Resources of the advantage in quantum illumination: Discord and entanglement

Technical Summary: Resources of the Advantage in Quantum Illumination: Discord and Entanglement

Problem Statement
Quantum illumination (QI) is a protocol where entangled photons are used to detect an object in a noisy environment, offering a quantum advantage over classical illumination even when initial entanglement is destroyed by noise. While it is established that quantum correlations are the source of this advantage, the precise operational roles of entanglement and quantum discord in mixed states remain ambiguous. Specifically, it is unclear whether entanglement or discord acts as a necessary or sufficient condition for achieving higher quantum advantage, particularly when the probe state is a mixed state rather than a pure Bell state. Previous studies have shown that for completely mixed environments, the quantum advantage equals the "discord of encoding," but the relationship between this advantage and the initial entanglement or discord of the probe state requires further delineation.

Methodology
The authors investigate this problem using a benchmark family of two-qubit mixed states known as Maximally Mixed Marginal (MMM) states (also called Bell diagonal states). These states are defined by the density matrix $\rho = \frac{1}{4}(I + \sum_{i=1}^3 c_i \sigma_i^A \otimes \sigma_i^B)$, where $|c_i| \leq 1$. This family encompasses a wide spectrum of quantum states, including Bell states, Werner states, $\alpha$-states, and $\beta$-states, allowing for a comprehensive analysis of quantum correlations.

The study proceeds as follows:

1. Setup: The probe is an MMM state, and the environment is a completely mixed state. The protocol involves sending a "signal" photon to a target (with reflectivity $\eta$) and retaining an "idler" photon for joint measurement.
2. Quantification: The authors calculate the Quantum Advantage (QA) as the difference in Holevo information between joint measurements (Bell basis) and local measurements. They compute two primary measures of quantum correlation for the initial probe state:
   • Entanglement: Measured via Entanglement of Formation (EoF), Relative Entropy of Entanglement ($E_{REE}$), and Bures measure of entanglement ($E_B$).
   • Discord: Measured via standard mutual-information-based quantum discord and Geometric Discord.
3. Conditional Extremal Analysis: To resolve the "broadened" (non-unique) relationship between advantage and correlation measures, the authors perform a conditional extremal analysis. They group MMM states into clusters with identical initial advantage and either identical initial discord or identical initial entanglement. Within each cluster, they identify the states with the maximum and minimum values of the opposing resource (e.g., max/min entanglement for fixed discord).
4. High-Noise Regime: An analytical expansion is performed for states near the completely mixed state (high-noise regime) to determine the functional dependence of advantage on initial discord.

Key Contributions and Results

• Validation of Discord of Encoding: The study confirms that for MMM states in a completely mixed environment, the quantum advantage is exactly equal to the discord of encoding ($\delta_{enc}$), consistent with prior theorems.
• Broadened Relations: The authors demonstrate that there is no one-to-one mapping between quantum advantage and initial entanglement or initial discord. States with identical initial entanglement (or discord) can yield different quantum advantages, indicating that the "usable" portion of discord depends on additional structural features of the state.
• Role of Entanglement (Sufficient but Not Necessary): Through conditional extremal analysis, the authors find that for a fixed initial discord, the maximum entanglement increases monotonically with the quantum advantage. However, the minimum entanglement does not show this trend (as separable states can still exhibit advantage).
  • Conclusion: Higher entanglement is a sufficient (but not necessary) resource for higher advantage.
• Role of Discord (Necessary but Not Always Sufficient): Conversely, for a fixed initial entanglement, the minimum discord increases monotonically with the quantum advantage. However, the maximum discord does not always increase with advantage (e.g., Werner states show high advantage with lower discord in certain regimes).
  • Conclusion: Higher discord is a necessary (but not always sufficient) resource for higher advantage.
• Robustness Across Measures: The authors repeat the analysis using alternative measures: Relative Entropy of Entanglement, Bures measure of entanglement, and Geometric Discord. The results remain qualitatively consistent, indicating that the observed resource-advantage relationship is a robust feature of the protocol within the MMM family, not an artifact of a specific quantifier.
• Linear Dependence in High-Noise Regime: In the limit where the probe state approaches a completely mixed state (high noise), the authors derive an analytical linear dependence: $QA \propto \delta_{in}$. Specifically, $\delta_{enc} = p_0 \eta^2 (1-p_0) \delta_{in}$. This highlights discord as the key resource for the resilience of quantum illumination to noise.

Significance
The paper claims to provide a precise operational delineation of the roles of entanglement and discord in quantum illumination. By moving beyond simple correlations to a conditional extremal analysis, the authors establish that:

1. Discord is the primary resource for noise resilience: In high-noise regimes, the linear scaling of advantage with initial discord underscores discord's role in maintaining protocol performance when entanglement is suppressed.
2. Entanglement is a booster, not a prerequisite: While high entanglement guarantees high advantage, it is not required, as separable discordant states can also outperform classical limits.
3. Generality: The consistency of these findings across conceptually distinct measures of quantum correlation suggests these are fundamental properties of the QI protocol for mixed states, rather than mathematical artifacts.

The authors modestly note that while MMM states cover a full spectrum of correlations, extending these specific results to general two-qubit mixed states remains an open problem for future work.