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arXiv:2405.16733

SIC-POVMs and the Knaster's Conjecture

S. B. Samuel, Z. Gedik

Published May 27, 2024• 0 citations
arXivHTMLPDF

Abstract

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 $(n^2-1)$ of the matrices have the same value of $Tr(ρ^k)$. 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 $Tr(ρ^3)$ on the Bloch sphere of $3$ and $4$ dimensional Hilbert spaces. In the $3$-dimensional Hilbert space, we generate $10^4$ generalized SIC-POVMs for randomly chosen $Tr(ρ^3)$ values such that all the elements are equivalent up to unitary transformations.

DOI
10.48550/arxiv.2405.16733

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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.
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 SIC-POVMs exist in all dimensions; however, the general proof of their existence is still an open problem. By mapping SIC-POVMs onto the generalized Bloch sphere, we prove two geometric existence statements associated with the SIC-POVM existence problem. First, we prove the Knaster’s conjecture for n vertices of an n-simplex and use that to prove the existence of a continuous family of general SIC-POVMs where the trace of the kth power of the operators is the same for (n2 − 1) of the elements. Furthermore, by using numerical methods, we show that in dimensions 3 and 4, a regular simplex can be constructed on the Bloch sphere such that all its vertices correspond to density matrices having the same trace of ρ3. In the three-dimensional Hilbert space, we generate 104 general SIC-POVMs for where all the elements have the same set of randomly chosen eigenvalues, indicating the continuous set of general SIC-POVMs can be constructed for all possible values of trace of ρ3.
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.
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.
The existence of pairwise equiangular complex lines [equivalently, a symmetric informationally complete positive operator-valued measure (SIC-POVM)] in -dimensional Hilbert space is known only for finitely many dimensions ⁠. We prove that, if there exists a set of real units in a certain ray class field (depending on ⁠) satisfying certain algebraic properties, a SIC-POVM exists, when 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 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 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.