Extreme bosonic linear channels

Hami Mehrabi, Ludovico Lami, Mark M. Wilde

In this paper, we develop an extension of the Central Limit Theorem (CLT) to the setting of bosonic quantum channels. This extension provides a deeper understanding of Gaussian bosonic channels as extremal objects. Using our CLT for bosonic quantum channels, we recover both the classical CLT and the CLT for bosonic quantum states, thereby offering a unified perspective that connects classical probability theory with continuous-variable quantum systems. Moreover, using our result, we can provide necessary uncertainty relations that every physical (possibly non-Gaussian) bosonic quantum channel must satisfy. As another application of our limit theorems, we derive tight lower bounds on the energy-constrained quantum capacity of linear bosonic channels by relating it to the capacity of their associated Gaussian bosonic channels, further reinforcing the role of Gaussian channels as extremal.

Alexander Semenovich Holevo

The set of all channels with fixed input and output is convex. We first give a convenient formulation of necessary and sufficient condition for a channel to be extreme point of this set in terms of complementary channel, a notion of big importance in quantum information theory. This formulation is based on the general approach to extremality of completely positive maps in an operator algebra due to Arveson. We then apply this formulation to prove the main result of this note: under certain nondegeneracy conditions, purity of the environment is necessary and sufficient for extremality of Bosonic linear (quasi-free) channel. It follows that Gaussian channel between finite-mode Bosonic systems is extreme if and only if it has minimal noise.

Walter E Thirring

Jeffrey H. Shapiro

Alessandro Falco, Giacomo De Palma

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.