Zauner's conjecture

Question

The conjecture is as follow: In $\mathbb{C}^{n}$, there exists $\{v_1,\cdots,v_{n^2}\}$ such that the following holds: $$ \left| \left \langle v_i, v_j \right \rangle \right| = \begin{cases} 1 & i = j\\ \frac{1}{n+1} & i \ne j\end{cases}$$ I have a prove for when $n = 2$, basically what I did is just assuming without loss of generality that one of the vectors is $\begin{bmatrix} 1 \\ 0 \end{bmatrix}$, and brute force the rest of the vectors. I'm curious where this construction fails when $n \ge 3$, or has the conjecture already been proven? I can't seem to find literature that it has been proven on the Internet though.

Answer 1

Here is a recent update, with a proof that Zauner's conjecture (the existence the existence of $n^2$ equiangular lines in $n$ complex dimensions) holds for $n\leq 67$: SIC-POVMs: A new computer study (2010).

It remains an open question whether the conjecture is true in all dimensions. Some insight into the difficulty of the conjecture is given in The Lie Algebraic Significance of Symmetric Informationally Complete Measurements (2011). See also this MO post and http://physicsoverflow.org/382

This blog from 2015 gives more background.