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Quantum Information and Computation

Galois automorphisms of a symmetric measurement

Published May 1, 2013• 17 citations
Publisher

Abstract

Symmetric Informationally Complete Positive Operator Valued Measures (usually referred to as SIC-POVMs or simply as SICs) have been constructed in every dimension $\le 67$. 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).

DOI
10.26421/qic13.7-8-8

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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 .

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Symmetric Informationally Complete Positive Operator Valued Measures (usually referred to as SIC-POVMs or simply as SICs) have been constructed in every dimension ≤67\le 67≤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 333 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 ggg-unitary, and show that every WH SIC fiducial is an eigenvector of a family of ggg-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).
We describe the structure of the extended Clifford group [defined to be the group consisting of all operators, unitary and antiunitary, which normalize the generalized Pauli group (or Weyl–Heisenberg group as it is often called)]. We also obtain a number of results concerning the structure of the Clifford group proper (i.e., the group consisting just of the unitary operators which normalize the generalized Pauli group). We then investigate the action of the extended Clifford group operators on symmetric informationally complete–positive operator valued measures (or SIC–POVMs) covariant relative to the action of the generalized Pauli group. We show that each of the fiducial vectors which has been constructed so far (including all the vectors constructed numerically by Renes et al.) is an eigenvector of one of a special class of order 3 Clifford unitaries. This suggests a strengthening of a conjecture of Zauner’s. We give a complete characterization of the orbits and stability groups in dimensions 2–7. Finally, we show that the problem of constructing fiducial vectors may be expected to simplify in the infinite sequence of dimensions 7,13,19,21,31,… . We illustrate this point by constructing exact expressions for fiducial vectors in dimensions 7 and 19.
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.
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.
We show that in prime dimensions not equal to 3, each group covariant symmetric informationally complete positive operator valued measure (SIC POVM) is covariant with respect to a unique Heisenberg–Weyl (HW) group. Moreover, the symmetry group of the SIC POVM is a subgroup of the Clifford group. Hence, two SIC POVMs covariant with respect to the HW group are unitarily or antiunitarily equivalent if and only if they are on the same orbit of the extended Clifford group. In dimension 3, each group covariant SIC POVM may be covariant with respect to three or nine HW groups, and the symmetry group of the SIC POVM is a subgroup of at least one of the Clifford groups of these HW groups, respectively. There may exist two or three orbits of equivalent SIC POVMs for each group covariant SIC POVM, depending on the order of its symmetry group. We then establish a complete equivalence relation among group covariant SIC POVMs in dimension 3, and classify inequivalent ones according to the geometric phases associated with fiducial vectors. Finally, we uncover additional SIC POVMs by regrouping of the fiducial vectors from different SIC POVMs which may or may not be on the same orbit of the extended Clifford group.
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 .