Covert Entanglement in Noisy Quantum Channels

• Covert entanglement generation is the process of creating undetectable quantum entanglement between remote parties, ensuring both secrecy and covertness.
• It utilizes noisy channel models like lossy bosonic channels and quantum hypothesis testing to guarantee that adversaries cannot discern active entanglement transmission.
• Practical protocols implement sparse and secret-key assisted coding to achieve an O(√n) scaling of generated ebits while balancing key consumption and reliability.

Covert entanglement generation refers to the creation of shared quantum entanglement between remote parties in such a way that any adversary monitoring the environment remains unable to detect whether entanglement transmission has occurred. This paradigm enhances the confidentiality of quantum networks by providing not only secrecy—the inaccessibility of entanglement states—but also covertness—the undetectability of entanglement distribution itself. The concept is now formalized within rigorous information-theoretic and quantum operational frameworks, especially in the setting of noisy quantum and bosonic channels.

1. Channel Models and the Covertness Constraint

Covert entanglement generation is typically examined over noisy quantum channels subject to active adversarial monitoring. The prototypical example is the lossy thermal-noise bosonic channel, relevant for photonic, microwave, and radio-frequency (RF) platforms (Anderson et al., 11 Jun 2025). In each channel use, Alice's input (mode $A$) interacts via a beamsplitter with an environment thermal mode $\hat\rho_{\rm th}(\bar n_B)$ of mean photon number $\bar n_B$ and transmissivity $\eta$. The outputs are: Bob’s mode ($B$), carrying the signal mode Alice wishes to entangle; Willie’s mode ($W$), captured by the adversary; and, in the Stinespring picture, an environment mode ($E$), with evolution represented by isometry $V^{\eta,\bar n_B}_{A\to BWE}$.

The adversary’s detection task is formalized via quantum hypothesis testing: distinguishing between the “innocent” background (e.g., vacuum input) and active transmission events based on the statistical difference (trace distance or relative entropy) between the joint output state sequences $\hat\rho_{W^n}^{(0)}$ and $\hat\rho_{W^n}^{(1)}$. Covert codes must ensure that the quantum relative entropy $$D(\hat\rho_{W^n}^{(1)}\,\|\,\hat\rho_{W^n}^{(0)})\leq \delta_C$$ for arbitrarily small $\delta_C$, so that Willie’s minimum error probability in detection satisfies $P_W^{(e)}\geq \frac12 - O(\sqrt{\delta_C})$ (Anderson et al., 11 Jun 2025, Kimelfeld et al., 26 Mar 2025).

2. Operational Definition and Coding Framework

A covert entanglement-generation code over $n$ channel uses is defined by three operational components:

• A maximally entangled reference-message state $$\ket{\Phi_{RM}} = \frac{1}{\sqrt{d}}\sum_{m=1}^{d}\ket{m}_R\ket{m}_M$$.
• An encoding isometry $U_{M\to A^n}$, mapping each message basis to a transmitted codeword $\ket{\phi_m}_{A^n}$.
• Bob’s decoding channel (CPTP map) $\mathcal{D}_{B^n \to \hat{M}}$.

Reliability demands high entanglement fidelity:

$$F(\ket{\Phi_{RM}}, \mathrm{id}_R\otimes \mathcal{D}\circ\mathcal{E}^{\otimes n} (\ket{\Phi\bra{\Phi}}_{RM}))\to 1.$$

Covertness is satisfied iff Willie's output state obeys: $$D(\hat\rho_{W^n}^{(1)}\,\|\,\hat\rho_{W^n}^{(0)})\leq \delta_C.$$ A rate $R_{\rm EG}$ is achievable if for any $\epsilon, \delta_C>0$ and sufficiently large $n$, a code exists generating $d = \exp(R_{\rm EG}\sqrt{n\delta_C})$ ebits under these constraints (Anderson et al., 11 Jun 2025, Kimelfeld et al., 26 Mar 2025).

3. The Square-Root Law and Capacity Characterization

Covert entanglement-generation exhibits the square-root law (SRL): the maximum number of ebits that can be generated covertly and reliably is $O(\sqrt{n})$ over $n$ channel uses. This mirrors the SRL discovered in classical covert communication. The derivation proceeds by observing that, for per-mode mean photon number $\bar n_s\ll 1$, the quantum relative entropy scales as $D\sim n\bar n_s^2$ and the achievable entanglement rate across modes scales linearly in $\bar n_s$, leading to an optimal scaling of ebits as $O(\sqrt{n\,\delta_C})$ (Anderson et al., 11 Jun 2025, Kimelfeld et al., 26 Mar 2025).

