I am interested in the existence of a set of vectors $\{ v\_{ij} \}\_{ij} \subseteq \mathbb{C}^N$ for $i \in \{1,\dots,N\}$, $j \in \{1,\dots,N+1\}$ such that $\left\vert v^\*\_{ij} v\_{ij'} \right\vert = 1/N$ for $j \ne j'$ and
$v^\*\_{ij} v\_{i'j} = \delta\_{ii'}$.


For real vectors, the condition $\left\vert v^\*\_{ij} v\_{ij'} \right\vert = 1/N$ is satisfied if $\{ v\_{ij} : i \}$ is a regular simplex for each $j$. I have verified numerically the existence of such solutions for $N \le 5$. For $N=3$ a solution is given by the [Compound of three tetrahedra](https://en.wikipedia.org/wiki/Compound_of_three_tetrahedra).


For $N=p^k$ with $p$ prime, a solution is given by a set of mutually unbiased bases. These are not known to exist when $N \ne p^k$ but my conditions are looser than the MUB conditions.


Numerical experiments show that such vectors exist for $N \le 6$. Do they exist for all $N$?



Set of orthogonal simplexes or partial mutually unbiased bases
Vern Paulsen has provided me an answer, and I reproduce it here with his permission. For $j \in \{1,\dots,N+1\}$ define the matrix with columns $U\_j = (v\_{1j};\dots;v\_{Nj})$. The condition $v^\*\_{ij} v\_{i'j} = \delta\_{ii'}$ is equivalent to $U\_j$ being a unitary matrix. The other condition, $\lvert v^\*\_{ij} v\_{ij'} \rvert=1/N$ for $j \ne j'$, is equivalent to the matrix product $U\_j^\* U\_{j'}$ having values of modulus $1/N$ along the diagonal.


Define $\omega = e^{2\pi i/N}$ and $\gamma = e^{2\pi i/(N+1)}$. Let $W$ be the Fourier transform matrix $W\_{ij} = \omega^{ij} / \sqrt{N}$. Define $V\_j = \mathrm{diag}(\gamma^j, \gamma^{2j}, \dots, \gamma^{Nj})$ and $U\_j = W V\_j W^\*$. Since $W$ and $V\_j$ are unitary, so is $U\_j$. And it can be seen that the diagonal entries of $U^\*\_j U\_{j'} = W V^\*\_j V\_{j'} W^\*$ are equal to $\mathrm{Tr}(V^\*\_j V\_{j'})/N = \sum\_{i=1}^N \gamma^{(j'-j)N}/N = -1/N$ when $j \ne j'$.


