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arXiv:2312.13400 · Symmetry Integrability and Geometry Methods and Applications

SICs and the Triangle Group (3,3,3)

Danylo Yakymenko

Published May 29, 2024• 0 citations
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Abstract

The problem of existence of symmetric informationally-complete positive operator-valued measures (SICs for short) in every dimension is known as Zauner's conjecture and remains open to this day. Most of the known SIC examples are constructed as an orbit of the Weyl-Heisenberg group action. It appears that in these cases SICs are invariant under the so-called canonical order-three unitaries, which define automorphisms of the Weyl-Heisenberg group. In this note, we show that those order-three unitaries appear in projective unitary representations of the triangle group $(3,3,3)$. We give a full description of such representations and show how it can be used to obtain results about the structure of canonical order-three unitaries. In particular, we present an alternative way of proving the fact that any canonical order-three unitary is conjugate to Zauner's unitary if the dimension $d>3$ is prime.

DOI
10.3842/sigma.2024.044

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The problem of existence of symmetric informationally-complete positive operator-valued measures (SICs for short) in every dimension is known as Zauner's conjecture and remains open to this day. Most of the known SIC examples are constructed as an orbit of the Weyl-Heisenberg group action. It appears that in these cases SICs are invariant under the so-called canonical order-three unitaries, which define automorphisms of the Weyl-Heisenberg group. In this note, we show that those order-three unitaries appear in projective unitary representations of the triangle group (3,3,3)(3,3,3)(3,3,3). We give a full description of such representations and show how it can be used to obtain results about the structure of canonical order-three unitaries. In particular, we present an alternative way of proving the fact that any canonical order-three unitary is conjugate to Zauner's unitary if the dimension d>3d>3d>3 is prime.
In this work, we generalise an object that has proved interesting to both the pure and applied mathematician alike. Specifically, symmetric informationally complete positive operator valued measures (SIC-POVMs) were first encountered in the study finite tight frames, where they provide examples of complex equiangular tight frames. Then, the quantum information theory community discovered their own utility for these objects, for they provided a particularly useful example of a generalised measurement of a quantum system, a type of positive operator valued measure. In particular, they have proven useful to quantum state tomography and for probing quantum foundations. SIC-POVMs are collections of rank-one orthogonal projections acting on finite-dimensional Hilbert spaces and operational SIC-POVMs are their arbitrary rank counterparts. In this thesis, we cover necessary background on the standard quantum formalism, define SIC-POVMs and outline how they have been constructed up until this point via the orbit of the Weyl-Heisenberg group, which motivates Zauner’s conjecture. We go on to discuss generalised operational SIC-POVMs and give some new theorems that classify them. We then show that an infinite family of operational SIC-POVMs can be obtained via the representation theory of the symmetric group. Possibly shedding light on the unproven Zauner’s conjecture, we prove that an element of this infinite family cannot be generated via an orbit of the Weyl-Heisenberg group. We finish by describing the relation between these operational SIC-POVMs and Bell inequalities.
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.
Five years ago, Lane Hughston showed that some of the symmetric informationally complete positive operator valued measures (SICs) in dimension 3 coincide with the Hesse configuration (a structure well known to algebraic geometers, which arises from the torsion points of a certain elliptic curve). This connection with elliptic curves is signalled by the presence of linear dependencies among the SIC vectors. Here we look for analogous connections between SICs and algebraic geometry by performing computer searches for linear dependencies in higher dimensional SICs. We prove that linear dependencies will always emerge in Weyl–Heisenberg orbits when the fiducial vector lies in a certain subspace of an order 3 unitary matrix. This includes SICs when the dimension is divisible by 3 or equal to 8 mod 9. We examine the linear dependencies in dimension 6 in detail and show that smaller dimensional SICs are contained within this structure, potentially impacting the SIC existence problem. We extend our results to look for linear dependencies in orbits when the fiducial vector lies in an eigenspace of other elements of the Clifford group that are not order 3. Finally, we align our work with recent studies on representations of the Clifford group.
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 SCI-POVMs exist in all dimensions; however, the general proof of their existence is still an open problem. The Bloch sphere representation of SIC-POVMs allows for a general geometric description of the set of operators, where they form the vertices of a regular simplex oriented based on a continuous function. We use this perspective of the SIC-POVMs to prove the Knaster's conjecture for the geometry of SIC-POVMs and prove the existence of a continuous family of generalized SIC-POVMs where (n2−1)(n^2-1)(n2−1) of the matrices have the same value of Tr(ρk)Tr(ρ^k)Tr(ρk). Furthermore, by using numerical methods, we show that a regular simplex can be constructed such that all its vertices map to the same value of Tr(ρ3)Tr(ρ^3)Tr(ρ3) on the Bloch sphere of 333 and 444 dimensional Hilbert spaces. In the 333-dimensional Hilbert space, we generate 10410^4104 generalized SIC-POVMs for randomly chosen Tr(ρ3)Tr(ρ^3)Tr(ρ3) values such that all the elements are equivalent up to unitary transformations.