For general quantum channels $\mathcal{N}_{A\to B}$, the covert entanglement capacity $C_{\rm EG}(\mathcal{N})$ is given by: $$\log T = \sqrt{n\,\delta} \; C_{\rm EG}(\mathcal{N}) + o(\sqrt{n}), \quad C_{\rm EG}(\mathcal{N}) = \frac{D(\sigma_1\|\sigma_0)}{\sqrt{\tfrac12\,\eta(\omega_1\|\omega_0)}}$$ where $\sigma_x$ ($\omega_x$) denote Bob’s (Willie’s) state given Alice’s input $x$, and $\eta(\omega_1\|\omega_0)$ is derived from the second-order (Umegaki) expansion (Kimelfeld et al., 26 Mar 2025). For the bosonic channel, a single-letter capacity expression is obtained: $L_{\rm EG} = c_{\rm cov}\;\times\;c_{\rm rel}$ with

$$c_{\rm cov} = \frac{\sqrt{2\eta\bar n_B(1+\eta \bar n_B)}}{1-\eta}, \qquad c_{\rm rel} = \eta\log\left(1+\frac{1}{(1-\eta)\bar n_B}\right)$$

(Anderson et al., 11 Jun 2025).

4. Achievable Protocols: Classical-to-Quantum Conversion and Sparse Coding

The most efficient covert entanglement-generation protocols begin with classical covert communication codes under secrecy constraint and “coherify” them (Kimelfeld et al., 26 Mar 2025, Anderson et al., 11 Jun 2025). The classical protocol uses random codebooks, position-based coding, and secret-key assistance. This is lifted to a quantum protocol by preparing the reference entangled state, coherently encoding message superpositions with secret key randomization, and applying quantum decoding (gentle POVM or sequential measurement) at Bob.

For bosonic channels, the achievable approach uses a QPSK coherent-state codebook interleaved with vacuum signals. Bob’s sequential decoding achieves reliability, while quantum relative entropy and resolvability techniques ensure covertness.

For practical photonic channels, single-rail and dual-rail qubit-based sparse coding are also analyzed. These schemes substitute ideal coherent-state codewords with a fraction $q = O(1/\sqrt{n})$ of non-vacuum qubit transmissions. The number of covertly generated ebits then scales as $q n R$, with $R$ the rate achieved by random stabilizer coding under induced Pauli noise (Anderson et al., 11 Jun 2025). Such practical schemes, however, fall short of the fundamental quantum-limit capacity.

5. Role of Secrecy, Key Consumption, and Tradeoffs

In the covert setting, reliable undetectable entanglement generation generally requires more secret key bits than ebits generated. The ratio

$$\frac{\log T}{\log |\mathcal{K}|} \approx \frac{D(\sigma_1||\sigma_0)}{D(\omega_1||\omega_0)} < 1$$

so that classical one-time pad secrecy is not always sufficient and covert binning + resolvability may be necessary, especially when Willie’s channel is as informative as Bob’s. Nevertheless, the presence of secrecy constraints does not reduce the leading-order rate of covert entanglement generation compared with classical covert communication: both exhibit the same $O(\sqrt{n})$ scaling, with secrecy incurring only an increase in key length (Kimelfeld et al., 26 Mar 2025).

6. Alternative Mechanisms: Relativistic and Field-Theoretic Covert Generation

Physical mechanisms for covert entanglement generation that do not rely on optical channel transmission have been explored in relativistic settings. The “covert quantum internet” scenario uses entanglement extraction via Unruh–DeWitt detectors coupled to the Minkowski vacuum (Bradler et al., 2017). Two localized detectors execute time-synchronized windowed interactions with a quantum field, creating entanglement in their internal states without excitations detectable above background vacuum fluctuations. Covertness is enforced by bounding the total quantum relative entropy between the field with and without detector interaction, requiring $N \lambda^4 \ll 1$ over $N$ pulses of coupling strength $\lambda$. The raw entanglement yield per pulse is $O(\lambda^2)$; after distillation, the overall ebit rate is $O(\lambda^4/T)$, revealing a direct tradeoff between stealth and throughput.

7. Limitations, Open Problems, and Connections to Covert Channel Theory

Certain limitations of covert entanglement generation are rigorously established. In particular, no “purely quantum” covert channel exists: positive covert capacity for entanglement generation requires the existence of a classical covert channel (i.e., nonzero covert classical capacity). Pre-shared entanglement can enhance, but not enable, covert communication over a channel that is otherwise useless for covert transmission (Mestel, 2022).

It remains undecidable to algorithmically determine the entangled covert capacity of an arbitrary interactive channel, though semidefinite-programming hierarchies provide upper bounds in specific formulations based on nonlocal games and min-entropy criteria (Mestel, 2022). A systematic theory for covert entanglement distribution in fully interactive and networked settings, explicit security proofs for new physical models, and exact classification of channels with quantum advantage in covert communication are accepted as open questions.

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