Spectrum estimation of a d-dimensional quantum state is possible with o(d²) copies—specifically O(d² (log log d / log d)²)—beating Keyl–Werner and full tomography.
On Quantum Estimation, Quantum Cloning and Finite Quantum de Finetti Theorems , ISBN=
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The Keyl-Werner algorithm is not optimal for spectrum estimation
Spectrum estimation of a d-dimensional quantum state is possible with o(d²) copies—specifically O(d² (log log d / log d)²)—beating Keyl–Werner and full tomography.