Memory capacity and lifetime thresholds for testing pure Gaussianity

How much quantum information must a sequential optical test retain, and how reliably must that information survive between source preparations? This paper gives preparation budgets and impossibility bounds that distinguish useful quantum storage from regimes where the classical sample order is unavoidable. Optical source and fully counted memory A fixed unknown normal state supplies independent, identical preparations on \(m\) bosonic modes. The null consists of all pure Gaussian states with total mean photon number at most one. Alternatives have half trace distance at least \(1/32\) from every pure Gaussian state; neither their energy nor the comparison family's energy is capped. Mixed states and infinite photon tails are included. Each visit accesses one entire original once. The total quantum dimension retained between visits is at most \(D\), while classical records and computation are unrestricted. At every actual waiting interval, the entire register independently survives unchanged with known probability \(a\), or is erased with a classical flag. The quantity \(k_a(D,m)\) is the smallest deterministic original-preparation cap achieving worst-case error at most \(1/3\). Failed preparations, discarded trials and reserved positions are all charged. A retained register of \(q\) qubits has \(D=2^q\). An exact threshold for statistical advantage Without retained quantum information, the optimal sample order is \(Θ(m)\). Along arbitrary growing-mode sequences, the precise criterion for improving that order is \[ k_a(D,m)=o(m) am→∞\ and\ aD/m→∞. \] The two conditions express separate capacity and survival requirements. Arbitrarily large storage cannot reduce the linear sample order when \(a=O(1/m)\), yet a survival probability tending to zero can still permit an advantage. Let \(A=max\{1,m^2/D\}\). At the fixed energy and separation above, constants \(c,C>0\), independent of \(m,D,a\), give \[ cmin\{m,A/a\}≤ k_a(D,m)≤ Cmin\{m,A/a+min(a⁻²,m2/3)\}, 0