Size and reliability limits of quantum memory for heterodyne uncertainty

Can a larger quantum memory compensate for a less reliable one? This paper answers that question for optical heterodyne quantum conditional entropy under a total photon budget and a strict physical memory dimension. It separates an uncertainty loss that additional storage can reduce from an erasure floor that no increase in dimension can remove. Preparation and noise model. The optical system has \(m\) modes and total mean photon number at most \(E\). Its joint preparation with a memory is optimized over all normal mixed states, including arbitrary optical coherence and infinite photon tails. The input memory has dimension at most \(d\). Independently, the entire register survives with probability \(1-p\) or is erased into an orthogonal readable failure state. The physical output therefore has \(d+1\) levels; a total \(q\)-qubit register permits \(d\le2^q-1\). Every event is retained, with no postselection. For heterodyne measure \(d^{2m}z/\pi^m\) and natural logarithms, the exact all-preparation reduction is \[V_{\rm er}(m,d,E,p)=\min_{\operatorname{rank}\rho\le d,\;\operatorname{Tr}(N\rho)\le E}\{W(\rho)-(1-p)S(\rho)\}.\] Here \(W\) is Wehrl entropy, \(S\) is von Neumann entropy and \(N\) is total photon number. The rank constraint follows from pure attainment of the original mixed optical–memory problem. The objective describes quantum side information; it is not automatically the uncertainty achieved by a classical readout of the memory. An exact reliability ceiling. Define \(g(x)=(1+x)\ln(1+x)-x\ln x\), \(S_*=m g(E/m)\), the ideal unlimited-memory value \(B_m(E)=m-E\ln(1+m/E)\), and \(a_E=1-[(1+E/m)\ln(1+m/E)]^{-1}\). For any target excess satisfying \(0<\varepsilon<a_ES_*\), \[p_{\rm crit}=\frac{\varepsilon}{m g(E/m)}.\] A finite register reaches \(V_{\rm er}\le B_m(E)+\varepsilon\) if and only if \(p<p_{\rm crit}\). At equality only an infinite memory attains the target. Above the ceiling the target is impossible even with unlimited memory. Finite storage and reliability pay different costs. Let \(S_d\) be the largest entropy of any rank-at-most-\(d\) optical state within the same photon budget, and \(\Delta_d=S_*-S_d\). For \(0\le p<a_E\), \[pS_*+(a_E-p)\Delta_d\le V_{\rm er}-B_m(E)\le pS_*+(1-p)\Delta_d.\] The first contribution survives at unlimited dimension, whereas \(\Delta_d\) tends to zero as storage grows. An exact finite Gibbs program computes \(S_d\), including partially occupied photon shells. The two bounds give necessary and sufficient resource tests rather than an exact full finite-dimensional frontier. The unlimited-memory floor is obtained from an explicit scalar thermal optimization. Above its energy-use crossover the attaining thermal source uses less than the allowed energy. At \(0<E\le10^{-4}\), \(2\le d\le m+1\) and \(p<1/8\), an explicit vacuum–one-photon entangled preparation remains globally optimal over the full preparation class. Finite examples also reserve the failure flag inside the stated register and distinguish heralded erasure from unheralded replacement. The analysis builds on Gaussian Wehrl extremality theorems established in earlier literature and on Zixuan He's previous results on ideal-memory spectrum shaping and low-photon resource laws. Its additional contribution is the optimized finite-memory erasure contract, the exact feasibility boundary and physically budgeted attaining sources. The integrated 14-page article contains the complete proofs, two figure groups, finite certificates and references.