while a passive eavesdropper, Warden Willie, who is assumed to be able to collect all the transmitted photons that do not reach Bob, has an average probability of detection error that is lower-bounded by

1/2-ε

for an arbitrarily small

ε>0

. We analyze the thermal noise and pure loss channels. The square root law holds for the thermal noise channel even if Willie employs a quantum-optimal measurement, while Bob is equipped with a standard coherent detection receiver. We also show that LPD communication is not possible on the pure loss channel. However, this result assumes Willie to possess an ideal receiver that is not subject to excess noise. If Willie is restricted to a practical receiver with a non-zero dark current, the square root law is achievable on the pure loss channel.

DOI: 10.48550/arxiv.1403.5616

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\mathcal{O}(\sqrt{n})

bits to Bob over

n

channel uses such that Bob's average codeword error probability is upper-bounded by an arbitrarily small

δ>0

1/2-ε

for an arbitrarily small

ε>0

\mathcal {O}(N)

bits of information over

N

channel uses, provided that the mean photon numbers of all users reaching the detecting end are equal. To prove this, a recursive upper bound is developed on the trace distance (TD) to measure the detection probability of the warden, which is tighter than the traditional quantum relative entropy measurement. However, if the mean photon number of any covert user deviates from the above condition, its covert capacity will degrade to

\mathcal {O}(\sqrt{N})

bits, whereas the capacities of other users will remain at

\mathcal {O}(N)

bits. This result is derived by exploiting the iterative relationship of the upper bound of TD and introducing a two-step detection strategy for the warden to handle random user activation uncertainty. Numerical experiments validate the theoretical findings, confirming the performance of covert transmission in both scenarios.

\propto\sqrt{n}

qubits can be transmitted covertly and reliably over

n

uses of an optical channel. Our achievability proof uses photonic dual-rail qubit encoding, which has been proposed for long-range repeater-based quantum communication and entanglement distribution. Our converse employs prior covert signal power limit results and adapts well-known methods to upper bound quantum capacity of optical channels. Finally, we believe that the gap between our lower and upper bounds for the number of reliable covert qubits can be mitigated by improving the quantum error correction codes and quantum channel capacity bounds.