Controlled-Memory Identification with Shared Generators: Dimension--Horizon Tradeoffs and Quantum--Classical Separation

Abstract: We determine the finite-horizon identification limits of a (d)-dimensional memory when the control interface is restricted to four shared, time-homogeneous generators. A writer encodes a message by applying a prefix word to a common root state. The verifier then applies a candidate-dependent query word followed by a shared terminal binary test, with each word of length at most (T). For fixed error tolerances (λ2>0) and (λ_1+λ_2<1), the optimal identification sizes satisfy (\ln N{\mathsf Q,4}=Θ{λ_1,λ_2}(\min{d^2,T})) for arbitrary CPTP generators and (\ln N{\mathsf S,4}=Θ{λ_1,λ_2}(\min{d,T})) for stochastic maps. Four unitary letters and four permutation letters attain these respective bounds, showing that a four-letter reversible alphabet reaches the static code-size orders in optimal quadratic and linear word length. In the fully dimension-limited regime, this yields (\ln N{\mathsf Q,4}=Θ(d^2)) versus (\ln N_{\mathsf S,4}=Θ(d)), establishing a quadratic quantum--classical separation in logarithmic code size. Low-overlap Grassmann and agreement-code packings supply the orbit endpoints, while exact inverse query words match each encoded state to its designated test. At the sharp boundary (λ_2=0), (0\leqλ_1<1), all four model classes satisfy (\ln N=Θ(\min{\ln d,T})). Thus any fixed positive false-acceptance tolerance activates the metric-packing regime, whereas zero false acceptance imposes an exclusive-witness constraint and yields the logarithmic-dimension boundary.