Codes and Expansions (CodEx) Seminar

Boumediene Et-Taoui (Université de Haute Alsace)
A survey on complex conference matrices

By a complex conference matrix (of order \(n\)) we will mean an \(n \times n\)-matrix \(C\), with zeroes on the diagonal, and complex entries with modulus one elsewhere, satisfying the condition \(C^\ast C = (n − 1)I\). Two complex conference matrices \(A\) and \(A'\) are equivalent if there exist unitary diagonal matrices \(D_1\) and \(D_2\) and a permutation matrix \(S\) such that \(A' = D_1SAST D_2\). Complex conference matrices have received considerable attention in the past few years due to their application in quantum information theory and in geometry.

In this talk we report, in the chronological order, all the known constructions of complex conference matrices. A \((n, k)\) equiangular Parseval frame is a set of \(n\) vectors in \(\mathbb{C}^k\) which generate equiangular lines in \(\mathbb{C}^k\) satisfying an additional property. We know from Holmes and Paulsen that the existence of an equiangular Parseval frame \((n, k)\) is equivalent to the existence of a Seidel matrix, a Hermitian matrix with diagonal entries all \(0\) and off-diagonal entries all of modulus \(1\), with two eigenvalues. In 2010 Duncan, Hoffman, Solazzo constructed complex Hermitian conference matrices of orders \(6\), \(10\), \(14\) and \(18\) and then deduced \((6,3)\), \((10,5)\), \((14,7)\) and \((18,9)\) complex equiangular Parseval frames. In 2013 I generalized their result by proving that for any integer \(k\) such that \(2k = p \alpha + 1 \equiv 2 (\mathrm{mod} 4)\), \(p\) prime, \(\alpha\) non-negative integer, there exists a family of \((2k, k)\) complex equiangular tight frames which depends on a complex number of modulus \(1\). However, Zauner constructed in his Phd thesis (2009) \((q + 1,(q + 1)/2)\) complex equiangular tight frames for any odd prime power \(q\), while I proved that the associated Seidel matrices of these frames are real symmetric conference matrices or the product by \(i\) of real skew symmetric conference matrices. This means that Zauner’s frames with \(q \equiv 1(\mathrm{mod} 4)\) are isometric to real frames, which is not the case of our family.

In 2017 I proved that for each order \(n\) for which there exists a real symmetric conference matrix of the Paley type there exists a complex symmetric conference matrix of order \(n − 1\). More precisely, for any integer \(k\) such that \(2k = p \alpha + 1 \equiv 2 (\mathrm{mod} 4)\), \(p\) odd prime, \(\alpha\) positive integer, I constructed a new, previously unknown infinite family of complex symmetric conference matrices of order \(2k−1\). In 2018 Blokhuis, Brehm and I generalize the above result to all real symmetric conference matrices. We also gave a new method which provides us with some complex symmetric conference matrices called dihedral, of even orders up to order \(18\). In 2021 jointly with Makhlouf we provided a new construction of complex skew-symmetric conference matrices. In addition we classified all complex conference matrices up to order \(5\). Furthermore we fully classified all the complex symmetric, skew-symmetric and Hermitian conference matrices of order \(6\).

Notice that complex conference matrices of order \(n\) are important for the construction of complex Hadamard matrices of order \(2n\) and also for the construction of \(n\)-sets of equi-isoclinic planes in Euclidean spaces.