Resources of the advantage in quantum Illumination: Discord and entanglement

TL;DR

The work investigates how entanglement and quantum discord contribute to the quantum advantage in quantum illumination by using Maximally Mixed Marginal (MMM) two-qubit states as the initial resource. It computes the advantage via Holevo information and demonstrates that the advantage equals the discord consumed during encoding ($\delta_{enc}$), while employing a conditional extremal analysis to reveal how entanglement and discord independently constrain performance. The findings show that higher entanglement is a sufficient (not necessary) resource for greater advantage, whereas higher discord is a necessary (not always sufficient) resource, with a striking linear relation between advantage and initial discord in the high-noise regime: $\delta_{enc} = p_0 \eta^2 (1-p_0) \delta_{in}$. These results offer an operational perspective on how different quantum correlations underpin illumination robustness and guide resource allocation in noisy sensing tasks, highlighting the pivotal role of discord beyond entanglement.

Abstract

We investigate the quantum advantage in quantum illumination using two-qubit mixed states as the initial resource. We show that in quantum illumination, the achievable advantage is determined by an interplay between initial entanglement and discord. First, we rigorously show that the quantum advantage for a given state equals the amount of discord consumed for illumination. Subsequently, we find that states with identical initial discord can lead to varying advantages, indicating that the usable portion of discord for illumination depends on additional structural features of the state. Then, we consider the relation between the advantage and both entanglement and discord by performing a conditional extremal analysis. To this end, for states clustered by identical advantage and initial discord, we compute the maximum and minimum initial entanglement within each cluster. We demonstrate that, for states with fixed initial discord, the maximum (and not minimum) entanglement increases by increment of the advantage. We conclude that for any given initial discord, higher entanglement is a sufficient (but not necessary) resource for higher advantage. On the other hand, for states clustered by identical advantage and initial entanglement, we compute the maximum and minimum initial discord in each group. Here, the minimum (and not always maximum) discord scales monotonically with advantage. It shows that, for fixed initial entanglement, higher discord is a necessary (but not always sufficient) resource for higher advantage. This result provides a refined, operational perspective on how different forms of quantum correlations govern the performance of the illumination protocol. We finally find a persistent linear dependence of the advantage on initial discord in the high-noise regime, highlighting discord as the key resource for resilience to noise in the protocol.

Figures (7)

• Figure 1: Probability of a correct guess, $p(there|yes)$, in terms of the reflectivity factor $\eta$. This probability for joint measurement is higher than that for local measurement.
• Figure 2: Mutual information $I_{mut}$ for the case of joint measurement and local measurement in terms of $\eta$. Difference between diagrams for joint measurement and local measurement is called quantum advantage.
• Figure 3: a) Entanglement of formation for different values of $c_1$, $c_2$ and $c_3$, is shown by different colors. b) Quantum discords are also shown by different colors. c) Entanglement of formation vs. Quantum discord for MMM states (purple dots) have been shown. $\alpha$-states (orange line), $\beta$-states (green line) and Werner states (red line)are also plotted. Depending on the entanglement value, in some region $\alpha$-states and in some regions Werner states are the upper bound. Also, $\beta$-states are the lower bound for the diagram.
• Figure 4: a) Quantum advantage for MMM-states for different values of $c_1$, $c_2$ and $c_3$ are shown by different colors. b) Quantum advantage vs. discord of encoding. It shows that they are equal for any given MMM state. The reflectivity factor here is $\eta=0.5$.
• Figure 5: Quantum advantage versus initial discord for $\eta=0.5$. a) Light blue dots are MMM states. $\alpha$-states, $\beta$-states and Werner states are plotted as orange, green and red line respectively. $\alpha$-states are the lower bound and $\beta$-states are the upper bound. There is a small horn near $\delta_{in}=0.33$ in the diagram which is related to the separable states. b) A similar diagram for only entangled states where the small horn in (a) disappears. Among a cluster of states in each mesh cell, states with maximum entanglement is chosen. Precision of the mesh is 1/80. The color bar shows the maximum entanglement among the states of each cluster. c) Corresponding to three columns in (b), for three given values of the initial discord, maximum entanglement vs. the quantum advantage is plotted. Entanglement increases with increasing quantum advantage.