How many orthonormal bases are needed to distinguish all pure quantum states?

TL;DR

This paper reviews recent findings on the minimal number of orthonormal bases required to uniquely identify all pure quantum states across different dimensions, highlighting known results and open problems.

Contribution

It summarizes and consolidates recent results on the minimal bases needed for pure state distinguishability in various dimensions, emphasizing unresolved cases.

Findings

For d=2, three bases suffice.

For d=3 and d>4, four bases are needed.

For d=4, the exact number remains an open problem.

Abstract

We collect some recent results that together provide an almost complete answer to the question stated in the title. For the dimension d=2 the answer is three. For the dimensions d=3 and d>4 the answer is four. For the dimension d=4 the answer is either three or four. Curiously, the exact number in d=4 seems to be an open problem.