Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2606.23535

How Stark units enter SIC overlaps

Ingemar Bengtsson, Gary McConnell

quant-phmath.NTSubmitted Jun 22, 2026• Announced Jun 23, 2026
arXivHTMLPDF

Abstract

It has been observed that the mutual scalar products of the vectors in a SIC-POVM are given by algebraic units, and at least in some cases by square roots of Stark units. The full picture is somewhat more complicated, especially if non-minimal SIC-POVMs are considered. We present a mixture of exact and numerical evidence suggesting that the overlap units are always products of integral powers of square roots of Stark units from ray class fields all of which are attached to the maximal ring of integers in the base field. In the non-minimal case a lattice of such ray class fields is involved. In every second dimension (counted in a certain way) some of the overlap units equal $\pm 1$, and we show that this follows from a special property of the ray class fields. Our observations are complementary to but consistent with the claim that the overlap units can be calculated directly from the Shintani--Faddeev modular cocycle.

Comments
43 pages, 6 figures, 13 tables

Related papers

SIC-POVMs from Stark units: Prime dimensions n 2 + 3

Marcus Appleby, Ingemar Bengtsson, Markus Grassl +2 more

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.

arXiv:2112.05552•Journal of Mathematical Physics•Nov 1, 2022•10 citations

A Conjecture on Almost Flat SIC-POVMs

Ingemar Bengtsson, Markus Grassl

A well supported conjecture states that SIC-POVMs -- maximal sets of complex equiangular lines -- with anti-unitary symmetry give rise to an identity expressing some of its overlaps as squares of the (rescaled) components of a suitably chosen fiducial vector. In number theoretical terms the identity essentially expresses Stark units as sums of products of pairs of square roots of Stark units. We investigate whether the identity is enough to determine these Stark units. The answer is no, but the failure might be quite mild.

quant-pharXiv:2512.13201•Dec 16, 2025•0 citations

SIC-POVMs and the Stark Conjectures

Gene S. Kopp

The existence of pairwise equiangular complex lines [equivalently, a symmetric informationally complete positive operator-valued measure (SIC-POVM)] in -dimensional Hilbert space is known only for finitely many dimensions ⁠. We prove that, if there exists a set of real units in a certain ray class field (depending on ⁠) satisfying certain algebraic properties, a SIC-POVM exists, when is an odd prime congruent to 2 modulo 3. We give an explicit analytic formula that we expect to yield such a set of units. Our construction uses values of derivatives of zeta functions at and is closely connected to the Stark conjectures over real quadratic fields. We verify numerically that our construction yields SIC-POVMs in dimensions 5, 11, 17, and 23, and we give the first exact SIC-POVM in dimension 23.

•Sep 15, 2021•32 citations

SIC-POVMs from Stark units: Dimensions n2 + 3 = 4p, p prime

Ingemar Bengtsson, Markus Grassl, Gary McConnell

The existence problem for maximal sets of equiangular lines (or SICs) in complex Hilbert space of dimension d remains largely open. In a previous publication we gave a conjectural algorithm for how to construct a SIC if d = n2 + 3 = p, a prime number. Perhaps the most surprising number-theoretical aspect of that algorithm is the appearance of Stark units in a key role: a single Stark unit from a ray class field extension of a real quadratic field serves as a seed from which the SIC is constructed. The algorithm can be modified to apply to all dimensions d = n2 + 3. Here we focus on the case when d = n2 + 3 = 4p, p prime, for two reasons. First, special measures have to be taken on the Hilbert space side of the problem when the dimension is even. Second, the degrees of the relevant ray class fields are “smooth” in a sense that facilitates exact calculations. As a result the algorithm becomes easier to explain. We give solutions for 17 different dimensions of this form, reaching d = 39 604. Several improvements relative to our previous publication are reported, but we cannot offer a proof that the algorithm works for any dimensions where it has not been tested.

arXiv:2403.02872•Journal of Mathematical Physics•Aug 1, 2025•4 citations

SIC-POVMs and the Stark Conjectures

Gene S. Kopp

Abstract The existence of $d^2$ pairwise equiangular complex lines [equivalently, a symmetric informationally complete positive operator-valued measure (SIC-POVM)] in $d$-dimensional Hilbert space is known only for finitely many dimensions $d$. We prove that, if there exists a set of real units in a certain ray class field (depending on $d$) satisfying certain algebraic properties, a SIC-POVM exists, when $d$ is an odd prime congruent to 2 modulo 3. We give an explicit analytic formula that we expect to yield such a set of units. Our construction uses values of derivatives of zeta functions at $s=0$ and is closely connected to the Stark conjectures over real quadratic fields. We verify numerically that our construction yields SIC-POVMs in dimensions 5, 11, 17, and 23, and we give the first exact SIC-POVM in dimension 23.

arXiv:1807.05877•International Mathematics Research Notices•Jun 19, 2019•19 citations

