
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