The entropy power inequality for quantum systems

Abstract

When two independent analog signals, X and Y are added together giving Z=X+Y, the entropy of Z, H(Z), is not a simple function of the entropies H(X) and H(Y), but rather depends on the details of X and Y's distributions. Nevertheless, the entropy power inequality (EPI), which states that exp [2H(Z)] \geq exp[2H(X)] + exp[2H(Y)], gives a very tight restriction on the entropy of Z. This inequality has found many applications in information theory and statistics. The quantum analogue of adding two random variables is the combination of two independent bosonic modes at a beam splitter. The purpose of this work is to give a detailed outline of the proof of two separate generalizations of the entropy power inequality to the quantum regime. Our proofs are similar in spirit to standard classical proofs of the EPI, but some new quantities and ideas are needed in the quantum setting. Specifically, we find a new quantum de Bruijin identity relating entropy production under diffusion to a divergence-based quantum Fisher information. Furthermore, this Fisher information exhibits certain convexity properties in the context of beam splitters.

Mathematical exposition with detailed proofs. 14 pages. v2: updated to match published version

References in corpus (4)

• Quantum f-divergences and error correction
• Introduction to quantum Fisher information
• Limits on classical communication from quantum entropy power inequalities
• Two Proofs of the Fisher Information Inequality via Data Processing Arguments

Cited by in corpus (48)

• A generalization of the Entropy Power Inequality to Bosonic Quantum Systems
• Fundamental limits on quantum dynamics based on entropy change
• Quantum Entropy and Central Limit Theorem
• Passive states optimize the output of bosonic Gaussian quantum channels
• The conditional entropy power inequality for Gaussian quantum states
• Gaussian states minimize the output entropy of one-mode quantum Gaussian channels
• Gaussian States Minimize the Output Entropy of the One-Mode Quantum Attenuator
• The conditional Entropy Power Inequality for bosonic quantum systems
• Entropy power inequalities for qudits
• Gaussian optimizers for entropic inequalities in quantum information
• Convergence rates for the quantum central limit theorem
• Capacities of Quantum Amplifier Channels
• The conditional entropy power inequality for quantum additive noise channels
• Geometric inequalities from phase space translations
• Strong converse rates for classical communication over thermal and additive noise bosonic channels
• Stabilizer Testing and Magic Entropy via Quantum Fourier Analysis
• Linear growth of the entanglement entropy for quadratic Hamiltonians and arbitrary initial states
• Passive states as optimal inputs for single-jump lossy quantum channels
• Quantum Brascamp-Lieb Dualities
• The One-Mode Quantum-Limited Gaussian Attenuator and Amplifier Have Gaussian Maximizers
• Bounds on Information Combining With Quantum Side Information
• New lower bounds to the output entropy of multi-mode quantum Gaussian channels
• Spectral density of mixtures of random density matrices for qubits
• On a Quantum Entropy Power Inequality of Audenaert, Datta and Ozols
• Upper bounds on the quantum capacity for general attenuator and amplifier
• The Entropy Power Inequality with quantum conditioning
• Quantum hypothesis testing in many-body systems
• Thinning, photonic beamsplitting, and a general discrete entropy power inequality
• Conditional quantum entropy power inequality for $d$-level quantum systems
• Contractivity properties of a quantum diffusion semigroup
• Quantum Ruzsa Divergence to Quantify Magic
• Uncertainty relations with quantum memory for the Wehrl entropy
• Coherent state coding approaches the capacity of non-Gaussian bosonic channels
• Upper bounds on the private capacity for bosonic Gaussian channels
• Universal upper bounds for Gaussian information capacity
• Multiparticle quantum interference in Bogoliubov bosonic transformations
• The squashed entanglement of the noiseless quantum Gaussian attenuator and amplifier
• A de Bruijn identity for discrete random variables
• The multi-mode quantum Entropy Power Inequality
• Quantum Rényi Entropy Functionals for Bosonic Gaussian Systems
• Towards Optimal Convergence Rates for the Quantum Central Limit Theorem
• The generalized strong subadditivity of the von Neumann entropy for bosonic quantum systems
• Fluctuation and dissipation in memoryless open quantum evolutions
• The multimode conditional quantum Entropy Power Inequality and the squashed entanglement of the multimode extreme bosonic Gaussian channels
• Entropy Power Inequality in Fermionic Quantum Computation
• Quantifying coherence with quantum addition
• Curvature-dimension conditions for symmetric quantum Markov semigroups
• Weighted $p$-Rényi Entropy Power Inequality: Information Theory to Quantum Shannon Theory