Transmitting Correlation for Data Transmission over the Bosonic Arbitrarily Varying Channel

Shared randomness is the central ingredient for stabilizing symmetrizable communication systems against arbitrarily varying jammers. Given the presence of the jammer, however, the question arises how this precious resource could have been distributed. Several works discuss the use of external sources for this task. In this work, we show, based on the most standard optical communication model, how the sender and receiver can employ either classically correlated thermal light or entangled two-mode squeezed states created at and transmitted by the sender to counter the jamming attack of an energy-limited jammer during the distribution phase. Both sender and receiver are only allowed to use homodyne detection in our model, and the sender has to obey a power limit as well.

💡 Research Summary

The paper investigates data transmission over a bosonic arbitrarily varying channel (AVC) when a power‑limited jammer is present. In classical AVC theory, a positive capacity can be achieved only if the legitimate parties share a small amount of common randomness; without it, a symmetrizable channel has zero capacity. The authors ask how such randomness can be generated in the first place, and they answer this question within the most standard optical communication model: a 50:50 beam‑splitter followed by homodyne detection on both sides.

The authors define three coding frameworks. (1) Deterministic codes (DC) allow the sender to use only displaced thermal states (S_N(\alpha)) with energy constraint (N+|\alpha|^2\le E) and the jammer to use displaced thermal states with constraint (N+|\beta|^2\le P). (2) Common‑randomness‑assisted codes (CRAC) consist of a family of DCs indexed by a shared random seed; the seed can be used to randomise the encoding map. (3) Correlated codes (CC) extend DCs by allowing the sender to generate a bipartite state (\rho_{RS}) whose reduced state on the transmitted mode satisfies the same energy constraint as a displaced thermal state, while the other mode is kept locally. The sender may perform homodyne measurements on the local mode and adapt future transmissions based on the measurement outcomes. Two families of (\rho_{RS}) are considered: (i) classically correlated two‑mode thermal states (obtained by mixing two independent thermal modes on a 50:50 beam‑splitter) and (ii) entangled two‑mode squeezed vacuum (TMSV) states.

The main theorem contains four statements:

1. If the jammer’s power (P) is at least the sender’s power (E) ((P\ge E)), any deterministic code has zero capacity. The proof uses the symmetry of the beam‑splitter: the jammer can mimic the sender’s signal, making the receiver’s measurement distribution independent of the transmitted message, which forces the error probability to stay above a constant (≥ ¼).
2. If the sender’s power is strictly smaller ((E<P)), deterministic codes achieve a strictly positive rate. The sender transmits coherent states (\pm\sqrt{E}); the jammer’s displaced thermal state adds a Gaussian shift with variance (N+1). Homodyne detection yields a binary channel with transition probability at least (1/2+ \nu(E,P)) where (\nu(E,P)=\frac14\operatorname{erf}!\big((\sqrt{E}-\sqrt{P})/\sqrt{P}+1\big)>0). Standard results on arbitrarily varying binary symmetric channels give a positive capacity lower bound.
3. For any positive (E,P), the common‑randomness‑assisted capacity (\bar C) is positive. By averaging over all jammer strategies and using a symmetric input distribution, the induced binary channel’s error probability can be bounded by (\epsilon(E,P)=\frac12+\varepsilon(E,P)) with (\varepsilon(E,P)=\sqrt{E\pi(1+P)}e^{-(\sqrt{E}-\sqrt{P})^2}>0). The capacity of an AVC with such a binary symmetric channel is at least (1-h(\epsilon)), which is strictly larger than zero.
4. For any positive (E,P), correlated codes also achieve a positive rate ((C_Q>0)). The authors analyse two concrete bipartite states. For the classically correlated thermal pair, the joint homodyne outcomes ((X_1,X_3)) are jointly Gaussian with a calculable covariance; a quadrant‑decoding rule maps each outcome to a bit pair. The probability of landing in the “correct” quadrant can be expressed through the bivariate normal CDF (\Phi_2) and is shown to exceed (1/2) by an amount that depends on the squeezing parameter (r) (or equivalently on the thermal photon number). For the entangled TMSV state, the covariance matrix after the beam‑splitter is derived explicitly; the correlation coefficient (\rho=C/(A+B)) can be made arbitrarily close to one by increasing the squeezing. Consequently the error probability decays exponentially with the squeezing, yielding a strictly positive achievable rate. The proof demonstrates that even purely classical correlations (thermal‑thermal) are sufficient; entanglement merely improves the bound.

The technical contribution lies in turning the abstract notion of “shared randomness” into a concrete physical resource that can be generated under realistic power constraints. The paper shows that homodyne detection alone suffices to extract the required correlation, avoiding more demanding photon‑counting or non‑Gaussian measurements. Moreover, the analysis bridges classical AVC theory (symmetrizability, random coding) with continuous‑variable quantum optics (Gaussian states, beam‑splitters, covariance matrices). The results suggest practical protocols for satellite‑to‑ground or fiber‑based quantum communication where the legitimate parties cannot pre‑share secret keys but can exchange low‑energy correlated light.

Limitations include the idealised assumptions of a perfect 50:50 beam‑splitter, lossless channels, and noiseless homodyne detectors. Realistic losses, detector inefficiencies, and phase noise could degrade the correlation and reduce the achievable rates. The jammer model is restricted to displaced thermal states; more general non‑Gaussian or adaptive attacks are not covered. Future work should address robustness to loss, develop error‑correction schemes tailored to the induced binary channels, and explore experimental demonstrations of the correlation‑distribution phase. Overall, the paper provides a clear pathway from theoretical AVC capacity results to implementable quantum‑optical protocols that generate the essential shared randomness on‑the‑fly.