Optimal lower bound for unentangled spectrum estimation

Establish a matching lower bound showing that the copy complexity in Theorem 1.2, namely O(d^3 min{(ε log d)^−4,(ε log d)^−2}) for estimating the spectrum of a d-dimensional quantum state to total variation error ε using adaptive single-copy measurements, is optimal among all adaptive single-copy protocols.

Background

The paper proves an upper bound for spectrum estimation under unentangled measurements with a factor of d more copies than in the entangled model. The authors conjecture that this upper bound is optimal even when measurements may be chosen adaptively.

The currently available unentangled lower bound is only the lower bound inherited from the more powerful entangled-measurement setting. Consequently, proving a genuinely unentangled lower bound matching the paper’s algorithm remains unresolved.