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Moment maps and Galois orbits for SIC-POVMs

Kael Dixon, Simon M Salamon

Published Dec 6, 2019• 0 citations
OpenAlex

Abstract

The equations that define covariant SIC-POVMs are interpreted in terms of moment maps. Attention is focussed on orbits of a cyclic subgroup of a maximal torus and their images in the moment polytope. In the SIC case, these images lie in an intersection of quadrics, which we describe explicitly. We then develop the conjectural relationships with number field theory by describing the structure of Galois orbits of overlap phases.

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Moment Maps and Galois Orbits in Quantum Information Theory

Kael Dixon, Simon M Salamon

SIC-POVMs are configurations of points or rank-one projections arising from the action of a finite Heisenberg group on $\mathbb{C}^d$. The resulting equations are interpreted in terms of moment maps by focusing attention on the orbit of a cyclic subgroup and the maximal torus in $\mathrm{U}(d)$ that contains it. The image of a SIC-POVM under the associated moment map lies in an intersection of real quadrics, which we describe explicitly. We also elaborate the conjectural description of the related number fields and describe the structure of Galois orbits of overlap phases.

arXiv:1912.03209•SIAM Journal on Applied Algebra and Geometry•Jan 1, 2020•1 citation

SIC-POVMs and orders of real quadratic fields

Gene S. Kopp, Jeffrey C. Lagarias

This paper concerns SIC-POVMs and their relationship to class field theory. SIC-POVMs are generalized quantum measurements (POVMs) described by d 2 equiangular complex lines through the origin in C d . Weyl–Heisenberg covariant SICs are those SIC-POVMs described by the orbit a single vector under a finite Weyl–Heisenberg group WH ( d ) . We relate known data on the structure and classification of Weyl–Heisenberg covariant SICs in low dimensions to arithmetic data attached to certain orders of real quadratic fields. For 4 ≤ d ≤ 90 , we show the number of known geometric equivalence classes of Weyl–Heisenberg covariant SICs in dimension d equals the cardinality of the ideal class monoid of (not necessarily invertible) ideal classes in the real quadratic order O Δ d of discriminant Δ d = ( d + 1 ) ( d − 3 ) ; we conjecture the equality extends to all d ≥ 4 . We prove that this conjecture implies the existence of more than one geometric equivalence class of Weyl–Heisenberg covariant SICs for every d > 22 . We conjecture that Galois multiplets of SICs may be put in one-to-one correspondence with the over-orders O ′ of O Δ d in such a way that the number of geometric classes in the multiplet equals the ring class number of O ′ . We test that conjecture against known data on exact SICs in low dimensions. We refine the “class field hypothesis” of Appleby, Flammia, McConnell, and Yard [8] to predict the exact class field over Q ( Δ d ) generated by the ratios of vector entries for the equiangular lines defining a Weyl–Heisenberg SIC. The refined conjectures use a recently developed class field theory for orders of number fields [36] . The refined class fields assigned to over-orders O ′ have a natural partial order under inclusion; the inclusions of these class fields fail to be strict in some cases. We characterize such cases and give a table of them for d < 500 .

arXiv:2407.08048•Journal of Number Theory•Jul 17, 2026•0 citations

The overlap phases of SIC-POVMs

Xin Lü

Abstract A symmetric informationally complete positive-operator-valued measure (SIC-POVM) is a special type of generalized quantum measurements that possesses a high degree of symmetry. It plays an important role in quantum tomography, quantum cryptography and foundations of quantum mechanics. The most important and difficult problem about SIC-POVM is to show its existence in every dimension n, which is suggested by many numerical evidences. In this paper, this existence problem of SIC-POVMs is investigated by considering their overlap phases. Specifically, the symmetric requirements of SIC-POVMs are encoded into the overlap phases between the fiducial project and the n 2 displacement operators. In this way, an equivalent condition of the existence of Weyl-Heisenberg group covariant SIC-POVMs is obtained in terms of these phases, so that a set of complex equations is established, which enables one to study the existence problem in a more explicit and direct fashion. In particular, we show that all the numbers appearing in the overlap phases have to be algebraic numbers. The significance of Zauner’s conjecture in our equivalent condition is also discussed, and therefore the SIC-POVMs in dimension four is constructed explicitly. Actually, our system of equations always admits a solution, the real difficulty lies in how to ensure these complex number solutions are of norm one, and therefore indeed phases. Our result shows that constructing SIC-POVMs from the overlap phases is possible, and symmetries can be applied directly to reduce the number of equations, so that it provide new insights into the difficult problem of SIC-POVMs, especially about the algebraic properties of these overlap phases.

