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arXiv:1706.00440 · Communications in Mathematical Physics

The Conditional Entropy Power Inequality for Bosonic Quantum Systems

Giacomo De Palma, Dario Trevisan

Published Jan 3, 2018• 30 citations
arXivHTMLPDF
DOI
10.1007/s00220-017-3082-8

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We prove the multimode conditional quantum Entropy Power Inequality for bosonic quantum systems. This inequality determines the minimum conditional von Neumann entropy of the output of the most general linear mixing of bosonic quantum modes among all the input states of the modes with given conditional entropies. Bosonic quantum systems constitute the mathematical model for the electromagnetic radiation in the quantum regime, which provides the most promising platform for quantum communication and quantum key distribution. We apply our multimode conditional quantum Entropy Power Inequality to determine new lower bounds to the squashed entanglement of a large family of bosonic quantum Gaussian states. The squashed entanglement is one of the main entanglement measures in quantum communication theory, providing the best known upper bound to the distillable key. Exploiting this result, we determine a new lower bound to the squashed entanglement of the multimode bosonic Gaussian channels that are extreme,i.e., that cannot be decomposed as a non-trivial convex combination of quantum channels. The squashed entanglement of a quantum channel provides an upper bound to its secret-key capacity,i.e., the capacity to generate a secret key shared between the sender and the receiver. Lower bounds to the squashed entanglement are notoriously hard to prove. Our results contribute to break this barrier and will stimulate further research in the field of quantum communication with bosonic quantum systems.
Abstract The conditional entropy power inequality is a fundamental inequality in information theory, stating that the conditional entropy of the sum of two conditionally independent vector-valued random variables each with an assigned conditional entropy is minimum when the random variables are Gaussian. We prove the conditional entropy power inequality in the scenario where the conditioning system is quantum. The proof is based on the heat semigroup and on a generalization of the Stam inequality in the presence of quantum conditioning. The entropy power inequality with quantum conditioning will be a key tool of quantum information, with applications in distributed source coding protocols with the assistance of quantum entanglement.
We develop a method for deriving Gaussian optimality results for bosonic quantum systems from inequalities for qubit systems. Inspired by Gross's derivation of the Gaussian logarithmic Sobolev inequality from an inequality on the two-point space, we use qubits as quantum two-point spaces and lift bosonic Weyl operators to pseudo-Weyl operators on large qubit systems. By controlling weighted norms and relative entropy in the fluctuation limit, we establish two transfer theorems that convert tensorized qubit inequalities into inequalities for bosonic Gaussian channels. As a first application, we prove that the sharp 222-logarithmic Sobolev inequality for the bosonic Ornstein--Uhlenbeck semigroup tensorizes without loss in the constant. We also establish a sharp output entropy inequality for tensor powers of a class of qubit channels. This includes a quantum Mrs.~Gerber's lemma determining the minimum output entropy of tensor powers of the qubit depolarizing channel at fixed input entropy. Applying our relative entropy transfer theorem, we obtain multimode constrained minimum output entropy bounds for Gaussian additive-noise channels and thermal attenuators in the heating range.