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 .
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 The Symmetric Informationally Complete Positive Operator-Valued Measures (SIC-POVMs) are known to exist in all dimensions ⩽ 151 and many larger dimensions as high as 39 604 . All known solutions with the exception of the Hoggar solutions are covariant with respect to the Weyl-Heisenberg group and in the case of dimension 3, it has been proven that all SIC-POVMs are Weyl-Heisenberg group covariant. In this work, we explore this by SIC-POVMs in dimensions 4–7 without the assumption of group covariance. We introduce two functions with which SIC-POVM Gram matrices can be generated without the group covariance constraint, and analytically show that the SIC-POVM Gram matrices exist on critical points of the surfaces defined by the two functions on a subspace of Hermitian matrices. In dimensions 4–7, all known SIC-POVM Gram matrices lie in disjoint continuous sets of solution. Thus, we define an equivalent class that relates Gram matrices based on the trivial symmetries of the two functions. In dimensions 4–7, we generated { 1.7 × 10 6 , 1.1 × 10 5 , 169 , 50 } Gram matrices, respectively. For each of the Gram matrices, we generate the symmetry group of their respective disjoint sets. In all cases, the symmetry group was isomorphic to a subgroup of the Clifford group containing the Weyl-Heisenberg group matrices and the order-3 unitaries. In dimensions 4–6, All constructed solutions belonged to a single equivalence class of Gram matrices whereas in dimension 7, we find two distinct families of SIC-POVMs. In dimensions 4 and 5, the absence of new classes of SIC-POVM strongly suggests that the functions do not have a non-trivial symmetry. Furthermore, in all the dimensions tested, the only equivalence classes found correspond to the distinct stabilizers of fiducial vectors of each dimension.
Symmetric Informationally Complete Positive Operator Valued Measures (usually referred to as SIC-POVMs or simply as SICs) have been constructed in every dimension ≤67. However, a proof that they exist in every finite dimension has yet to be constructed. In this paper we examine the Galois group of SICs covariant with respect to the Weyl-Heisenberg group (or WH SICs as we refer to them). The great majority (though not all) of the known examples are of this type. Scott and Grassl have noted that every known exact WH SIC is expressible in radicals (except for dimension 3 which is exceptional in this and several other respects), which means that the corresponding Galois group is solvable. They have also calculated the Galois group for most known exact examples. The purpose of this paper is to take the analysis of Scott and Grassl further. We first prove a number of theorems regarding the structure of the Galois group and the relation between it and the extended Clifford group. We then examine the Galois group for the known exact fiducials and on the basis of this we propose a list of nine conjectures concerning its structure. These conjectures represent a considerable strengthening of the theorems we have actually been able to prove. Finally we generalize the concept of an anti-unitary to the concept of a g-unitary, and show that every WH SIC fiducial is an eigenvector of a family of g-unitaries (apart from dimension 3).
Symmetric Informationally Complete Positive Operator Valued Measures (usually referred to as SIC-POVMs or simply as SICS) have been constructed in every dimension up to 67. However, a proof that they exist in every finite dimension has yet to be constructed. In this paper we examine the Galois group of SICs covariant with respect to the Weyl-Heisenberg group (or WH SICs as we refer to them). The great majority (though not all) of the known examples are of this type. Scott and Grassl have noted that every known exact WH SIC is expressible in radicals (except for dimension 3 which is exceptional in this and several other respects), which means that the corresponding Galois group is solvable. They have also calculated the Galois group for most known exact examples. The purpose of this paper is to take the analysis of Scott and Grassl further. We first prove a number of theorems regarding the structure of the Galois group and the relation between it and the extended Clifford group. We then examine the Galois group for the known exact fiducials and on the basis of this we propose a list of nine conjectures concerning its structure. These conjectures represent a considerable strengthening of the theorems we have actually been able to prove. Finally we generalize the concept of an anti-unitary to the concept of a g-unitary, and show that every WH SIC fiducial is an eigenvector of a family of g-unitaries (apart from dimension 3).
Zauner's conjecture asserts that d2 equiangular lines exist in all d complex dimensions. In quantum theory, the d2 lines are dubbed a SIC, as they define a favoured standard informationally complete quantum measurement called a SIC-POVM. This note supplements A. J. Scott and M. Grassl [J. Math. Phys. 51 (2010), 042203] by extending the list of published numerical solutions. We provide a putative complete list of Weyl-Heisenberg covariant SICs with the known symmetries in dimensions d≤90, a single solution with Zauner's symmetry for every d≤121 and solutions with higher symmetry for d=124,143,147,168,172,195,199,228,259 and 323.
Abstract The existence of d2 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.
The paper concerns Weyl-Heisenberg covariant SIC-POVMs (symmetric informationally complete positive operator valued measures) and full sets of MUBs (mutually unbiased bases) in prime dimension. When represented as vectors in generalized Bloch space a SIC-POVM forms a d^2-1 dimensional regular simplex (d being the Hilbert space dimension). By contrast, the generalized Bloch vectors representing a full set of MUBs form d+1 mutually orthogonal d-1 dimensional regular simplices. In this paper we show that, in the Weyl-Heisenberg case, there are some simple geometrical relationships between the single SIC-POVM simplex and the d+1 MUB simplices. We go on to give geometrical interpretations of the minimum uncertainty states introduced by Wootters and Sussman, and by Appleby, Dang and Fuchs, and of the fiduciality condition given by Appleby, Dang and Fuchs.