Journal of Physics Conference Series•Nov 1, 2020•0 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

How Stark units enter SIC overlaps

Ingemar Bengtsson, Gary McConnell

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.

quant-pharXiv:2606.23535•Jun 23, 2026

Surveying points in the complex projective plane

Lane P. Hughston, Simon M Salamon

We classify SIC-POVMs of rank one in CP^2, or equivalently sets of nine equally-spaced points in CP^2, without the assumption of group covariance. If two points are fixed, the remaining seven must lie on a pinched torus that a standard moment mapping projects to a circle in R^3. We use this approach to prove that any SIC set in CP^2 is isometric to a known solution, given by nine points lying in triples on the equators of the three 2-spheres each defined by the vanishing of one homogeneous coordinate. We set up a system of equations to describe hexagons in CP^2 with the property that any two vertices are related by a cross ratio (transition probability) of 1/4. We then symmetrize the equations, factor out by the known solutions, and compute a Groebner basis to show that no SIC sets remain. We do find new configurations of nine points in which 27 of the 36 pairs of vertices of the configuration are equally spaced.

arXiv:1410.5862•Oct 21, 2014•2 citations
The equations that define covariant SIC-POVMs are interpreted in terms of moment maps. Attention is focussed on orbits of a cyclic subgroup of a maximal torus and their images in the moment polytope. In the SIC case, these images lie in an intersection of quadrics, which we describe explicitly. We then develop the conjectural relationships with number field theory by describing the structure of Galois orbits of overlap phases.
SIC-POVMs are configurations of points or rank-one projections arising from the action of a finite Heisenberg group on Cd\mathbb{C}^dCd. The resulting equations are interpreted in terms of moment maps by focusing attention on the orbit of a cyclic subgroup and the maximal torus in U(d)\mathrm{U}(d)U(d) that contains it. The image of a SIC-POVM under the associated moment map lies in an intersection of real quadrics, which we describe explicitly. We also elaborate the conjectural description of the related number fields and describe the structure of Galois orbits of overlap phases.
This paper concerns SIC-POVMs and their relationship to class field theory. SIC-POVMs are generalized quantum measurements (POVMs) described by d 2 equiangular complex lines through the origin in C d . Weyl–Heisenberg covariant SICs are those SIC-POVMs described by the orbit a single vector under a finite Weyl–Heisenberg group WH ( d ) . We relate known data on the structure and classification of Weyl–Heisenberg covariant SICs in low dimensions to arithmetic data attached to certain orders of real quadratic fields. For 4 ≤ d ≤ 90 , we show the number of known geometric equivalence classes of Weyl–Heisenberg covariant SICs in dimension d equals the cardinality of the ideal class monoid of (not necessarily invertible) ideal classes in the real quadratic order O Δ d of discriminant Δ d = ( d + 1 ) ( d − 3 ) ; we conjecture the equality extends to all d ≥ 4 . We prove that this conjecture implies the existence of more than one geometric equivalence class of Weyl–Heisenberg covariant SICs for every d > 22 . We conjecture that Galois multiplets of SICs may be put in one-to-one correspondence with the over-orders O ′ of O Δ d in such a way that the number of geometric classes in the multiplet equals the ring class number of O ′ . We test that conjecture against known data on exact SICs in low dimensions. We refine the “class field hypothesis” of Appleby, Flammia, McConnell, and Yard [8] to predict the exact class field over Q ( Δ d ) generated by the ratios of vector entries for the equiangular lines defining a Weyl–Heisenberg SIC. The refined conjectures use a recently developed class field theory for orders of number fields [36] . The refined class fields assigned to over-orders O ′ have a natural partial order under inclusion; the inclusions of these class fields fail to be strict in some cases. We characterize such cases and give a table of them for d < 500 .
Abstract A symmetric informationally complete positive-operator-valued measure (SIC-POVM) is a special type of generalized quantum measurements that possesses a high degree of symmetry. It plays an important role in quantum tomography, quantum cryptography and foundations of quantum mechanics. The most important and difficult problem about SIC-POVM is to show its existence in every dimension n, which is suggested by many numerical evidences. In this paper, this existence problem of SIC-POVMs is investigated by considering their overlap phases. Specifically, the symmetric requirements of SIC-POVMs are encoded into the overlap phases between the fiducial project and the n 2 displacement operators. In this way, an equivalent condition of the existence of Weyl-Heisenberg group covariant SIC-POVMs is obtained in terms of these phases, so that a set of complex equations is established, which enables one to study the existence problem in a more explicit and direct fashion. In particular, we show that all the numbers appearing in the overlap phases have to be algebraic numbers. The significance of Zauner’s conjecture in our equivalent condition is also discussed, and therefore the SIC-POVMs in dimension four is constructed explicitly. Actually, our system of equations always admits a solution, the real difficulty lies in how to ensure these complex number solutions are of norm one, and therefore indeed phases. Our result shows that constructing SIC-POVMs from the overlap phases is possible, and symmetries can be applied directly to reduce the number of equations, so that it provide new insights into the difficult problem of SIC-POVMs, especially about the algebraic properties of these overlap phases.
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.
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 classify SIC-POVMs of rank one in CP^2, or equivalently sets of nine equally-spaced points in CP^2, without the assumption of group covariance. If two points are fixed, the remaining seven must lie on a pinched torus that a standard moment mapping projects to a circle in R^3. We use this approach to prove that any SIC set in CP^2 is isometric to a known solution, given by nine points lying in triples on the equators of the three 2-spheres each defined by the vanishing of one homogeneous coordinate. We set up a system of equations to describe hexagons in CP^2 with the property that any two vertices are related by a cross ratio (transition probability) of 1/4. We then symmetrize the equations, factor out by the known solutions, and compute a Groebner basis to show that no SIC sets remain. We do find new configurations of nine points in which 27 of the 36 pairs of vertices of the configuration are equally spaced.