Quantum illumination is a technique for detecting the presence of a target in a noisy environment by means of a
quantum probe. We prove that the two-mode squeezed vacuum state is the optimal probe for quantum illumination
in the scenario of asymmetric discrimination, where the goal is to minimize the decay rate of the probability of a
false positive with a given probability of a false negative. Quantum illumination with two-mode squeezed vacuum
states offers a 6 dB advantage in the error probability exponent compared to illumination with coherent states.
Whether more advanced quantum illumination strategies may offer further improvements had been a longstanding
open question. Our fundamental result proves that nothing can be gained by considering more exotic quantum
states, such as, e.g., multimode entangled states. Our proof is based on a fundamental entropic inequality for the
noisy quantum Gaussian attenuators. We also prove that without access to a quantum memory, the optimal probes
for quantum illumination are the coherent states.