Central Limit Theorem for Bosonic Quantum Channels

TL;DR

This paper extends the Central Limit Theorem to bosonic quantum channels, unifying classical and quantum results, and provides new bounds on quantum capacity related to Gaussian channels.

Contribution

It introduces a CLT for bosonic quantum channels, connecting classical and quantum probability, and derives bounds on quantum capacity using Gaussian channel extremality.

Findings

Unified perspective on classical and quantum CLT

Necessary uncertainty relations for bosonic channels

Lower bounds on energy-constrained quantum capacity

Abstract

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.