Unified Capacity Bound for Single-Copy Quantum State Representation

A central question in quantum information and quantum machine learning is whether quantum states provide a fundamentally stronger representation of classical data. We establish a unified capacity bound and no-go theorem for a broad class of single-copy quantum models. In the noiseless setting, we prove that their representational capacity is controlled by a purely classical invariant or the sign-rank of the target data relationships. Such an algebraic bound holds irrespective of the quantum states' internal entanglement, limiting the advantage of such models to at most quadratic in the effective feature space dimension. On realistic hardware, capacity is further curtailed by noise as any task whose sign-rank exceeds the rank of the device's quantum channel is unrealizable, yielding an algorithm-independent, physically grounded no-go theorem that is directly testable through the channel's superoperator rank. This merged result unifies algebraic and physical limitations, predicts staircase-like capacity collapses as channel rank drops, and provides a practical benchmark for near-term devices. It delineates where single-copy models cannot deliver structural gains and points to alternative routes beyond this paradigm as necessary for genuine quantum advantages.