Gaussian optimizers for entropic inequalities in quantum information

Abstract

We survey the state of the art for the proof of the quantum Gaussian optimizer conjectures of quantum information theory. These fundamental conjectures state that quantum Gaussian input states are the solution to several optimization problems involving quantum Gaussian channels. These problems are the quantum counterpart of three fundamental results of functional analysis and probability: the Entropy Power Inequality, the sharp Young's inequality for convolutions, and the theorem "Gaussian kernels have only Gaussian maximizers." Quantum Gaussian channels play a key role in quantum communication theory: they are the quantum counterpart of Gaussian integral kernels and provide the mathematical model for the propagation of electromagnetic waves in the quantum regime. The quantum Gaussian optimizer conjectures are needed to determine the maximum communication rates over optical fibers and free space. The restriction of the quantum-limited Gaussian attenuator to input states diagonal in the Fock basis coincides with the thinning, which is the analog of the rescaling for positive integer random variables. Quantum Gaussian channels provide then a bridge between functional analysis and discrete probability.

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• Quantum optimal transport with quantum channels
• Rényi and Tsallis entropies: three analytic examples
• Rényi and Tsallis entropies of the Aharonov-Bohm ring in uniform magnetic fields
• The conditional entropy power inequality for quantum additive noise channels
• Linear growth of the entanglement entropy for quadratic Hamiltonians and arbitrary initial states
• The One-Mode Quantum-Limited Gaussian Attenuator and Amplifier Have Gaussian Maximizers
• New lower bounds to the output entropy of multi-mode quantum Gaussian channels
• Time-local optimal control for parameter estimation in the Gaussian regime
• Accessible information of a general quantum Gaussian ensemble
• The Entropy Power Inequality with quantum conditioning
• One-dimensional pseudoharmonic oscillator: classical remarks and quantum-information theory
• Ricci curvature of quantum channels on non-commutative transportation metric spaces
• The squashed entanglement of the noiseless quantum Gaussian attenuator and amplifier
• Computable limits of optical multiple-access communications
• Complexity of quantum tomography from genuine non-Gaussian entanglement
• A Meta Logarithmic-Sobolev Inequality for Phase-Covariant Gaussian Channels
• The generalized strong subadditivity of the von Neumann entropy for bosonic quantum systems