Covert Entanglement Generation over Bosonic Channels

TL;DR

This paper investigates covert generation of entanglement over lossy thermal-noise bosonic channels, showing a square-root scaling law $L_{ m EG}\sqrt{n}$ for the number of covert ebits across $n$ channel uses and providing a single-letter expression for the optimal $L_{ m EG}$.The authors adapt strategies from finite-dimensional covert entanglement results to the bosonic setting, deriving an achievability demonstration that leverages a pre-shared secret and position-based coding, and they establish a corresponding converse bound that aligns with the non-covert classical capacity bounds for these channels.Beyond the optimal scheme, they analyze practical implementations using single- and dual-rail photonic qubits, deriving achievable rates and highlighting a substantial gap to the theoretical optimum due to the complexity of decoupling and channel-infinite-dimensional effects.The work advances covert quantum communication by linking entanglement-generation performance to the same fundamental product of covert and reliable constants that govern classical covert capacities, while also identifying key directions for closing practical gaps through advanced bosonic codes and tighter converses.

Abstract

We explore covert entanglement generation over the lossy thermal-noise bosonic channel, which is a quantum-mechanical model of many practical settings, including optical, microwave, and radio-frequency (RF) channels. Covert communication ensures that an adversary is unable to detect the presence of transmissions, which are concealed in channel noise. We show that a square root law (SRL) for covert entanglement generation similar to that for classical communication: $L_{\rm EG}\sqrt{n}$ entangled bits (ebits) can be generated covertly and reliably over $n$ uses of a bosonic channel. We report a single-letter expression for optimal $L_{\rm EG}$ as well as an achievable method. We additionally analyze the performance of covert entanglement generation using single- and dual-rail photonic qubits, which may be more practical for physical implementation.

Figures (6)

• Figure 1: Bosonic channel model. a) is the lossy thermal-noise bosonic channel $\mathcal{E}_{A\to BW}^{\eta,\bar{n}_\mathrm{B}}$, parameterized by $\eta\in[0,1]$ representing loss, and thermal state $\hat{\rho}_{\rm th}(\bar{n}_\mathrm{B})$ with mean thermal photon number $\bar{n}_\mathrm{B}$ representing the channel noise. The channel has input subsystem at Alice $A$ and output subsystems $B$ and $W$ at Bob and Willie, respectively. b) is the isometry $V_{A \to BWE}^{\eta,\bar{n}_\mathrm{B}}$ in the Stinespring dilation of the bosonic channel $\mathcal{E}_{A\to BW}^{\eta,\bar{n}_\mathrm{B}}$. The TMS block is a two-mode squeezer with gain $G=1+\bar{n}_B$ and $\ket{0}$ is a vacuum state.
• Figure 2: System model for covert secrecy and covert entanglement generation. Alice either has an input and transmits, or she is quiet. When she has an input, she encodes and modulates a state $\hat{\rho}_{A^n}$ of system $A^n$ before transmitting it over $n$ uses of the bosonic channel $\mathcal{E}^{\eta,\bar{n}_\mathrm{B}}_{A\to BW}$ depicted in Fig. \ref{['fig:therm-iso-channels']}\ref{['fig:thermal-channel']}. Bob receives state $\hat{\rho}_{B^n}$, demodulates and decodes to estimate the input. Warden Willie receives state $\hat{\rho}_{W^n}$ and uses it to decide between hypotheses $H_0$ and $H_1$ corresponding to a quiet or transmitting Alice. For covert secrecy, Alice uses position-based coding with a pre-shared secret key $k$, unknown to Willie, to encode the message $m$ into a coherent-state codeword modulated using quadrature phase-shift keying (QPSK). Bob employs sequential decoding with the pre-shared secret $k$ to estimate $m$. For entanglement generation, Alice prepares a maximally-entangled state $\ket{\Phi}\bra{\Phi}_{RM}$. She encodes the state of system $M$ as a superposition of the codewords from the aforementioned secrecy codebook in system $A^n$. Bob constructs a coherent version of the sequential decoding scheme used in the secrecy construction to recover a state entangled with the reference system $R$.
• Figure 3: Construction of achievable covert quantum entanglement generation over the bosonic channel $\mathcal{E}_{A\to BW}^{\eta,\bar{n}_\textrm{B}}$. Alice first prepares a Bell state $\hat{\Phi}_{RM}$. She keeps the reference subsystem $R$ and transmits the state of subsystem $M$ of $\hat{\Phi}_{RM}$ as follows: given a pre-shared secret $\mathbf{x}$, she applies a QECC corresponding to the number $w(\mathbf{x})$ of non-innocent output states, the output of the QECC is $\hat{\rho}_{R\check{A}^{w(\mathbf{x})}}$. She then applies Pauli gates defined by a pre-shared secret sequence $\mathbf{y}$, followed by sparse coding that "spreads" these $w(\mathbf{x})$ non-innocent states across $n$ channel uses by inserting innocent input states according to locations in $\mathbf{x}$. Alice then modulates and transmits the resulting state $\hat{\rho}_{RA^n}$. Bob receives $\hat{\rho}_{RB^n}$, sub-selects the systems containing the non-innocent states using $\mathbf{x}$, obtaining $\hat{\rho}_{RB^{w(\mathbf{x})}}$ and demodulates by projecting them onto the qubit basis, yielding $\hat{\rho}_{R\check{B}^{w(\mathbf{x})}}$. He then inverts the Pauli twirling operation using $\mathbf{y}$ and decodes to obtain a state $\hat{\tilde{\Phi}}_{R\check{M}}$ entangled with Alice's system $R$. Willie performs an optimal hypothesis test on whether transmission occurred.
• Figure 4: Total number of covert ebits as a function of $\bar{n}_\mathrm{B}$ for a) $\eta = 0.95$, b) $\eta = 0.8$, and c) $\eta = 0.65$. In each subfigure, $n = 10^8$ and $\delta = 0.05$. The black line is the optimal rate from Theorem \ref{['thm:covert-capacity']}, while the dotted red and dash-dotted blue lines correspond to the single- and dual-rail rates. The yellow dashed line shows the asymptotic convergence of the optimal rate as $\bar{n}_\mathrm{B} \to \infty$.
• Figure 5: Total number of covert ebits as a function of $\eta$ for a) $\bar{n}_\mathrm{B}=10^{-6}$, b) $\bar{n}_\mathrm{B}=10^{-3}$, and c) $\bar{n}_\mathrm{B}=10^{-1}$, where $n=60\times10^9$ and $\delta=0.05$ for each. The black line is the optimal rate from Theorem \ref{['thm:covert-capacity']}, while the dotted red and dash-dotted blue lines correspond to the single- and dual-rail rates.

Theorems & Definitions (17)

• Definition 3.1
• Definition 3.2
• Theorem 1
• Lemma 1
• Lemma 2
• Definition 3.3
• Definition 3.4
• Theorem 2
• Remark 1
• Lemma 3