Quantum limits and one-qubit advantage in optical predictive compression

What should an optical memory retain when only a limited amount of source information can be kept? This paper gives the complete quantum information frontier for a fixed Gaussian source and relevance channel, and identifies when a single retained qubit preserves correlations that every classical record loses. It also turns that advantage into a concrete test of whether received light belongs to its stored source preparation. The physical experiment and the complete frontier A fixed Gaussian state on \(A\) is prepared with its actual purifier \(C\). A normal encoder acts on \(C\) to produce memory \(M\), while a prescribed Gaussian channel maps \(A\) to the relevant optical output \(B\). The source-information cost and useful correlation are \(r=I(A:M)\) and \(v=I(B:M)\), respectively. All logarithms are natural. The paper determines \[ CQ(R)=N:\,I(A:M)≤ RI(B:M). \] For every fixed faithful finite-mode Gaussian source and finite-mode Gaussian relevance channel, optimization over arbitrary normal quantum encoders reduces to Gaussian supporting values and a mixture of at most two Gaussian branches. The branch flag is physically retained. The unrestricted frontier is a supremum with no imposed memory dimension or memory-energy cap; an explicit finite approximation gives an error uniform in the information budget. Hard finite-dimensional implementations are treated separately. Why one qubit can preserve the leading low-brightness information For a thermal source with mean photon number \(n\) and a thermal attenuator with transmissivity \(0<η<1\), write the output occupation as \(η n+b\), where \(b≥0\) is the environment's added thermal occupation. With \(g(x)=(x+1)log(x+1)-xlog x\), the exact optimum over all normal classical records is \[ Ccl(n,R)=g(η n+b)-g(η u_R+b), u_R=g⁻¹\!(max\{g(n)-R,0\}). \] A legal encoder applies vacuum loss with transmissivity \(λ\) to the purifier and then coherently retains two levels, resetting the entire higher-photon tail. For fixed \(b>0\) and \(0<λ<1/2\), at the common budget \(R_n=g(n)+g(λ n)-g((1-λ)n)\), its actual relevance satisfies \[ v_2=ηλ/1+b\,nlog(1/n)+O(n), Ccl(n,R_n)=2ληlog(1+1/b)\,n+O\!(n/log(1/n)). \] The full optical tail is included in the resource accounting. At \(n=0.01\), \(η=b=0.5\), \(λ=0.2\) and \(R=0.023875\), the certified predictive-mutual-information advantage exceeds 85%. The leading correlation is quantified by memory-side discord and can survive when the final output–memory state is separable. The vacuum-environment endpoint is evaluated separately for the same encoder. A same-resource correlation decision The receiver distinguishes matched source pairs from independent systems with identical marginals. For \(k\) uses, the retained predictive memory obeys \( K_k≤2^k\) and \(I(A^k:K_k)≤ kR\), including all retained flags. Both competitors may make arbitrary joint binary measurements on the received optical modes and their retained memory. The received modes are common observation access, outside the predictive-memory cap. Let \(α_k\) be the missed-correlation probability and \(β_k\) the false-match probability. For every classical block encoder, even with an unrestricted classical alphabet, the fixed-error converse is \[ α_k≤ε β_k≥exp\![-k c_nR+log2/1-ε], c_n=ηlog(1+1/(η n+b))/log(1+1/n). \] At the finite parameters above and \(ε=0.1\), the product-qubit encoder achieves an asymptotic false-match exponent of 0.0049 nats per pair, whereas every classical block strategy has exponent below 0.003138386231. This uses one qubit per source pair and identical measurement permissions. It is an asymptotic discrimination guarantee, distinct from the 85% single-copy information comparison; it specifies neither a finite-sample threshold nor a laboratory receiver circuit. The integrated paper contains the complete proofs, resource accounting and figures. The exact thermal classical comparator, the finite-memory construction and the block-testing connection have explicit proof dependencies separate from the general Gaussian-frontier reduction.