Finite-rank approximation with exact position records and energy

Can a finite-rank quantum state approximate a given optical state while reproducing an entire continuous measurement record and its original mean energy exactly? This paper provides a constructive affirmative answer, with quantitative rank, trace-distance and entropy control. The main result concerns one oscillator \(A=L^2( R)\), with \([Q,P]=i\) and photon operator \(N=(Q^2+P^2-1)/2\), coupled to a finite memory \(B= C^b\). The input is any normal state \(ρAB\) with mean photons \(E=Tr(Nρ_A)<∞\). Its complete position–memory record \(G_ρ(q)\) retains both the position probability and the conditional quantum state of the memory. For every integer \(L≥0\) and \(0<a≤1\), there is a state \(ω\) satisfying \[G_ω=G_ρ, Tr(Nω_A)=E, rankω≤ b(L+2), 12\|ρ-ω\|_1/L+1+a.\] Thus the approximants converge inside the same exact-record, exact-energy constraint set. A purification needs a new register of dimension at most \(b(L+2)\); retaining both the original memory and this register costs total dimension at most \(b^2(L+2)\). The construction preserves a positive head of the original state and replaces its actual positive tail. If the head has rank at most \(r\) and the tail mass is \(p\), at most \(b\) replacement vectors suffice, giving rank at most \(r+b\) and half trace-distance error at most \(p\) before energy restoration. A position-dependent unitary with \(\|U-I\|≤ a\) then restores the exact energy without changing rank, spectrum or the measured record. With natural logarithms, \[|S(ρ)-S(ω)|≤ h_2(p)+p\,[g(E/p)+log b],\] where \(h_2\) is binary entropy, \(g(x)=(1+x)log(1+x)-xlog x\), and the zero-tail case is understood continuously. The same estimate controls quantum conditional entropy. Horizontal matrix purification removes kinetic overhead while handling zeros, changes of rank and the full real line. Scalar position distributions admit the corresponding construction for any fixed finite number of modes. The approximation is chosen from the known source, and finite rank refers to full wavefunctions rather than finite photon support. The integrated 12-page PDF contains two figures, complete proofs, and attribution of the classical kinetic, purification and phase identities used.