A Complete Solution to the Attainability of the Quantum Fisher Information for Qutrit systems

Quantum multiparameter estimation sets the ultimate limits on how accurately information can be extracted from a quantum system. While quantum resources can enhance precision, the simultaneous estimation of multiple parameters is more nuanced as the optimal measurements may not commute. Accordingly, it is of fundamental importance to understand when the ultimate limit of quantum multiparameter estimation, set by the quantum Fisher information (QFI), is attainable. Such conditions are known when entangling measurements across asymptotically many copies of the quantum state are performed. However, general conditions for the attainability of the QFI with measurements on a finite number of copies remain unknown, and this problem was recently listed as one of five open problems in quantum information. Here we present results which mark a major step towards answering this question. First we present the gap persistence theorem, reducing the question of whether the QFI is attainable with a measurement on any finite number of copies of the quantum state to the question of attainability with single-copy measurements - a significantly simpler problem. Second, we demonstrate that the QFI may be arbitrarily far from the true precision. Finally, we present necessary and sufficient conditions for the attainability of the QFI for almost full-rank density matrices. This is the first non-trivial setting for which conditions on the attainability of the QFI are known, and completely solves the problem for qutrit systems.