Quantum illumination with an adversarial return phase: minimax reduction and the worst-case value of entanglement

We study target detection by quantum illumination (QI) when the return phase is unknown and may be chosen adversarially. The problem is a minimax quantum hypothesis test: for a fixed probe, the target picks a distribution of phase shifts and the radar picks the measurement. For any phase-covariant return channel and phase-invariant null hypothesis, which includes the bosonic thermal-loss model, the uniform phase together with the Helstrom test of the phase-averaged pair is an explicit saddle point. The worst-case performance therefore equals the known-phase performance of the probe pinched onto sectors of fixed total signal excitation number. With a phase drawn independently for each pulse, an entangled number-correlated probe performs exactly as well as its dephased, classically correlated counterpart. With a phase common to M ≥ 2 pulses, coherences inside the sectors survive. For a discrete-variable model we obtain the value in closed form without background, 1 − ½(1 − N_S κ)^M, and the exact two-pulse gain over the dephased probe with background, which is positive only above a memory-coherence threshold. For two pulses, the best independently prepared separable probe is exactly the dephased probe, so this gain is a strict advantage over every such separable probe above the threshold. Classical correlations between pulses can, however, outperform the entangled probe; the entangled version of the same pattern distribution again does better. The results hold both when background reaches the receiver under each hypothesis as modeled and when it is matched so that the vacuum return carries no signature. We bound the minimax Chernoff exponent between an achievable block-pinching exponent and the known-phase exponent and evaluate the bounds for discrete-variable and bosonic models.