The conditional Entropy Power Inequality for bosonic quantum systems

Abstract

We prove the conditional Entropy Power Inequality for Gaussian quantum systems. This fundamental inequality determines the minimum quantum conditional von Neumann entropy of the output of the beam-splitter or of the squeezing among all the input states where the two inputs are conditionally independent given the memory and have given quantum conditional entropies. We also prove that, for any couple of values of the quantum conditional entropies of the two inputs, the minimum of the quantum conditional entropy of the output given by the conditional Entropy Power Inequality is asymptotically achieved by a suitable sequence of quantum Gaussian input states. Our proof of the conditional Entropy Power Inequality is based on a new Stam inequality for the quantum conditional Fisher information and on the determination of the universal asymptotic behaviour of the quantum conditional entropy under the heat semigroup evolution. The beam-splitter and the squeezing are the central elements of quantum optics, and can model the attenuation, the amplification and the noise of electromagnetic signals. This conditional Entropy Power Inequality will have a strong impact in quantum information and quantum cryptography. Among its many possible applications there is the proof of a new uncertainty relation for the conditional Wehrl entropy.

References in corpus (4)

• Gaussian states in continuous variable quantum information
• The Wehrl entropy has Gaussian optimizers
• The multi-mode quantum Entropy Power Inequality
• Multimode Gaussian optimizers for the Wehrl entropy and quantum Gaussian channels

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• The conditional entropy power inequality for quantum additive noise channels
• Bosonic quantum communication across arbitrarily high loss channels
• Linear growth of the entanglement entropy for quadratic Hamiltonians and arbitrary initial states
• Quantum Brascamp-Lieb Dualities
• New lower bounds to the output entropy of multi-mode quantum Gaussian channels
• Upper bounds on the quantum capacity for general attenuator and amplifier
• The Entropy Power Inequality with quantum conditioning
• Refined diamond norm bounds on the emergence of objectivity of observables
• RLD Fisher Information Bound for Multiparameter Estimation of Quantum Channels
• Uncertainty relations with quantum memory for the Wehrl entropy
• Universal upper bounds for Gaussian information capacity
• Upper bounds on the private capacity for bosonic Gaussian channels
• The squashed entanglement of the noiseless quantum Gaussian attenuator and amplifier
• Quantum Rényi Entropy Functionals for Bosonic Gaussian Systems
• The generalized strong subadditivity of the von Neumann entropy for bosonic quantum systems
• The multimode conditional quantum Entropy Power Inequality and the squashed entanglement of the multimode extreme bosonic Gaussian channels
• Curvature-dimension conditions for symmetric quantum Markov semigroups
• Weighted $p$-Rényi Entropy Power Inequality: Information Theory to Quantum Shannon Theory