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arXiv:2310.06418

Construction of $\varepsilon_{d}$-ASIC-POVMs via $2$-to-$1$ PN functions and the Li bound

Cao, Meng, Xiantao Deng

Published Oct 10, 2023• 0 citations
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Abstract

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 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 $\varepsilon_{d}$-ASIC-POVMs in dimension $d=q$ and a class of $\varepsilon_{d}$-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 $\varepsilon_{q}$-ASIC-POVMs. We show that the set of vectors corresponding to the $\varepsilon_{q}$-ASIC-POVM forms a biangular frame. The construction of $\varepsilon_{q+1}$-ASIC-POVMs is based on a multiplicative character sum estimate called the Li bound. We show that the set of vectors corresponding to the $\varepsilon_{q+1}$-ASIC-POVM forms an asymptotically optimal codebook. We characterize "how close" the $\varepsilon_{q}$-ASIC-POVMs (resp. $\varepsilon_{q+1}$-ASIC-POVMs) are from being SIC-POVMs of dimension $q$ (resp. dimension $q+1$). Finally, we explain the significance of constructing $\varepsilon_{d}$-ASIC-POVMs.

DOI
10.48550/arxiv.2310.06418

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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
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.
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.
We construct the set of all general (i.e. not necessarily rank 1) symmetric informationally complete (SIC) positive operator valued measures (POVMs). In particular, we show that any orthonormal basis of a real vector space of dimension corresponds to some general SIC POVM and vice versa. Our constructed set of all general SIC POVMs contains weak SIC POVMs for which each POVM element can be made arbitrarily close to a multiple times the identity. On the other hand, it remains open if for all finite dimensions our constructed family contains a rank 1 SIC POVM.
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.
Abstract The existence of d2d^2d2 pairwise equiangular complex lines [equivalently, a symmetric informationally complete positive operator-valued measure (SIC-POVM)] in ddd-dimensional Hilbert space is known only for finitely many dimensions ddd. We prove that, if there exists a set of real units in a certain ray class field (depending on ddd) satisfying certain algebraic properties, a SIC-POVM exists, when ddd 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=0s=0s=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.