Abstract In quantum information theory, symmetric informationally complete positive operator-valued measures (SIC-POVMs) are related to quantum state tomography, quantum cryptography and foundational studies. It is difficult to construct SIC-POVMs and remains unknown whether SIC-POVMs exist for infinitely dimension. Therefore, researchers have proposed the solution to construct approximately symmetric informationally complete positive operator-valued measures (ASIC-POVMs). This paper proposes a construction of ASIC-POVMs using character sums over Galois rings. The dimension of this ASIC-POVMs is q-1, where q is a prime power. Also, our constructions include partial results about ASIC-POVMs constructed by X. Cao et al (X. Cao, J. Mi and S. Xu: Two constructions of approximately symmetric informationally complete positive operator-valued measures. J. Math. Phys., 58(6), 062201 (2017)). Mathematics Subject Classification (2020) 81P15 · 81P45
A symmetric informationally complete positive operator-valued measure (SIC-POVM) is a POVM in [Formula: see text] consisting of [Formula: see text] positive operators of rank one such that all of whose Hermite inner products are equal. SIC-POVMs are important in quantum information theory, which have many applications in quantum state tomography, quantum cryptography and basic research in quantum mechanics. However, it is very difficult to construct SIC-POVMs. Therefore, many scholars have focused on approximately symmetric informationally complete positive operator-valued measures (ASIC-POVMs) for which the Hermite inner products are close to equal. In this paper, two new constructions of ASIC-POVMs are provided by using character sums and some special functions over finite fields.
Symmetric informationally complete positive operator-valued measures (SIC-POVMs) have many applications in quantum information. However, it is not easy to construct SIC-POVMs and there are only a few known classes of them, and we do not even know whether there exists an infinite class of them, thus constructing approximately symmetric informationally complete positive operator-valued measures (ASIC-POVMs) has its own meaning. In this paper, we use character sums over finite fields to present two constructions of ASIC-POVMs. We show that there are some classes of infinite families of ASIC-POVMs by using some special functions over finite fields.
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
d2 subnormalized projectors with equal pairwise fidelity. However, it is difficult to construct SIC-POVMs, and it is not even clear whether there exists an infinite family of SIC-POVMs. To realize some possible applications in quantum information processing, Klappenecker et al. [37] introduced an approximate version of SIC-POVMs called approximately symmetric informationally complete POVMs (ASIC-POVMs). In this paper, we construct a class of
εd-ASIC-POVMs in dimension
d=q and a class of
εd-ASIC-POVMs in dimension
d=q+1, respectively, where
q is a prime power. We prove that all
2-to-
1 perfect nonlinear (PN) functions can be used for constructing
εq-ASIC-POVMs. We show that the set of vectors corresponding to the
εq-ASIC-POVM forms a biangular frame. The construction of
εq+1-ASIC-POVMs is based on a multiplicative character sum estimate called the Li bound. We show that the set of vectors corresponding to the
εq+1-ASIC-POVM forms an asymptotically optimal codebook. We characterize "how close" the
εq-ASIC-POVMs (resp.
εq+1-ASIC-POVMs) are from being SIC-POVMs of dimension
q (resp. dimension
q+1). Finally, we explain the significance of constructing
εd-ASIC-POVMs.
A general symmetric-informationally-complete (GSIC)--positive-operator-valued measure (POVM) is known to provide an optimal quantum state tomography among minimal IC POVMs with a fixed average purity. In this paper we provide a general construction of a GSIC POVM by means of a complete orthogonal basis (COB), also interpreted as a normal quasiprobability representation. A spectral property of a COB is shown to play a key role in the construction of SIC POVMs and also for the bound of the mean-square error of the state tomography. In particular, a necessary and sufficient condition to construct a SIC POVM for any
d is constructively given by the power of traces of a COB. We give three simple constructions of COBs from which one can systematically obtain GSIC POVMs.
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 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.