The quantum memory cost of optical correlations

How much quantum memory is needed to restore an unknown optical pulse together with its correlations? This paper gives matching storage laws showing that the answer depends on the actual joint preparation, even when the optical marginal is a positive mixture of coherent states. Fix an optical mode number \(m\), a known real \(k\)-dimensional displacement subspace with \(1\le k\le2m\), and a positive mean photon budget \(U_0\). The source is any probability mixture of coherent states in that subspace with actual mean photons at most \(U_0\). One normal encoder and decoder must work for the entire unknown source class, and every transmitted message or flag counts toward the same register dimension. The reference is inaccessible to both maps. Let \(\delta\) be the required half trace-distance error and \(q_{\min}(\delta)\) the least encoded qubit count. As \(\delta\downarrow0\), \[q_{\min}(\delta)=\begin{cases}\dfrac{k}{2}\log_2(1/\delta)+O(1),&\text{optical marginal or joint coherent preparation},\\k\log_2(1/\delta)+O(1),&\text{arbitrary reference extension or classical-bit reference}.\end{cases}\] A single inaccessible classical bit already witnesses the larger leading cost. To recover the smaller cost for correlations, the joint source must actually be prepared as \(\omega_{AB}=\int |Jx\rangle\langle Jx|\otimes\tau_B(x)\,\mathrm d\mu(x)\), with \(\int|x|^2\mathrm d\mu\le U_0\). Here \(J\) is the known displacement embedding and \(\tau_B(x)\) may be noncommuting normal reference states. The constants are uniform in the reference dimension and the orientation of \(J\); the codec may depend on the known \(J\). The mechanism is the difference between the probability of a rare bright component and the square-root amplitude of its retained correlations. Exact optical Gram matrices and a reference-coherence witness establish the converse for arbitrary normal decoders, including infinite-rank outputs. A common finite optical subspace supplies the attaining compression schemes while retaining the full photon tails. These are single-input universal memory-dimension laws at fixed \(m,k,U_0\), with every communication resource counted. They complement established known-source and many-copy compression theory. The integrated 10-page PDF contains two figures, complete proofs and references.