A set of d 2 vectors in a Hilbert space of dimension d is called equiangular if each pair of vectors encloses the same angle. The projection operators onto these vectors define a POVM which is distinguished by its high degree of symmetry. Measures of this kind are called symmetric informationally complete , or SIC POVMs for short, and could be applied for quantum state tomography. Despite its simple geometrical description, the problem of constructing SIC POVMs or even proving their existence seems to be very hard. It is our purpose to introduce two applications of discrete Wigner functions to the analysis of the problem at hand. First, we will present a method for identifying symmetries of SIC POVMs under Clifford operations. This constitutes an alternative approach to a structure described before by Zauner and Appleby. Further, a simple and geometrically motivated construction for an SIC POVM in dimensions two and three is given (which, unfortunately, allows no generalization). Even though no new structures are found, we hope that the re-formulation of the problem may prove useful for future inquiries.
A set of d 2 vectors in a Hilbert space of dimension d is called equiangular if each pair of vectors encloses the same angle. The projection operators onto these vectors define a POVM which is distinguished by its high degree of symmetry. Measures of this kind are called symmetric informationally complete , or SIC POVMs for short, and could be applied for quantum state tomography. Despite its simple geometrical description, the problem of constructing SIC POVMs or even proving their existence seems to be very hard. It is our purpose to introduce two applications of discrete Wigner functions to the analysis of the problem at hand. First, we will present a method for identifying symmetries of SIC POVMs under Clifford operations. This constitutes an alternative approach to a structure described before by Zauner and Appleby. Further, a simple and geometrically motivated construction for an SIC POVM in dimensions two and three is given (which, unfortunately, allows no generalization). Even though no new structures are found, we hope that the re-formulation of the problem may prove useful for future inquiries.
We consider the existence in arbitrary finite dimensions d of a positive operator valued measure (POVM) comprised of d2 rank-one operators all of whose operator inner products are equal. Such a set is called a “symmetric, informationally complete” POVM (SIC–POVM) and is equivalent to a set of d2 equiangular lines in Cd. SIC–POVMs are relevant for quantum state tomography, quantum cryptography, and foundational issues in quantum mechanics. We construct SIC–POVMs in dimensions two, three, and four. We further conjecture that a particular kind of group-covariant SIC–POVM exists in arbitrary dimensions, providing numerical results up to dimension 45 to bolster this claim.
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 purpose of this project is first to survey all known simple constructions of SIC-POVMs and second to explore these solutions for additional symmetries. We focus on three main constructions: the Weyl-Heisenberg construction; constructions using Hadamard matrices; and Hoggar's construction. We then explore fiducial vectors in dimensions 7 and 19 which have additional attractive number-theoretic properties. Finally, we generate geometric configurations of the Majorana states of some SIC-POVMs in dimensions 3 and 4. We then plot the Majorana vectors of those SIC-POVMs in real dimension 3; these are then found to have special symmetry. In the process of generating the Majorana states, two potentially new SIC-POVMs in dimension three have been found.
This paper is an announcement of our result and the detailed version will be submitted to somewhere.(See [2].)POVMs (positive operator valued measure) on quantum systems are considered as a measurement in quantum physics.If POVMs have a good condition, then we can determine a state by results of a measurement.In this case, the POVM is called informationally complete and the process is called quantum state tomography.First, we introduce SIC-POVMs (symmetric informationally complete POVM).SIC-POVMs is generated by vectors in Cn whose the absolute values of inner products of each vectors are same.Zauner conjectured that there exist such n2 vectors in Cn for any n .But the existence is only proved when n≤15 and n=19 , 24, 35,48[5, 8].Next, we introduce conditional SIC-POVMs.A state of a quantum system is a density matrix which has several parameters.When a few parameters are known, then SIC-POVM is not the best measurement to determine the state.Hence we need another POVM and it is a conditional SIC-POVM.Conditional SIC-POVMs are also generated by vectors in Cn .But the existence of conditional SIC-POVMs depends on the system.We will discuss the details in Sect. 4. 2 Preliminaries Definition 2.1 ρ∈Mn(C) is called a density matrix (or state) if ρ≥0 and ′b(ρ)=1.For any density matrix ρ , we can define a state ρ^ by ρ^(X)=h(ρX) .Conversely, any state is written by the above form.Therefore, there exists a one-t (≻ one correspondence between density matrices and states.Definition 2.2 A set of positive operators {Pi}i=1k⊂Mn(C) is called a positive operator valued measure (POVM) if Pi≥0(1≤i≤k) and ∑i=1kP=I.If P1 is a projection, then {Pi}i=1k is called a projection valued measure (PVM) .
The notion of Symmetric Informationally Complete Positive Operator-Valued Measures (SIC-POVMs) arose in physics as a kind of optimal measurement basis for quantum systems. However the question of their existence is equivalent to that of the existence of a maximal set of \emph{complex equiangular lines}. That is to say, given a complex Hilbert space of dimension d, what is the maximal number of (complex) lines one can find which all make a common (real) angle with one another, in the sense that the inner products between unit vectors spanning those lines all have a common absolute value? A maximal set would consist of d2 lines all with a common angle of arccosd+11. The same question has been posed in the real case and some partial answers are known. But at the time of writing no unifying theoretical result has been found in the real or the complex case: some sporadic low-dimensional numerical constructions have been converted into algebraic solutions but beyond this very little is known. It is conjectured that such structures always arise as orbits of certain fiducial vectors under the action of the Weyl (or generalised Pauli) group. In this paper we point out some new construction methods in the lowest dimensions (d=2 and d=3). We should mention that the SIC-POVMs so constructed are all unitarily equivalent to previously known SIC-POVMs.
Abstract A symmetric informationally complete positive-operator-valued measure (SIC-POVM) is a special type of generalized quantum measurements that possesses a high degree of symmetry. It plays an important role in quantum tomography, quantum cryptography and foundations of quantum mechanics. The most important and difficult problem about SIC-POVM is to show its existence in every dimension n, which is suggested by many numerical evidences. In this paper, this existence problem of SIC-POVMs is investigated by considering their overlap phases. Specifically, the symmetric requirements of SIC-POVMs are encoded into the overlap phases between the fiducial project and the n 2 displacement operators. In this way, an equivalent condition of the existence of Weyl-Heisenberg group covariant SIC-POVMs is obtained in terms of these phases, so that a set of complex equations is established, which enables one to study the existence problem in a more explicit and direct fashion. In particular, we show that all the numbers appearing in the overlap phases have to be algebraic numbers. The significance of Zauner’s conjecture in our equivalent condition is also discussed, and therefore the SIC-POVMs in dimension four is constructed explicitly. Actually, our system of equations always admits a solution, the real difficulty lies in how to ensure these complex number solutions are of norm one, and therefore indeed phases. Our result shows that constructing SIC-POVMs from the overlap phases is possible, and symmetries can be applied directly to reduce the number of equations, so that it provide new insights into the difficult problem of SIC-POVMs, especially about the algebraic properties of these overlap phases.