Optimal entanglement weights for heterodyne uncertainty with finite quantum memory

A finite quantum memory is specified by more than its entanglement entropy. This paper determines how to shape its Schmidt weights and allocate optical photons to minimize the quantum conditional uncertainty of a prescribed heterodyne measurement. The resulting resource laws let researchers compare a uniformly weighted entangled source with a shaped source, and assess when sharing one memory across optical modes is more efficient than independently preparing mode–memory pairs. Complete preparation model and sharp leading resource law. Let \(Vm,D(E)\) be the minimum of \(H_μ(Z|B)=h_μ(Z)-χ(Z:B)\) over all normal, possibly mixed optical–memory states with total mean photon number at most \(E\) and total accessible memory dimension at most \(D\). Here \(Z\) is the outcome of an \(m\)-mode heterodyne measurement, \(B\) is the retained quantum memory, \(χ\) is Holevo information, and \(dμ=π⁻ᵐd²ᵐz\). Accessible flags count toward the same memory budget. For every fixed \(m\) and every joint limit \(D,E→∞\), \[Vm,D(E)~Λ_m(E/T_D)/T_D, T_D=(m!D)1/m.\] The function \(Λ_m\) is an explicitly characterized constrained radial ground-state energy. Its Kummer and Bessel profiles give legal attaining preparations. The converse retains arbitrary coherent inputs and their photon tails, rather than optimizing only a diagonal ansatz. Why shaping changes the resource exponent. At the common photon ceiling \(E=T_D\), optimally shaped sources have uncertainty of order \(D-1/m\), whereas final flat Schmidt spectra have optimal order \(D-1/(2m)\), even after optimizing the occupied optical subspace. Flat weights create a sharp support boundary; tapering the weights replaces its boundary-layer cost by a smaller gradient cost. To reach a small target uncertainty \(ε\), this changes the leading memory requirement from \(2mlog_2(1/ε)\) to \(mlog_2(1/ε)\), and the attainable photon scaling from quadratic to linear in \(1/ε\). This improvement is compatible with nearly maximal entanglement entropy. For the explicit canonical saturated construction, \[S(ρB,D)=ln D-c_m+o(1), \|ρB,D-I_D/D\|_1 L_m>0.\] The full trace-norm limits are approximately \(L_1=0.8771\) and \(L_2=1.1170\); the half-trace-distance convention halves these numbers. Thus a vanishing relative entropy deficit does not make the optimized spectrum uniform. Collective preparation at fully counted resources. For two modes in the energy-sufficient asymptotic regime, sharing quantum memory reduces the leading conditional entropy by about \(10.24%\) relative to the optimized independent mode–memory preparation class. That comparator includes local state optimization, integer memory and energy allocations, mixtures and charged accessible flags; it is not the class of all separable joint states. A separate finite construction uses seven coherent optical inputs, three retained qubits and \(0.21\) input photons. Under the specified loss, phase and stochastic memory-fault contract, its retained photon budget is at most \(0.200781\), and its conditional entropy is strictly lower than that of the same independent comparator by more than \(0.00547535\) nats. The integrated article gives the preparation circuit, one-sided numerical bounds and complete same-source error accounting. Interpretation. The objective uses quantum side information. A classical measurement of the memory generally has accessible information below \(χ(Z:B)\), so the conditional-entropy advantages do not by themselves assert the same improvement in estimation precision or classical readout performance. The finite circuit is a rigorously specified source model, not a qualification of an implemented laboratory apparatus. The single PDF contains the article, five figures, full proofs and references.