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arXiv:1304.7031 · Journal of Mathematical Physics

The conditional entropy power inequality for Gaussian quantum states

Robert König

Published Feb 1, 2015• 31 citations
arXivHTMLPDF

Abstract

We propose a generalization of the quantum entropy power inequality involving conditional entropies. For the special case of Gaussian states, we give a proof based on perturbation theory for symplectic spectra. We discuss some implications for entanglement-assisted classical communication over additive bosonic noise channels.
DOI
10.1063/1.4906925

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We propose a generalization of the quantum entropy power inequality involving condi-tional entropies. For the special case of Gaussian states, we give a proof based on perturba-tion theory for symplectic spectra. We discuss some implications for entanglement-assisted classical communication over additive bosonic noise channels. 1 Classical and quantum entropy-power inequalities The entropy power inequality, proposed by Shannon [27] and later established with increasing rigor by Stam [29] and Blachman [5], has become a fundamental tool in classical informa-tion theory. Shannon’s original application of the entropy power inequality is a lower bound on the capacity of an additive (but potentially non-Gaussian) noise channel [27, Theorem 18]. However, the usefulness of the entropy power inequality is especially evident in multi-terminal information theory. Among the most well-known applications are the characterization of the Gaussian broadcast channel by Bergman [4], the Gaussian two-description problem by Ozarow [25] and the quadratic Gaussian CEO problem by Oohama [24]. In these multi-user settings, Fano’s inequality by itself is insufficient to characterize different tradeoffs. A more recent application proposed by Liu and Viswanath [23] uses entropy power inequalities to solve certain optimization problems. The entropy power inequality lower bounds the differential entropy of the convolution of two independent random variables X, Y taking values in Rn. Its covariance-preserving version states that H( λX + 1 − λY) ≥ λH(X) + (1 − λ)H(Y) for all 0 ≤ λ ≤ 1. (1) Eq. (1) can be shown [22, 32] to be equivalent to the more commonly used statement e2H(X+Y)/n ≥ e2H(X)/n + e2H(Y)/n. The latter explains the terminology as e2H(X)/n is the power, i.e., variance of a Gaussian ran-dom variable with identical entropy as H(X). Inequalities such as (1) are closely related to Log-Sobolev inequalities (see e.g., [31]) as well as Brunn-Minkowski-type inequalities [6]. Gen-eralizations to free probability [30] and quantum states have been considered. 1 ar
We prove the quantum conditional entropy power inequality for quantum additive noise channels. This inequality lower bounds the quantum conditional entropy of the output of an additive noise channel in terms of the quantum conditional entropies of the input state and the noise when they are conditionally independent given the memory. We also show that this conditional entropy power inequality is optimal in the sense that we can achieve equality asymptotically by choosing a suitable sequence of Gaussian input states. We apply the conditional entropy power inequality to find an array of information-theoretic inequalities for conditional entropies which are the analogs of inequalities which have already been established in the unconditioned setting. Furthermore, we give a simple proof of the convergence rate of the quantum Ornstein-Uhlenbeck semigroup based on entropy power inequalities.
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 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.
This dissertation deals with entropic inequalities for bosonic channels in quantum information theory. Such inequalities have applications in foundational questions of quantum communication and can be used to make statements about the convergence of quantum dynamical semigroups. We prove, among other results, an inequality for the output entropy of a quantum channel which models additive classical noise and apply this inequality to derive bounds on the classical capacity of a general family of non-Gaussian quantum channels.