When Unknown Mode Layouts Cost Quantum Memory: A Sharp Reliability Threshold

Entropy sets the usual storage rate; reliability determines which rare structures must also be stored. Two optical sources can have the same limiting entropy rate yet require different quantum memories once storage failure must decay exponentially. This paper determines when missing mode-grouping information creates that extra cost, and how much classical layout information is necessary to remove it. Source and task. The source consists of rare shared blocks, \[T_m=\big[(1-w_m)\sigma_0^{\otimes m}+w_m\sigma_1^{\otimes m}\big]^{\otimes n},\qquad M=mn,\qquad n/m\to\kappa>0,\quad nw_m\to\lambda>0.\] The known one-mode densities are distinct and faithful, with a common exponential photon moment. An unknown permutation rearranges the individual modes. Storage uses one common subspace for every possible layout, preserves the accepted quantum component coherently, and measures error by rejection probability. The known-layout comparison supplies the grouping, while the realized branch labels remain unavailable in both cases. 1. Exact quantum-memory and reliability formulas. Define \[\tau_\theta=(1-\theta)\sigma_0+\theta\sigma_1,\qquad \Psi(\alpha)=\max_{0\le\theta\le1}\ln\operatorname{Tr}\tau_\theta^\alpha.\] For a memory of dimension at most \(e^{MR}\), the optimal worst-layout rejection exponent is \[E_{\mathrm{lay}}(R)=\sup_{0<\alpha\le1}\frac{(1-\alpha)R-\Psi(\alpha)}{\alpha}.\] Equivalently, at prescribed rejection probability \(\epsilon_m=e^{-cM+o(M)}\), \(c>0\), the minimum common-subspace dimension obeys \[\lim_{m\to\infty}\frac{\ln d_m^{\mathrm{lay}}(\epsilon_m)}M=R_{\mathrm{lay}}(c)=\inf_{0<\alpha<1}\frac{\alpha c+\Psi(\alpha)}{1-\alpha}.\] The powers are evaluated on the actual mixture density operator, including noncommuting conditional states. The comparison parameter \(\theta\) allows the formula to capture rare macroscopic branch populations whose probabilities are subexponential on the total-mode scale. 2. A sharp optical threshold. For additive Gaussian noise with covariance increment \(\operatorname{diag}(v,u)\), \(0n_0.\] For \(y>1/2\), the critical reliability exponent is \[n_c=n_0+\frac{(2y-1)(\nu_0^2-1/4)}{\nu_0},\qquad c_c=\beta(n_c-n_0)-g(n_c)+g(n_0)>0.\] It separates two storage decisions: \[R_{\mathrm{lay}}(c)=R_{\mathrm{known}}(c)\quad(0 R_{\mathrm{known}}(c)\quad(c>c_c).\] For \(0