Dimension towers of SICs. I. Aligned SICs and embedded tight frames

Marcus Appleby, Ingemar Bengtsson, Irina Dumitru +1 more

Algebraic number theory relates SIC-POVMs in dimension d > 3 to those in dimension d(d − 2). We define a SIC in dimension d(d − 2) to be aligned to a SIC in dimension d if and only if the squares of the overlap phases in dimension d appear as a subset of the overlap phases in dimension d(d − 2) in a specified way. We give 19 (mostly numerical) examples of aligned SICs. We conjecture that given any SIC in dimension d, there exists an aligned SIC in dimension d(d − 2). In all our examples, the aligned SIC has lower dimensional equiangular tight frames embedded in it. If d is odd so that a natural tensor product structure exists, we prove that the individual vectors in the aligned SIC have a very special entanglement structure, and the existence of the embedded tight frames follows as a theorem. If d − 2 is an odd prime number, we prove that a complete set of mutually unbiased bases can be obtained by reducing an aligned SIC to this dimension.

arXiv:1707.09911•Journal of Mathematical Physics•Nov 1, 2017•21 citations
It has been observed that the mutual scalar products of the vectors in a SIC-POVM are given by algebraic units, and at least in some cases by square roots of Stark units. The full picture is somewhat more complicated, especially if non-minimal SIC-POVMs are considered. We present a mixture of exact and numerical evidence suggesting that the overlap units are always products of integral powers of square roots of Stark units from ray class fields all of which are attached to the maximal ring of integers in the base field. In the non-minimal case a lattice of such ray class fields is involved. In every second dimension (counted in a certain way) some of the overlap units equal ±1\pm 1±1, and we show that this follows from a special property of the ray class fields. Our observations are complementary to but consistent with the claim that the overlap units can be calculated directly from the Shintani--Faddeev modular cocycle.
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.
A well supported conjecture states that SIC-POVMs -- maximal sets of complex equiangular lines -- with anti-unitary symmetry give rise to an identity expressing some of its overlaps as squares of the (rescaled) components of a suitably chosen fiducial vector. In number theoretical terms the identity essentially expresses Stark units as sums of products of pairs of square roots of Stark units. We investigate whether the identity is enough to determine these Stark units. The answer is no, but the failure might be quite mild.
The existence of pairwise equiangular complex lines [equivalently, a symmetric informationally complete positive operator-valued measure (SIC-POVM)] in -dimensional Hilbert space is known only for finitely many dimensions ⁠. We prove that, if there exists a set of real units in a certain ray class field (depending on ⁠) satisfying certain algebraic properties, a SIC-POVM exists, when is an odd prime congruent to 2 modulo 3. We give an explicit analytic formula that we expect to yield such a set of units. Our construction uses values of derivatives of zeta functions at and is closely connected to the Stark conjectures over real quadratic fields. We verify numerically that our construction yields SIC-POVMs in dimensions 5, 11, 17, and 23, and we give the first exact SIC-POVM in dimension 23.
The existence problem for maximal sets of equiangular lines (or SICs) in complex Hilbert space of dimension d remains largely open. In a previous publication we gave a conjectural algorithm for how to construct a SIC if d = n2 + 3 = p, a prime number. Perhaps the most surprising number-theoretical aspect of that algorithm is the appearance of Stark units in a key role: a single Stark unit from a ray class field extension of a real quadratic field serves as a seed from which the SIC is constructed. The algorithm can be modified to apply to all dimensions d = n2 + 3. Here we focus on the case when d = n2 + 3 = 4p, p prime, for two reasons. First, special measures have to be taken on the Hilbert space side of the problem when the dimension is even. Second, the degrees of the relevant ray class fields are “smooth” in a sense that facilitates exact calculations. As a result the algorithm becomes easier to explain. We give solutions for 17 different dimensions of this form, reaching d = 39 604. Several improvements relative to our previous publication are reported, but we cannot offer a proof that the algorithm works for any dimensions where it has not been tested.
Abstract The existence of d2d^2d2 pairwise equiangular complex lines [equivalently, a symmetric informationally complete positive operator-valued measure (SIC-POVM)] in ddd-dimensional Hilbert space is known only for finitely many dimensions ddd. We prove that, if there exists a set of real units in a certain ray class field (depending on ddd) satisfying certain algebraic properties, a SIC-POVM exists, when ddd is an odd prime congruent to 2 modulo 3. We give an explicit analytic formula that we expect to yield such a set of units. Our construction uses values of derivatives of zeta functions at s=0s=0s=0 and is closely connected to the Stark conjectures over real quadratic fields. We verify numerically that our construction yields SIC-POVMs in dimensions 5, 11, 17, and 23, and we give the first exact SIC-POVM in dimension 23.
Algebraic number theory relates SIC-POVMs in dimension d > 3 to those in dimension d(d − 2). We define a SIC in dimension d(d − 2) to be aligned to a SIC in dimension d if and only if the squares of the overlap phases in dimension d appear as a subset of the overlap phases in dimension d(d − 2) in a specified way. We give 19 (mostly numerical) examples of aligned SICs. We conjecture that given any SIC in dimension d, there exists an aligned SIC in dimension d(d − 2). In all our examples, the aligned SIC has lower dimensional equiangular tight frames embedded in it. If d is odd so that a natural tensor product structure exists, we prove that the individual vectors in the aligned SIC have a very special entanglement structure, and the existence of the embedded tight frames follows as a theorem. If d − 2 is an odd prime number, we prove that a complete set of mutually unbiased bases can be obtained by reducing an aligned SIC to this dimension.