On some special cases of the Entropy Photon-Number Inequality

We show that the Entropy Photon-Number Inequality (EPnI) holds where one of the input states is the vacuum state and for several candidates of the other input state that includes the cases when the state has the eigenvectors as the number states and either has only two non-zero eigenvalues or has arbitrary number of non-zero eigenvalues but is a high entropy state. We also discuss the conditions, which if satisfied, would lead to an extension of these results.Comment: 12 pages, no figure

A generalization of the Entropy Power Inequality to Bosonic Quantum Systems

In most communication schemes information is transmitted via travelling modes of electromagnetic radiation. These modes are unavoidably subject to environmental noise along any physical transmission medium and the quality of the communication channel strongly depends on the minimum noise achievable at the output. For classical signals such noise can be rigorously quantified in terms of the associated Shannon entropy and it is subject to a fundamental lower bound called entropy power inequality. Electromagnetic fields are however quantum mechanical systems and then, especially in low intensity signals, the quantum nature of the information carrier cannot be neglected and many important results derived within classical information theory require non-trivial extensions to the quantum regime. Here we prove one possible generalization of the Entropy Power Inequality to quantum bosonic systems. The impact of this inequality in quantum information theory is potentially large and some relevant implications are considered in this work

New lower bounds to the output entropy of multi-mode quantum Gaussian channels

We prove that quantum thermal Gaussian input states minimize the output entropy of the multi-mode quantum Gaussian attenuators and amplifiers that are entanglement breaking and of the multi-mode quantum Gaussian phase contravariant channels among all the input states with a given entropy. This is the first time that this property is proven for a multi-mode channel without restrictions on the input states. A striking consequence of this result is a new lower bound on the output entropy of all the multi-mode quantum Gaussian attenuators and amplifiers in terms of the input entropy. We apply this bound to determine new upper bounds to the communication rates in two different scenarios. The first is classical communication to two receivers with the quantum degraded Gaussian broadcast channel. The second is the simultaneous classical communication, quantum communication and entanglement generation or the simultaneous public classical communication, private classical communication and quantum key distribution with the Gaussian quantum-limited attenuator

Passive states optimize the output of bosonic Gaussian quantum channels

An ordering between the quantum states emerging from a single mode gauge-covariant bosonic Gaussian channel is proven. Specifically, we show that within the set of input density matrices with the same given spectrum, the element passive with respect to the Fock basis (i.e. diagonal with decreasing eigenvalues) produces an output which majorizes all the other outputs emerging from the same set. When applied to pure input states, our finding includes as a special case the result of A. Mari, et al., Nat. Comm. 5, 3826 (2014) which implies that the output associated to the vacuum majorizes the others

The Entropy Power Inequality with quantum conditioning

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

The One-Mode Quantum-Limited Gaussian Attenuator and Amplifier Have Gaussian Maximizers

We determine the p->q norms of the Gaussian one-mode quantum-limited attenuator and amplifier and prove that they are achieved by Gaussian states, extending to noncommutative probability the seminal theorem "Gaussian kernels have only Gaussian maximizers" (Lieb in Invent Math 102(1):179-208, 1990). The quantum-limited attenuator and amplifier are the building blocks of quantum Gaussian channels, which play a key role in quantum communication theory since they model in the quantum regime the attenuation and the noise affecting any electromagnetic signal. Our result is crucial to prove the longstanding conjecture stating that Gaussian input states minimize the output entropy of one-mode phase-covariant quantum Gaussian channels for fixed input entropy. Our proof technique is based on a new noncommutative logarithmic Sobolev inequality, and it can be used to determine the p->q norms of any quantum semigroup.Comment: Annales Henri Poincar\'e (2018