The memory cost of quantum illumination

Quantum illumination uses a retained idler to improve detection of a weak optical return in thermal noise. For a source designer, the practical resource question is how much ideal quantum storage is needed for a specified improvement. This work gives the exact asymptotic answer when the signal-photon budget and the total retained-register dimension are constrained independently. Exact finite-memory frontier Consider a thermal attenuator with transmissivity \(0<η<1\) and vacuum-output background occupation \(b>0\), tested against the replacer that returns the same thermal state. For \(n\) parallel uses, every admissible probe obeys a mean total signal-photon ceiling \(nN\) and a strict bound \( R≤ enP\) on the entire retained register, including retained source records. The memory rate is \(P/ln2\) ideal qubits per use. With vanishing miss probability, the optimal false-alarm exponent is \[Ξ(N,P)=ηln(1+1/b)N+G\!(min\{P,g(N)\}).\] Here all logarithms are natural, \(g(x)=(x+1)ln(x+1)-xln x\), with \(g(0)=0\), and \[G(p)=g(b)+p-F\!(g⁻¹(p)).\] The function \(F(x)\) is the entropy of the full complementary output for a thermal input of mean occupation \(x\); the complement includes the purification of the thermal environment. The first term in \(Ξ\) is the optimum without retained memory, and \(G\) is the additional detection benefit. The converse covers arbitrary normal mixed probes, correlations across channel uses and collective measurements. Matching sources satisfy the actual finite-register and energy bounds: they allocate photons between signal–idler correlations and a known displacement, and approach the frontier through finite-support preparations. How much storage is useful? The exact saturation rate and initial marginal value are \[Pₛₐₜ(N)=g(N), G'(0^+)=η/1+b.\] Because \(G\) is strictly increasing and strictly concave, at a positive signal budget \(N>0\) the minimum rate retaining a fixed fraction \(0<r<1\) of the full additive memory benefit is \[P_r(N)=G⁻¹\!(rG(g(N)))