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arXiv:1811.00947 · Journal of Mathematical Physics

Simplified exact SICS

Marcus Appleby, Ingemar Bengtsson

Published Jun 1, 2019• 14 citations
arXivHTMLPDF

Abstract

In the standard basis, exact expressions for the components of SIC vectors (belonging to a symmetric informationally complete positive operator valued measure or POVM) are typically very complicated. We show that a simple transformation to a basis adapted to the symmetries of a fiducial SIC vector can result in a massive reduction in complexity. We rely on a conjectural number theoretic connection between SICs in dimension dj and SICs in dimension dj+1 = dj(dj − 2). We focus on the sequence 5, 15, 195, … We rewrite Zauner’s exact solution for the SIC in dimension 5 to make its simplicity manifest and use our adapted basis to convert numerical solutions in dimensions 15 and 195 to exact solutions. Comparing to the known exact solutions in dimension 15, we find that the simplification achieved is dramatic. The proof that the exact vectors are indeed SIC fiducial vectors, also in dimension 195, is guided by the standard ray class hypothesis about the algebraic number fields generated by the SICs. In the course of the calculation, we introduce SIC adapted generators for the ray class field. We conjecture that our result generalizes to every dimension in the particular sequence we consider.

DOI
10.1063/1.5081508

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In the standard basis, exact expressions for the components of SIC vectors (belonging to a symmetric informationally complete positive operator valued measure or POVM) are typically very complicated. We show that a simple transformation to a basis adapted to the symmetries of a fiducial SIC vector can result in a massive reduction in complexity. We rely on a conjectural number theoretic connection between SICs in dimension dj and SICs in dimension dj+1 = dj(dj − 2). We focus on the sequence 5, 15, 195, … We rewrite Zauner’s exact solution for the SIC in dimension 5 to make its simplicity manifest and use our adapted basis to convert numerical solutions in dimensions 15 and 195 to exact solutions. Comparing to the known exact solutions in dimension 15, we find that the simplification achieved is dramatic. The proof that the exact vectors are indeed SIC fiducial vectors, also in dimension 195, is guided by the standard ray class hypothesis about the algebraic number fields generated by the SICs. In the course of the calculation, we introduce SIC adapted generators for the ray class field. We conjecture that our result generalizes to every dimension in the particular sequence we consider.
Abstract Recently, several intriguing conjectures have been proposed connecting symmetric informationally complete quantum measurements (SIC POVMs, or SICs) and algebraic number theory. These conjectures relate the SICs to their minimal defining algebraic number field. Testing or sharpening these conjectures requires that the SICs are expressed exactly, rather than as numerical approximations. While many exact solutions of SICs have been constructed previously using Gröbner bases, this method has probably been taken as far as is possible with current computer technology (except in special cases where there are additional symmetries). Here, we describe a method for converting high-precision numerical solutions into exact ones using an integer relation algorithm in conjunction with the Galois symmetries of an SIC. Using this method, we have calculated 69 new exact solutions, including nine new dimensions, where previously only numerical solutions were known—which more than triples the number of known exact solutions. In some cases, the solutions require number fields with degrees as high as 12 288. We use these solutions to confirm that they obey the number-theoretic conjectures, and address two questions suggested by the previous work.
In complex vector spaces maximal sets of equiangular lines, known as SICs, are related to real quadratic number fields in a dimension dependent way. If the dimension is of the form n^2+3, the base field has a fundamental unit of negative norm, and there exists a SIC with anti-unitary symmetry. We give eight examples of exact solutions of this kind, for which we have endeavoured to make them as simple as we can---as a belated reply to the referee of an earlier publication, who claimed that our exact solution in dimension 28 was too complicated to be fit to print. An interesting feature of the simplified solutions is that the components of the fiducial vectors largely consist of algebraic units.
The purpose of this project is first to survey all known simple constructions of SIC-POVMs and second to explore these solutions for additional symmetries. We focus on three main constructions: the Weyl-Heisenberg construction; constructions using Hadamard matrices; and Hoggar's construction. We then explore fiducial vectors in dimensions 7 and 19 which have additional attractive number-theoretic properties. Finally, we generate geometric configurations of the Majorana states of some SIC-POVMs in dimensions 3 and 4. We then plot the Majorana vectors of those SIC-POVMs in real dimension 3; these are then found to have special symmetry. In the process of generating the Majorana states, two potentially new SIC-POVMs in dimension three have been found.
The notion of Symmetric Informationally Complete Positive Operator-Valued Measures (SIC-POVMs) arose in physics as a kind of optimal measurement basis for quantum systems. However the question of their existence is equivalent to that of the existence of a maximal set of \emph{complex equiangular lines}. That is to say, given a complex Hilbert space of dimension ddd, what is the maximal number of (complex) lines one can find which all make a common (real) angle with one another, in the sense that the inner products between unit vectors spanning those lines all have a common absolute value? A maximal set would consist of d2d^2d2 lines all with a common angle of arccos⁡1d+1\arccos{\frac{1}{\sqrt{d+1}}}arccosd+1​1​. The same question has been posed in the real case and some partial answers are known. But at the time of writing no unifying theoretical result has been found in the real or the complex case: some sporadic low-dimensional numerical constructions have been converted into algebraic solutions but beyond this very little is known. It is conjectured that such structures always arise as orbits of certain fiducial vectors under the action of the Weyl (or generalised Pauli) group. In this paper we point out some new construction methods in the lowest dimensions (d=2d=2d=2 and d=3d=3d=3). We should mention that the SIC-POVMs so constructed are all unitarily equivalent to previously known SIC-POVMs.
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 uK of norm −1. Let ZK denote the ring of integers of K; then, pZK splits into two ideals: p and p′. The initial entry of the fiducial is the square ξ2 of a geometric scaling factor ξ, which lies in one of the fields K(uK). 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 ZK with modulus p∞1, where ∞1 is one of the real places of K.
The problem of finding symmetric informationally complete positive-operator-valued-measures (SIC-POVMs) has been solved numerically for all dimensions ddd up to 67 [A. J. Scott and M. Grassl, J. Math. Phys. 51, 042203 (2010)], but a general proof of existence is still lacking. For each dimension, it was shown that it is possible to find a SIC-POVM that is generated from a fiducial state upon application of the operators of the Heisenberg-Weyl group. We draw on the numerically determined fiducial states to study their phase-space features, as displayed by the characteristic function and the Wigner, Bargmann, and Husimi representations, adapted to a Hilbert space of finite dimension. We analyze the phase-space localization of fiducial states, and observe that the SIC-POVM condition is equivalent to a maximal delocalization property. Finally, we explore the consequences in phase space of the conjectured Zauner symmetry. In particular, we construct a Hermitian operator commuting with this symmetry that leads to a representation of fiducial states in terms of eigenfunctions with definite semiclassical features.