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 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.
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 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.
Symmetric informationally complete (SIC) positive operator valued measures (POVMs) are a class of quantum measurements which, in addition to being informationally complete, satisfy three conditions: (1) every POVM element is rank one, (2) the Hilbert-Schmidt inner product between any two distinct elements is constant, and (3) the trace of each element is constant. The third condition is often overlooked, since it may give the impression that it follows trivially from the second. We show that this condition cannot be removed, as it leads to two distinct values for the trace of an element of the POVM. This observation has led us to define a broader class of measurements which we call semi-SIC POVMs. In dimension two, we show that semi-SIC POVMs exist, and we construct the entire family. In higher dimensions, we characterize key properties and applications of semi-SIC POVMs, and note that the proof of their existence remains open.
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.
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) of the matrices have the same value of 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) on the Bloch sphere of 3 and 4 dimensional Hilbert spaces. In the 3-dimensional Hilbert space, we generate 104 generalized SIC-POVMs for randomly chosen Tr(ρ3) values such that all the elements are equivalent up to unitary transformations.