102,956 research outputs found
An information-theoretic proof of Nash's inequality
We show that an information-theoretic property of Shannon's entropy power,
known as concavity of entropy power, can be fruitfully employed to prove
inequalities in sharp form. In particular, the concavity of entropy power
implies the logarithmic Sobolev inequality, and Nash's inequality with the
sharp constant
The conditional entropy power inequality for quantum additive noise channels
We prove the quantum conditional Entropy Power Inequality for quantum
additive noise channels. This inequality lower bounds the quantum conditional
entropy of the output of an additive noise channel in terms of the quantum
conditional entropies of the input state and the noise when they are
conditionally independent given the memory. We also show that this conditional
Entropy Power Inequality is optimal in the sense that we can achieve equality
asymptotically by choosing a suitable sequence of Gaussian input states. We
apply the conditional Entropy Power Inequality to find an array of
information-theoretic inequalities for conditional entropies which are the
analogues of inequalities which have already been established in the
unconditioned setting. Furthermore, we give a simple proof of the convergence
rate of the quantum Ornstein-Uhlenbeck semigroup based on Entropy Power
Inequalities.Comment: 26 pages; updated to match published versio
Geometric inequalities from phase space translations
We establish a quantum version of the classical isoperimetric inequality
relating the Fisher information and the entropy power of a quantum state. The
key tool is a Fisher information inequality for a state which results from a
certain convolution operation: the latter maps a classical probability
distribution on phase space and a quantum state to a quantum state. We show
that this inequality also gives rise to several related inequalities whose
counterparts are well-known in the classical setting: in particular, it implies
an entropy power inequality for the mentioned convolution operation as well as
the isoperimetric inequality, and establishes concavity of the entropy power
along trajectories of the quantum heat diffusion semigroup. As an application,
we derive a Log-Sobolev inequality for the quantum Ornstein-Uhlenbeck
semigroup, and argue that it implies fast convergence towards the fixed point
for a large class of initial states.Comment: 37 pages; updated to match published versio
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