Covert communication conceals from an adversary not the content of a transmission but its very existence. Over the lossy thermal-noise bosonic channel — the quantum-mechanical model of optical links — reliable covert communication of qubits is known to obey a square root law: only on the order of n\sqrt{n} qubits can be transmitted covertly and reliably over nn channel uses, and matching n\sqrt{n} achievability and converse bounds have been proved. The constant prefactor, however — the covert quantum capacity of the channel — remains undetermined: the best achievable lower bounds and the existing converse upper bound differ substantially, and the authors of the original result believe the converse is loose. The open problem is to characterize this capacity, i.e., to close the achievability–converse gap for the optimal number of covertly and reliably transmissible qubits over the lossy thermal-noise bosonic channel, in both the unassisted setting and the setting with a two-way covert classical channel, and in particular to decide whether the current energy-constrained data-processing converse is tight or can be improved.

Background

Covert (low-probability-of-detection) communication hides the fact that a transmission is taking place at all: a warden, conventionally named Willie, observes his channel output and must decide whether the legitimate sender Alice is transmitting or silent, and the protocol requires Willie's optimal detection advantage to vanish. Classical covert communication over noisy channels is governed by the square root law (SRL): over nn channel uses, at most and (with a pre-shared secret key sublinear in nn) at least on the order of n\sqrt{n} bits can be sent both reliably and covertly, so the asymptotically meaningful figure of merit is the constant prefactor of n\sqrt{n}, called the covert capacity.

The setting here is quantum. The lossy thermal-noise bosonic channel is the standard quantum model of optical (and microwave/RF) links: each channel use is a mode of the electromagnetic field, and the channel is a beamsplitter with transmittance η∈(0,1)\eta \in (0,1) coupling Alice's input mode to a thermal-environment mode of mean photon number nˉB≥0\bar{n}_{B} \ge 0; the legitimate receiver Bob holds the transmitted output while Willie may collect everything that does not reach Bob. When Alice is quiet she inputs the vacuum state (the innocent state), so Willie must discriminate between a thermal state and whatever he receives during transmission. Covertness is quantified by requiring Willie's optimal minimum error probability in this hypothesis test to remain at least 1/2−δ1/2 - \delta for a fixed covertness level δ∈(0,1/2)\delta \in (0, 1/2) (a closely related convention in the literature instead bounds the quantum relative entropy between Willie's two possible output states); reliability means the states Bob decodes are close in trace distance to what Alice sent.

Three lines of prior work frame the question. First, for purely classical data the covert capacity of this same bosonic channel is fully characterized: without entanglement assistance Lno-EAnL_{\text{no-EA}}\sqrt{n} covert bits are achievable and optimal (Gagatsos, Bullock, and Bash, 2020), and entanglement assistance improves the scaling to nlog⁡n\sqrt{n}\log n. Second, covert quantum key distribution — generating a classical secret covertly — was initiated by Arrazola and Scarani (2016), who showed covert transmission of qubits over optical channels is possible but that covert QKD consumes more key than it generates, and Tahmasbi and Bloch (2020) proved covert secret-key expansion over bosonic channels is possible when the legitimate parties have a channel advantage, with a converse showing it is impossible when Willie fully controls the channel. Third, even without any covertness constraint the quantum capacity of the thermal-loss bosonic channel (the rate at which it communicates unknown quantum states reliably) is itself unknown beyond bounds: it involves a regularized coherent information — the optimization of the coherent information IcI_{\mathrm{c}}, namely the entropy of the output minus the entropy exchanged with the environment, maximized over large blocks — for which no single-letter formula exists, and the sharpest available statements are upper bounds from energy-constrained data-processing arguments (Sharma, Wilde, Adhikari, and Takeoka, 2018).

