quantum states | Mamuta memuāri

I contracted the SIC-POVM sickness a month ago after reading a paper by Gelo Tabia and Marcus Appleby who explore the qutrit case in great detail. I got over it in a week or so just to contract it again today when a paper by Gilad Gour came out. It has a very promising title that ends with “…symmetric informationally complete measurements exist in all dimensions”. Unfortunately, it begins with “General…”. Let’s see what the paper is about. Update: Chris Ferrie has pointed out to me that a similar result has already been obtained by Marcus Appleby (see Sect. 4 of this paper).

Intuition

The geometric intuition of this result can be summarized as follows:

Choose an arbitrary orthonormal basis for the linear space of all $d \times d$ Hermitian matrices.
Construct a simplex out of these basis vectors in some specific way (see below).
Shrink it sufficiently small so that it fits inside the positive-semidefinite cone.

The last step always works, because the convex body formed by all density matrices contains a ball around the maximally mixed state. The required amount of shrinking is determined by parameter $t$ in Theorem 1 (it is also related via Eq. (7) to parameter $a$ that measures the “purity” of the resulting SIC-POVM).

Details and example

The second step can be described more precisely as follows. Let $\{f_1, \dotsc, f_{d^2-1}\}$ be an arbitrary orthonormal basis of $\mathbb{R}^{d^2-1}$ (think of it as the space of all $d \times d$ traceless Hermitian matrices). Then a regular simplex with vertices $p_1, \dotsc, p_{d^2}$ can be obtained as follows:

$p_i := f - d(d+1) f_i$

$p_{d^2} := (d+1) f$

where $i = 1, \dotsc, d^2-1$ and $f$ is the sum of all $f_i$ (these expressions correspond to Eq. (5) in the paper). For example, if $d=2$ and $f_i := e_i$ is the i-th standard basis vector, then vectors $p_i$ are the rows of the following matrix:

$P = \begin{pmatrix} -5 & 1 & 1 \\ 1 & -5 & 1 \\ 1 & 1 & -5 \\ 3 & 3 & 3 \end{pmatrix}$

They look like this in 3D (the orange arrows):
Simplex
Note that

$P \cdot P^{\mathsf{T}}= \begin{pmatrix} 27 & -9 & -9 & -9 \\ -9 & 27 & -9 & -9 \\ -9 & -9 & 27 & -9 \\ -9 & -9 & -9 & 27 \end{pmatrix}$

hence vectors $p_1, p_2, p_3, p_4$ indeed form a simplex in $\mathbb{R}^3$ . In fact, this construction works in any dimension — there is nothing special about it being of the form $d^2-1$ .

Open questions

The hard question (which is still open) is how to choose the basis $f_i$ so that you need to shrink the simplex as little as possible. In other words, you want the matrices associated to $p_i$ to have rank one. All we know about this case is that the matrices $F_i$ associated to the basis vectors $f_i$ must have certain eigenvalues given by Eq. (11) in the paper. How close do we get to this if $F_i$ are generalized Pauli matrices or Haar random?

It would be interesting to know if anything extra can be said about the matrices $F_i$ if the SIC-POVM obeys Weyl-Heisenberg symmetry (in prime dimensions this is without loss of generality due to Huangjun Zhu).