Geometric inequalities from phase space translations

Abstract

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.

37 pages; updated to match published version

References in corpus (6)

• Limits on classical communication from quantum entropy power inequalities
• Gaussian States Minimize the Output Entropy of the One-Mode Quantum Attenuator
• On a Quantum Entropy Power Inequality of Audenaert, Datta and Ozols
• Thinning, photonic beamsplitting, and a general discrete entropy power inequality
• Contractivity properties of a quantum diffusion semigroup
• The multi-mode quantum Entropy Power Inequality

Cited by in corpus (9)

• Gaussian optimizers for entropic inequalities in quantum information
• The conditional entropy power inequality for quantum additive noise channels
• Linear growth of the entanglement entropy for quadratic Hamiltonians and arbitrary initial states
• New lower bounds to the output entropy of multi-mode quantum Gaussian channels
• The Entropy Power Inequality with quantum conditioning
• Contractivity properties of a quantum diffusion semigroup
• Coherent state coding approaches the capacity of non-Gaussian bosonic channels
• The generalized strong subadditivity of the von Neumann entropy for bosonic quantum systems
• A Meta Logarithmic-Sobolev Inequality for Phase-Covariant Gaussian Channels