Against this background, Anderson, Eyre, Dailey, Rozpędek, and Bash (2024) established the SRL for covert quantum communication — transmission of unknown qubit states, as opposed to classical bits or keys — over the lossy thermal-noise bosonic channel. Their achievability uses dual-rail photonic qubits (a logical qubit encoded as a single photon delocalized over two modes, an encoding proposed for long-range quantum repeaters), random quantum codes decoded at the hashing-bound rate (the achievable rate 1−H(p⃗)1 - H(\vec{p}) for Pauli channels, where HH is the Shannon entropy of the Pauli error distribution), and a sparse-coding scheme driven by a pre-shared secret. Their covertness analysis assumed the Alice-to-Willie channel is entanglement-breaking — a channel whose output is separable from any reference system for every input; for thermal-loss channels this holds when the thermal noise is large enough relative to the transmittance — because it yields a closed form for the χ2\chi^{2}-divergence (a divergence Dχ2(ρ^∥σ^)=tr(ρ^2σ^−1)−1D_{\chi^{2}}(\hat{\rho}\|\hat{\sigma}) = \mathrm{tr}(\hat{\rho}^{2}\hat{\sigma}^{-1}) - 1 used to bound Willie's distinguishing power); Alice can also force the condition by adding loss or noise, at a cost in rate. Their converse combines the covert photon-number limits with energy-constrained quantum-capacity upper bounds of the data-processing type (discarding the pure-loss part of the channel decomposition), giving M(n)=O(n)M(n) = O(\sqrt{n}) covert qubits over nn uses. Both bounds scale as n\sqrt{n}, but the gap between the constants is, in the authors' words, fairly large.

Subsequent work by the same group sharpened the picture: the entanglement-breaking assumption was removed on the achievability side via Pauli twirling (randomizing errors so that the channel behaves as a Pauli channel and Willie's average view becomes maximally mixed), giving an improved lower bound valid for arbitrary memoryless quantum channels; the covert entanglement-generation capacity of the bosonic channel was meanwhile fully characterized (it equals the classical covert capacity Lno-EAL_{\text{no-EA}}). Nevertheless, as the authors state in their most recent treatment (December 2025), the improved lower bound for covert quantum communication still does not match the converse, which they posit is too loose. Determining the exact constant — the covert quantum capacity — is the open problem.

Problem Statement

Consider the lossy thermal-noise bosonic channel with transmittance η∈(0,1)\eta \in (0,1) from Alice to Bob and thermal-environment mean photon number nˉB≥0\bar{n}_{B} \ge 0, where the adversary Willie receives the complementary (lost) output. Alice either transmits over nn uses of the channel or stays silent, inputting vacuum when silent. Transmission is covert if Willie's optimal minimum error probability for discriminating transmission from silence over his nn-use output is at least 1/2−δ1/2 - \delta for a fixed covertness level δ∈(0,1/2)\delta \in (0, 1/2) (the literature also employs a closely related covertness criterion requiring the quantum relative entropy between Willie's two possible output states to be bounded; the two criteria are related through the trace distance and Pinsker-type inequalities but are not identical at the same δ\delta); it is reliable if the qubit states Bob decodes are vanishingly close in trace distance to the states Alice sent as n→∞n \to \infty. Alice and Bob may pre-share a classical secret sublinear in nn (scaling as O(n)O(\sqrt{n}) bits), per the standard covert-coding resource model, and the count of transmitted qubits may be taken in expectation over the protocol's internal randomization, following the source's convention.

The problem is to determine the covert quantum capacity of this channel: the supremal constant CQ(η,nˉB,δ)>0C_{Q}(\eta, \bar{n}_{B}, \delta) > 0 such that CQn(1−o(1))C_{Q}\sqrt{n}(1 - o(1)) qubits can be transmitted both reliably and covertly over nn uses. A complete answer requires closing the gap between the two existing bounds:

  1. Achievability: construct codes and physical encodings (e.g., improved photonic qubit modulations, bosonic codes, or entanglement-generation-plus-teleportation schemes) that attain the optimal constant — or prove that the best known achievable rates are already optimal.
  2. Converse: prove a matching upper bound on the number of reliably and covertly transmissible qubits; in particular, decide whether the existing converse of the 2024 source — which combines the covert photon-number limit with energy-constrained quantum-capacity (data-processing) bounds — is tight, and if it is loose, derive the improved converse that closes the gap.

The same characterization is asked for the variant in which Alice and Bob are additionally assisted by a two-way covert classical channel (with the covert classical bits consumed quantified). The answer should be expressed explicitly in terms of the channel parameters η\eta, nˉB\bar{n}_{B} and the covertness level δ\delta, and should be consistent with the known non-covert quantum-capacity bounds and the fully characterized covert classical (Lno-EAL_{\text{no-EA}}) and covert entanglement-generation capacities of the same channel in the boundary regimes (e.g., nˉB=0\bar{n}_{B} = 0, η→0\eta \to 0, η→1\eta \to 1).

