Abstract. We give an overview of some remarkable connections between symmetric informationally complete measurements (SIC-POVMs, or SICs) and algebraic number theory, in particular, a connection with Hilbert's 12 th problem. The paper is meant to be intelligible to a physicist who has no prior knowledge of either Galois theory or algebraic number theory.
Let K be a real quadratic field. For certain K with sufficiently small discriminant we produce explicit unit generators for specific ray class fields of K using a numerical method that arose in the study of complete sets of equiangular lines in C d (known in quantum information as symmetric informationally complete measurements or SICs). The construction in low dimensions suggests a general recipe for producing unit generators in infinite towers of ray class fields above arbitrary K and we summarise this in a conjecture. Such explicit generators are notoriously difficult to find, so this recipe may be of some interest.In a forthcoming paper we shall publish promising results of numerical comparisons between the logarithms of these canonical units and the values of L-functions associated to the extensions, following the programme laid out in the Stark Conjectures.
We study the correlation structure of separable and classical states in $2\times2$-~and~$2\times3$-dimensional quantum systems with fixed spectra. Even for such simple systems the maximal correlation - as measured by mutual information - over the set of unitarily accessible separable states is highly non-trivial to compute; however for the $2\times2$ case a particular class of spectra admits full analysis and allows us to contrast classical states with more general separable states. We analyse a particular entropic binary relation on the set of spectra and prove for the qubit-qutrit case that this relation alone picks out a unique classical maximum state for mutual information. Moreover the $2\times3$ case is the largest system with such a property.
We propose a recipe for constructing a fiducial vector for a symmetric informationally complete positive operator valued measure (SIC-POVM) in a complex Hilbert space of dimension of the form d = n2 + 3, focusing on prime dimensions d = p. Such structures are shown to exist in 13 prime dimensions of this kind, the highest being p = 19 603. The real quadratic base field K (in the standard SIC-POVM terminology) attached to such dimensions has fundamental units u K of norm −1. Let [Formula: see text] denote the ring of integers of K; then, [Formula: see text] splits into two ideals: [Formula: see text] and [Formula: see text]. The initial entry of the fiducial is the square ξ2 of a geometric scaling factor ξ, which lies in one of the fields [Formula: see text]. Strikingly, each of the other p − 1 entries of the fiducial vector is a product of ξ and the square root of a Stark unit. These Stark units are obtained via the Stark conjectures from the value at s = 0 of the first derivatives of partial L-functions attached to the characters of the ray class group of [Formula: see text] with modulus [Formula: see text], where ∞1 is one of the real places of K.
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