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:1807.05877 · International Mathematics Research Notices

SIC-POVMs and the Stark Conjectures

Gene S. Kopp

Published Jun 19, 2019• 19 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1093/imrn/rnz153

Related papers

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: 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

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

Symmetric Informationally Complete Positive Operator Valued Measure and Zauner conjecture

Stefan Joka

In this paper, we show that in Hilbert space of any finite dimension N, there are N^2 unit vectors which constitute Symmetric Informationally Complete Positive Operator Valued Measure (SIC-POVM).

quant-pharXiv:2601.13475•Jan 21, 2026•0 citations

Symmetric informationally complete positive operators-valued measures and the Knaster’s conjecture

S. B. Samuel, Z. Gedik

Symmetric Informationally Complete Positive Operator-Valued Measures (SIC-POVMs) have been constructed in many dimensions using the Weyl–Heisenberg group. In the quantum information community, it is commonly believed that SIC-POVMs exist in all dimensions; however, the general proof of their existence is still an open problem. By mapping SIC-POVMs onto the generalized Bloch sphere, we prove two geometric existence statements associated with the SIC-POVM existence problem. First, we prove the Knaster’s conjecture for n vertices of an n-simplex and use that to prove the existence of a continuous family of general SIC-POVMs where the trace of the kth power of the operators is the same for (n2 − 1) of the elements. Furthermore, by using numerical methods, we show that in dimensions 3 and 4, a regular simplex can be constructed on the Bloch sphere such that all its vertices correspond to density matrices having the same trace of ρ3. In the three-dimensional Hilbert space, we generate 104 general SIC-POVMs for where all the elements have the same set of randomly chosen eigenvalues, indicating the continuous set of general SIC-POVMs can be constructed for all possible values of trace of ρ3.

Journal of Mathematical Physics•Sep 1, 2025•1 citation

On the Existence of Algebraic Equiangular Lines

Igor Van Loo, Frédérique Oggier

We consider real and complex equiangular lines, generated by unit vectors. We show that, for an arbitrary dimension $d$, if there exists a set of $d^2$ equiangular unit vectors in $\mathbb{C}^d$, then there must exist a set of $d^2$ equiangular unit vectors with all of their coefficients in a number field. This result is motivated by the question of constructing SIC-POVMs in quantum physics and conjectures around them. We discuss applications of our techniques to the case of real equiangular lines and consequences of the above results.

quant-pharXiv:2603.09128•Journal of Algebra and Its Applications•Mar 11, 2026•0 citations
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.
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.
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 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.
Symmetric Informationally Complete Positive Operator-Valued Measures (SIC-POVMs) have been constructed in many dimensions using the Weyl–Heisenberg group. In the quantum information community, it is commonly believed that SIC-POVMs exist in all dimensions; however, the general proof of their existence is still an open problem. By mapping SIC-POVMs onto the generalized Bloch sphere, we prove two geometric existence statements associated with the SIC-POVM existence problem. First, we prove the Knaster’s conjecture for n vertices of an n-simplex and use that to prove the existence of a continuous family of general SIC-POVMs where the trace of the kth power of the operators is the same for (n2 − 1) of the elements. Furthermore, by using numerical methods, we show that in dimensions 3 and 4, a regular simplex can be constructed on the Bloch sphere such that all its vertices correspond to density matrices having the same trace of ρ3. In the three-dimensional Hilbert space, we generate 104 general SIC-POVMs for where all the elements have the same set of randomly chosen eigenvalues, indicating the continuous set of general SIC-POVMs can be constructed for all possible values of trace of ρ3.
We consider real and complex equiangular lines, generated by unit vectors. We show that, for an arbitrary dimension ddd, if there exists a set of d2d^2d2 equiangular unit vectors in Cd\mathbb{C}^dCd, then there must exist a set of d2d^2d2 equiangular unit vectors with all of their coefficients in a number field. This result is motivated by the question of constructing SIC-POVMs in quantum physics and conjectures around them. We discuss applications of our techniques to the case of real equiangular lines and consequences of the above results.