[Paper Review] Quantum-noise limited communication with low probability of detection

Boulat A. Bash, Saikat Guha+2 more|arXiv (Cornell University)|Mar 22, 2014

Quantum Information and Cryptography8 references3 citations

TL;DR

This paper demonstrates that quantum-noise limited communication with low probability of detection (LPD) is achievable over lossy optical channels when the eavesdropper (Willie) experiences even minimal excess noise—such as thermal noise or dark current. By using coherent-state on-off keying and exploiting Willie’s measurement noise, Alice can transmit 𝒪(√n) bits reliably to Bob over n channel uses while keeping Willie’s detection error probability arbitrarily close to 1/2, ensuring covert communication even under quantum-optimal eavesdropping.

ABSTRACT

We demonstrate the achievability of a square root limit on the amount of information transmitted reliably and with low probability of detection (LPD) over the single-mode lossy bosonic channel if either the eavesdropper's measurements or the channel itself is subject to the slightest amount of excess noise. Specifically, Alice can transmit $\mathcal{O}(\sqrt{n})$ bits to Bob over $n$ channel uses such that Bob's average codeword error probability is upper-bounded by an arbitrarily small $δ>0$ 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.

Motivation & Objective

• To establish fundamental limits of low probability of detection (LPD) communication over optical channels, particularly in the presence of quantum noise and practical receiver constraints.
• To investigate whether reliable LPD communication is possible when Willie uses a quantum-optimal measurement, while Bob employs a standard coherent detection receiver.
• To analyze the role of excess noise—such as thermal noise or dark current—in enabling LPD communication on both thermal noise and pure loss channels.
• To show that LPD communication is not possible on a pure loss channel with an ideal receiver, but becomes feasible when Willie’s receiver has non-zero dark current.
• To derive the square root law scaling for covert information transmission under realistic physical constraints on the adversary’s detection capability.

Proposed method

• Alice uses coherent-state on-off keying (OOK) modulation with average power q|α|² to transmit information over n channel uses.
• Bob employs a direct detection receiver to decode the signal, leveraging knowledge of the secret codebook to minimize decoding error probability.
• Willie’s detection is modeled as a binary hypothesis test: H₀ (no transmission) vs. H₁ (transmission), with his observation modeled as a Bernoulli process affected by dark current or thermal noise.
• The classical relative entropy (CRE) is used to lower-bound Willie’s average error probability, ensuring it remains close to 1/2, thus achieving low detectability.
• A Taylor series expansion of the relative entropy around zero signal power is used to derive an upper bound on the divergence, enabling the derivation of the power constraint for covert transmission.
• The key power constraint is derived as q|α|² = 4ε / (√n (1−η)) × √(p_d / (1−p_d)), which ensures Willie’s detection error probability is bounded below by 1/2−ε.

Experimental results

Research questions

• RQ1 Can reliable LPD communication be achieved over a thermal noise optical channel when Willie uses a quantum-optimal joint-detection receiver?
• RQ2 Is LPD communication possible on a pure loss channel if Willie has an ideal receiver capable of perfect detection of silence?
• RQ3 What is the fundamental scaling limit of covert information transmission over optical channels when Willie’s receiver is limited by practical noise such as dark current?
• RQ4 How does the presence of excess noise—either in the channel or in Willie’s receiver—enable the square root law scaling of covert communication?
• RQ5 Under what conditions does the square root law hold when Bob uses a standard coherent or direct detection receiver, and Willie uses a quantum-optimal measurement?

Key findings

• On the thermal noise channel, Alice can transmit 𝒪(√n) bits reliably to Bob over n channel uses while ensuring Willie’s average detection error probability is lower-bounded by 1/2−ε for any ε>0, even under quantum-optimal measurement.
• The square root law holds for the thermal noise channel regardless of whether Willie uses a joint-detection quantum measurement or a classical receiver, provided the channel introduces excess noise.
• On the pure loss channel with an ideal receiver (no dark current), LPD communication is not possible because Willie can perfectly detect when Alice is silent.
• However, when Willie’s receiver has a non-zero dark current (p_d > 0), the square root law becomes achievable, allowing Alice to transmit 𝒪(√n) covert bits.
• The required average signal power scales as 𝒪(1/√n), specifically q|α|² = 4ε / (√n (1−η)) × √(p_d / (1−p_d)), ensuring both reliability and low detectability.
• The results show that even minimal excess noise—whether from thermal environment or local receiver dark current—enables the square root law, making LPD communication feasible in realistic optical communication scenarios.