Chernoff Information Bottleneck for Covert Quantum Target Sensing

1. Problem Statement

The paper addresses the challenge of covert quantum target sensing, specifically in the context of LiDAR and Radar systems. The core problem involves a sensing party (Alice) attempting to detect or range a target while avoiding detection by an adversarial party (Eve).

• The Conflict: In standard quantum metrology, high-energy probes are often used to maximize precision. However, in covert scenarios, increasing probe energy makes detection by Eve easier. Therefore, the probe energy must be strictly constrained.
• The Limitation of Classical Probes: Classical coherent states (lasers) require high energy to achieve a meaningful signal-to-noise ratio (SNR) in the presence of thermal background noise. Under strict energy constraints, classical probes often fail to provide sufficient information for the sender while remaining undetectable.
• The Quantum Opportunity: Quantum resources (specifically entangled states) offer potential advantages in low-energy regimes. However, previous quantum sensing protocols often relied on phase-sensitive measurements and quantum memories, which are technologically difficult to implement in realistic, turbulent environments (e.g., atmospheric scattering).
• The Gap: There is a need for a framework that evaluates quantum advantages in covert sensing using photon counting (intensity measurements), which is the state-of-the-art for practical LiDAR, without relying on complex quantum memories or phase coherence.

2. Methodology

The authors introduce a novel framework based on the Chernoff Information Bottleneck to quantify the trade-off between sensing performance and covertness.

A. The Chernoff Information Bottleneck

Instead of using standard relative entropy, the authors utilize Chernoff Information ($\xi$), which characterizes the asymptotic exponential decay rate of the probability of error ($p_{err} \propto e^{-M\xi}$) in hypothesis testing, where $M$ is the number of modes (repetitions).

They define two metrics:

1. $\xi^{(A)}$: Alice's Chernoff information for distinguishing the target's presence/position (Sensing ability).
2. $\xi^{(E)}$: Eve's Chernoff information for distinguishing Alice's probe from the background (Detection ability).

The Covert Information ($I_C$) is defined as an optimization problem:
$$I_C(d, S) := \max_{\rho_0 \in S} \xi^{(A)} \quad \text{subject to} \quad \xi^{(E)} \leq d$$
Where $S$ is the set of allowed probe states, and $d$ is the threshold for Eve's detection capability.

B. The Scaling Law and Parameter $\gamma$

The authors analyze the relationship between Alice's and Eve's performance via the covert information curve $\xi^{(A)} = I_C(\xi^{(E)})$. They approximate this curve near zero as:
$$I_C(d) \approx \lambda d^\gamma$$
The exponent $\gamma$ is the critical figure of merit:

• $\gamma = 1$ (Linear): Efficient covert sensing is impossible. Increasing the number of modes $M$ does not improve the trade-off; the error probability for Alice cannot be made arbitrarily small without Eve detecting the probe.
• $\gamma < 1$ (Sublinear): Efficient covert sensing is possible. As $M \to \infty$, Alice's error probability $\epsilon \to 0$ while Eve's detection probability remains bounded, regardless of the smallness of the constraint $\delta$.

C. Experimental Model

• Scenario: Target ranging (m-hypothesis test) in the optical regime.
• Measurements: Photon counting (intensity detection), ignoring phase information due to environmental turbulence and speckle noise.
• Probes Compared:
  1. Classical: Coherent states ($|\alpha\rangle$).
  2. Quantum: Entangled Two-Mode Squeezed Vacuum (TMSV) states, where the signal is sent to the target and the idler is retained (though the paper notes the marginal distribution mimics a multi-thermal state, avoiding the need for active quantum memory in the specific ranging protocol analyzed).
• Environment: Modeled as a beam splitter with reflectance $\kappa$, followed by incoherent mixing with a multi-mode thermal background ($\mu_B$).

3. Key Contributions

1. Framework Development: The paper establishes the Chernoff Information Bottleneck as a rigorous tool for analyzing covert sensing, moving beyond simple relative entropy bounds to asymptotic error rates.
2. Quantum Advantage in Photon Counting: It demonstrates that quantum advantage in covert sensing is achievable even with photon counting (intensity-only) measurements, removing the requirement for phase-sensitive interferometry or quantum memories.
3. Theoretical Proof of Impossibility for Classical Probes: The authors prove that for classical coherent states, the scaling parameter $\gamma \approx 1$ (linear). This implies that under strict covertness constraints, classical strategies cannot achieve arbitrarily low error rates for ranging, regardless of the number of modes used.
4. Demonstration of Quantum Feasibility: For entangled TMSV probes, the scaling parameter is found to be $\gamma \approx 1/2$ (sublinear). This proves that quantum probes can achieve high-precision ranging with vanishingly small error probabilities while maintaining covertness, provided the number of modes $M$ is sufficiently large.

4. Results

• Scaling Analysis:
  • Classical (Coherent): $\xi^{(A)} \propto \mu$ and $\xi^{(E)} \propto \mu^2$ (in the low-energy limit). This leads to a linear relationship $\xi^{(A)} \propto \xi^{(E)}$, resulting in $\gamma \approx 1$. Consequently, increasing $M$ does not help; the system is fundamentally limited.
  • Quantum (TMSV): The entangled nature of the probe allows $\xi^{(A)}$ to scale more favorably relative to $\xi^{(E)}$. The analysis yields $\gamma \approx 1/2$.
• Error Probability:
  • For the quantum case, the probability of error for Alice scales as $\epsilon \sim \exp(-M^{1-\gamma} \dots)$. With $\gamma < 1$, increasing $M$ drives the error to zero exponentially fast while keeping Eve's detection probability low.
  • For the classical case, the error probability cannot be driven to zero without violating the covertness constraint.
• Numerical Simulations:
  • Figures 2 and 3 illustrate the "relevance-complexity" curves. The quantum curve (blue) is strictly sublinear, lying above the classical curve (red) in the relevant region.
  • Even when Eve's collection efficiency is reduced (making it harder for her to detect), the classical probe still fails to achieve efficient covert sensing due to the linear scaling ($\gamma \approx 1$), whereas the quantum probe succeeds.

5. Significance

• Practical Relevance: By focusing on photon counting rather than phase-sensitive measurements, the proposed protocol is compatible with current commercial LiDAR technology (e.g., single-photon avalanche diodes). This bridges the gap between theoretical quantum advantage and near-term practical applications.
• Security Implications: The work provides a fundamental limit on classical covert sensing, showing that classical radar/LiDAR systems are inherently vulnerable to detection if they attempt to operate at the low energies required for secrecy.
• Quantum Advantage: It identifies a specific regime (low energy, high background, photon counting) where quantum entanglement is not just a marginal improvement but the only viable option for effective covert sensing.
• LiDAR/Radar Integration: The findings suggest that integrating quantum sources (TMSV) into existing LiDAR architectures could revolutionize stealth sensing capabilities for defense and security applications.

In summary, the paper mathematically proves that entanglement enables a sublinear scaling of the sensing-covertness trade-off, allowing for perfect covert ranging in the asymptotic limit, a feat that is fundamentally impossible for classical coherent light sources under the same constraints.