Minimum Quantum Storage from Incomplete Channel Observations

How much quantum storage must a hidden implementation retain when experimental observations leave its structure undetermined? This paper establishes exact minimum quantum-storage requirements across all implementations compatible with specified channel observations. The resource is the worst-input expected logarithm of retained quantum dimension across one controlled temporal cut, with unlimited classical records available at no cost. For the channel dimension measure introduced by Cope and Uola, the paper proves their suggested minimax equality relating minimum storage to output Schmidt measure and extends it to every nonempty compact convex set of compatible channels. One common input enforces the resource minimum across all compatible implementations. A finite optimal implementation exists, although an optimal entangled probe need not. For qubit channels, four measured coefficients determine an exact, attained minimum without assuming channel covariance or unitality. An explicit five-Kraus construction attains 99/169, while a separate analytical certificate proves that every compatible implementation must use nontrivial quantum storage with at least that probability on the same specified input. A second exact formula uses only two Pauli-transfer coefficients and yields a minimum storage requirement of 0.58 when both coefficients are jointly constrained to the interval [0.79, 0.81]. The paper also presents semidefinite optimization over compatible experimental uncertainty regions. The results require calibrated quantum inputs and outputs, concealed preparation labels, and accounting for every quantum carrier across the storage cut. They constrain storage resources without reconstructing the hidden implementation or certifying environmental non-Markovianity. Motivated by Shadow Theory’s source/readout distinction, the proofs use standard quantum mechanics and quantum-information mathematics. Version 1.1 includes complete proofs, explicit attaining constructions, worked examples, and discussion of experimental assumptions and limitations. No experimental data are reported.