SIC-POVM sickness

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:

  1. Choose an arbitrary orthonormal basis for the linear space of all d \times d Hermitian matrices.
  2. Construct a simplex out of these basis vectors in some specific way (see below).
  3. 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).

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