# Fundamental limits of quantum-secure covert communication over bosonic channels

## TL;DR

This paper establishes the fundamental limits of quantum-secure covert communication over bosonic channels, deriving the square root law constant and demonstrating optimal modulation schemes for achieving covert capacity.

## Contribution

It derives the expression for the square root law constant in bosonic channels and proves the optimality of QPSK modulation for covert communication.

## Key findings

- The square root law governs covert communication over bosonic channels.
- QPSK modulation achieves the optimal covert capacity constant.
- BPSK is sub-optimal for covert communication despite being optimal in non-covert scenarios.

## Abstract

We investigate the fundamental limit of quantum-secure covert communication over the lossy thermal noise bosonic channel, the quantum-mechanical model underlying many practical channels. We assume that the adversary has unlimited quantum information processing capabilities as well as access to all transmitted photons that do not reach the legitimate receiver. Given existence of noise that is uncontrolled by the adversary, the square root law (SRL) governs covert communication: up to c*sqrt{n} covert bits can be transmitted reliably in n channel uses. Attempting to surpass this limit results in detection with unity probability as n approaches infinity. Here we present the expression for c, characterizing the SRL for the bosonic channel. We also prove that discrete-valued coherent state quadrature phase shift keying (QPSK) constellation achieves the optimal c, which is the same as that achieved by a circularly-symmetric complex-valued Gaussian prior on coherent state amplitude. Finally, while binary phase shift keying (BPSK) achieves the Holevo capacity for non-covert bosonic channels in the low received signal-to-noise ratio regime, we show that it is strictly sub-optimal for covert communication.