References

  1. Evan J. D. Anderson, Christopher K. Eyre, Isabel M. Dailey, Filip Rozpędek, and Boulat A. Bash, 'Covert Quantum Communication Over Optical Channels', arXiv:2401.06764 (2024), DOI: 10.48550/arXiv.2401.06764, https://arxiv.org/abs/2401.06764; conference version 'Square Root Law for Covert Quantum Communication over Optical Channels', Proc. 2024 IEEE International Conference on Quantum Computing and Engineering (QCE), DOI: 10.1109/QCE60285.2024.00211
  2. Evan J. D. Anderson, Michael S. Bullock, Filip Rozpędek, and Boulat A. Bash, 'Achievability of Covert Quantum Communication', arXiv:2501.13103 (2025), DOI: 10.48550/arXiv.2501.13103, https://arxiv.org/abs/2501.13103 (presented at the 2025 IEEE International Symposium on Information Theory)
  3. Evan J. D. Anderson, Michael S. Bullock, Ohad Kimelfeld, Christopher K. Eyre, Filip Rozpędek, Uzi Pereg, and Boulat A. Bash, 'Covert Entanglement Generation over Bosonic Channels', arXiv:2506.09474 (2025), DOI: 10.48550/arXiv.2506.09474, https://arxiv.org/abs/2506.09474; journal version accepted in IEEE Journal on Selected Areas in Communications (2025), DOI: 10.1109/JSAC.2025.3642074
  4. Ohad Kimelfeld, Boulat A. Bash, and Uzi Pereg, 'Covert Entanglement Generation and Secrecy', arXiv:2503.21002 (2025), DOI: 10.48550/arXiv.2503.21002, https://arxiv.org/abs/2503.21002; also in Proc. 2025 IEEE Information Theory Workshop (ITW), DOI: 10.1109/ITW62417.2025.11240442
  5. Christos N. Gagatsos, Michael S. Bullock, and Boulat A. Bash, 'Covert Capacity of Bosonic Channels', IEEE Journal on Selected Areas in Information Theory, vol. 1, no. 2, pp. 555–567 (2020), DOI: 10.1109/JSAIT.2020.3017199, arXiv:2002.06733, https://arxiv.org/abs/2002.06733
  6. Juan Miguel Arrazola and Valerio Scarani, 'Covert Quantum Communication', Physical Review Letters 117, 250503 (2016), DOI: 10.1103/PhysRevLett.117.250503, arXiv:1604.05438, https://arxiv.org/abs/1604.05438
  7. Mehrdad Tahmasbi and Matthieu R. Bloch, 'Toward Undetectable Quantum Key Distribution over Bosonic Channels', IEEE Journal on Selected Areas in Information Theory, vol. 1, no. 2, pp. 585–598 (2020), DOI: 10.1109/JSAIT.2020.3017212, arXiv:1904.12363, https://arxiv.org/abs/1904.12363
  8. Kunal Sharma, Mark M. Wilde, Sushovit Adhikari, and Masahiro Takeoka, 'Bounding the energy-constrained quantum and private capacities of phase-insensitive bosonic Gaussian channels', New Journal of Physics 20, 063025 (2018), DOI: 10.1088/1367-2630/aac11a, arXiv:1708.07257, https://arxiv.org/abs/1708.07257
  9. Abbas Arghavani, Alessandro V. Papadopoulos, Vahid Azimi Mousolou, Giuseppe Nebbione, and Shahid Raza, 'Robust Covert Quantum Communication under Bounded Channel Uncertainty', arXiv:2604.13116 (2026), DOI: 10.48550/arXiv.2604.13116, https://arxiv.org/abs/2604.13116
  10. Abbas Arghavani, Shahid Raza, Maryam Amiri, and Alessandro V. Papadopoulos, 'A Risk-Aware Framework for Covert Quantum Communication under Stochastic Channel Uncertainty', arXiv:2605.18928 (2026), DOI: 10.48550/arXiv.2605.18928, https://arxiv.org/abs/2605.18928