Sample-Efficient Tomography of a Class of Mixed States with Extensive Entanglement and Magic

Abstract: Full tomography of a generic many-qubit quantum state requires exponentially many copies, while suitable structural constraints can make reconstruction sample-efficient. Existing approaches exploit, for example, limited entanglement structure, low magic, or constrained state-preparation circuits. Here we consider a class of mixed states that can simultaneously exhibit extensive entanglement and extensive magic. Specifically, we introduce Clifford-encoded block-product (CEBP) states, obtained by applying an unknown global Clifford unitary to a tensor product of arbitrary mixed states supported on unknown blocks of bounded size. We show that, for a fixed block size, CEBP states can be reconstructed using polynomially many copies by exploiting Clifford-preserved Pauli correlations to recover the latent structure and reduce the remaining problem to local tomography. Our result demonstrates that sample-efficient tomography can arise from bounded complexity in a latent frame even when the physical state is highly entangled, highly magical, and mixed, and suggests complexity modulo structured transformations as a broader organizing principle for quantum-state learnability.