Related papers: SICs: Extending the list of solutions
Symmetric informationally complete measurements (SICs) are elegant, celebrated and broadly useful discrete structures in Hilbert space. We introduce a more sophisticated discrete structure compounded by several SICs. A SIC-compound is…
A well supported conjecture states that SIC-POVMs -- maximal sets of complex equiangular lines -- with anti-unitary symmetry give rise to an identity expressing some of its overlaps as squares of the (rescaled) components of a suitably…
This is a survey of some very old knowledge about Mutually Unbiased Bases (MUB) and Symmetric Informationally Complete POVMs (SIC). In prime dimensions the former are closely tied to an elliptic normal curve symmetric under the Heisenberg…
The Symmetric Informationally Complete Positive Operator-Valued Measures (SIC-POVMs) are known to exist in all dimensions $\leq 151$ and few higher dimensions as high as $1155$. All known solutions with the exception of the Hoggar solutions…
Coherence, treated as a resource in quantum information theory, is a basis-dependent quantity. Looking for states that have constant coherence under canonical changes of basis yields highly symmetric structures in state space. For the case…
We present a conjectured family of SIC-POVMs which have an additional symmetry group whose size is growing with the dimension. The symmetry group is related to Fibonacci numbers, while the dimension is related to Lucas numbers. The…
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).
In complex vector spaces maximal sets of equiangular lines, known as SICs, are related to real quadratic number fields in a dimension dependent way. If the dimension is of the form $n^2+3$ the base field has a fundamental unit of negative…
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…
In the standard basis exact expressions for the components of SIC vectors (belonging to a symmetric informationally complete POVM) are typically very complicated. We show that a simple transformation to a basis adapted to the symmetries of…
We introduce a method to determine whether a given generalised quantum measurement is isolated or it belongs to a family of measurements having the same prescribed symmetry. The technique proposed reduces to solving a linear system of…
Recently, several intriguing conjectures have been proposed connecting symmetric informationally complete quantum measurements (SIC POVMs, or SICs) and algebraic number theory. These conjectures relate the SICs and their minimal defining…
Informationally complete measurements are a dramatic discovery of quantum information science, and the symmetric IC measurements, known as SICs, are in many ways optimal among them. Close study of three of the "sporadic SICs" reveals an…
Symmetric informationally complete positive operator-valued measures (SIC-POVMs) in finite dimension $d$ are a particularly attractive case of informationally complete POVMs (IC-POVMs), which consist of $d^{2}$ subnormalized projectors with…
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…
Informationally complete (IC) measurements are fundamental tools in quantum information processing, yet their physical implementation remains challenging. By the Naimark extension theorem, an IC measurement may be realized by a von Neumann…
We show that in prime dimensions not equal to three, 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…
The problem of finding symmetric informationally complete POVMs (SIC-POVMs) has been solved numerically for all dimensions $d$ up to 67 (A.J. Scott and M. Grassl, {\it J. Math. Phys.} 51:042203, 2010), but a general proof of existence is…
We give a holomorphic quartic polynomial in the overlap variables whose zeros on the torus are precisely the Weyl-Heisenberg SICs (symmetric informationally complete positive operator valued measures). By way of comparison, all the other…
Line systems passing through the origin of the $d$ dimensional Euclidean space admitting exactly two distinct angles are called biangular. It is shown that the maximum cardinality of biangular lines is at least $2(d-1)(d-2)$, and this…