Give an explicit example of a $d = 4$ SIC-POVM

Question

For $q=e^{2 \pi i/3}$, the set of $d^2$ vectors ($d=3$) \begin{equation} \left( \begin{array}{ccc} 0 & 1 & -1 \\ 0 & 1 & -q \\ 0 & 1 & -q^2 \\ -1 & 0 & 1 \\ -q & 0 & 1 \\ -q^2 & 0 & 1 \\ 1 & -1 & 0 \\ 1 & -q & 0 \\ 1 & -q^2 & 0 \\ \end{array} \right) \end{equation}

forms a SIC-POVM (symmetric, informationally complete, positive operator-valued measure), as noted in eq. (1) of https://arxiv.org/abs/1109.6514 .

I would similarly like to have a 16-vector counterpart for $d=4$ (to use for entanglement detection--per https://arxiv.org/abs/1805.03955). (There is clearly a huge amount of interesting related literature on such topics, but an attempt of mine to find an explicit d=4 counterpart somewhere within it has so far not been successful.)

Answer 1

As indicated by Danylo in his anwser, eq. (32) in arXiv: 1103.2030 presents the sixteen vectors ("ignoring overall phases and normalisation") \begin{equation} \left( \begin{array}{cccc} x & 1 & 1 & 1 \\ x & 1 & -1 & -1 \\ x & -1 & 1 & -1 \\ x & -1 & -1 & 1 \\ i & x & 1 & -i \\ i & x & -1 & i \\ -i & x & 1 & i \\ -i & x & -1 & -i \\ i & i & x & -1 \\ i & -i & x & 1 \\ -i & i & x & 1 \\ -i & -i & x & -1 \\ i & 1 & -i & x \\ i & -1 & i & x \\ -i & 1 & i & x \\ -i & -1 & -i & x \\ \end{array} \right), \end{equation} where \begin{equation} x=\sqrt{2+\sqrt{5}}. \end{equation} These form "a set of 16 SIC-vectors covariant under the Heisenberg group".

To now normalize the vectors we need to multiply them by $\frac{1}{\sqrt{5+\sqrt{5}}}$. The resultant sixteen vectors $|\psi_i\rangle$ satisfy the desired relation ($d=4$) \begin{equation} |\langle\psi_i||\psi_j\rangle|^2 = \frac{d \delta_{ij}+1}{d+1}, i,j=1,2,\ldots, d^2 , \end{equation} given in eq. (7) in arXiv: 1805.03955, giving us the explicit set requested in the question.

Answer 2

You can find it here Symmetric Informationally Complete Quantum Measurements or here SIC-POVMs: A new computer study, in the appendix B.

Update

Given a single fiducial vector $v = (a_1,a_2,a_3,a_4)^T \in \mathbb{C}^4$ it's pretty easy to write down all SIC-POVM vectors. They are just $C^kS^lv$ for $k,l \in \{0..3\}$, where $C$ and $S$ are clock and shift matrices given by $$ C = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & i & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -i \\ \end{pmatrix}, ~~~ S = \begin{pmatrix} 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ \end{pmatrix}. $$

So $C^kS^lv = (a_{1-l},i^ka_{2-l},(-1)^ka_{3-l},(-i)^ka_{4-l})^T$, where $a_0 = a_4$, $a_{-1} = a_3$, etc.
Note that a phase of any SIC-POVM vector doesn't matter.

Update 2

A simple formula for all 16 vectors can be found here, eq (32)
The non-normalized fiducial vector is just $$ \left( \begin{array}{c} \sqrt{2+\sqrt{5}} \\ 1 \\ 1 \\ 1 \end{array} \right) $$ but we must change the matrices $C$ and $S$ to correctly generate 16 vectors: $$ C = e^{i\pi/4}\begin{pmatrix} 0 & 1 & 0 & 0 \\ -i & 0 & 0 & 0 \\ 0 & 0 & 0 & -i \\ 0 & 0 & -1 & 0 \\ \end{pmatrix}, ~~~ S = e^{i\pi/4}\begin{pmatrix} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ -i & 0 & 0 & 0 \\ 0 & i & 0 & 0 \\ \end{pmatrix}. $$