Conditional SIC-POVMs

In this paper, we examine a generalization of the symmetric informationally complete POVMs. SIC-POVMs are the optimal measurements for full quantum tomography, but if some parameters of the density matrix are known, then the optimal SIC POVM should be orthogonal to a subspace. This gives the concept of the conditional SIC-POVM. The existence is not known in general, but we give a result in the special case when the diagonal is known of the density matrix. © 1963-2012 IEEE

Examples of conditional SIC-POVMs

The state of a quantum system is a density matrix with several parameters. The concern herein is how to recover the parameters. Several possibilities exist for the optimal recovery method, and we consider some special cases. We assume that a few parameters are known and that the others are to be recovered. The optimal positive-operator-valued measure (POVM) for recovering unknown parameters with an additional condition is called a conditional symmetric informationally complete POVM (SIC-POVM). In this paper, we study the existence or nonexistence of conditional SIC-POVMs. We provide a necessary condition for existence and some examples.ArticleQUANTUM INFORMATION PROCESSING. 14(10):3983-3999 (2015)journal articl

Examples of conditional SIC-POVMs

The state of a quantum system is a density matrix with several parameters. The concern herein is how to recover the parameters. Several possibilities exist for the optimal recovery method, and we consider some special cases. We assume that a few parameters are known and that the others are to be recovered. The optimal positive-operator-valued measure (POVM) for recovering unknown parameters with an additional condition is called a conditional symmetric informationally complete POVM (SIC-POVM). In this paper, we study the existence or nonexistence of conditional SIC-POVMs. We provide a necessary condition for existence and some examples.ArticleQUANTUM INFORMATION PROCESSING. 14(10):3983-3999 (2015)journal articl

Optimal Single Qubit Tomography: Realization of Locally Optimal Measurements on a Quantum Computer

Quantum bits, or qubits, are the fundamental building blocks of present quantum computers. Hence, it is important to be able to characterize the state of a qubit as accurately as possible. By evaluating the qubit characterization problem from the viewpoint of quantum metrology, we are able to find optimal measurements under the assumption of good prior knowledge. We implement these measurements on a superconducting quantum computer. Our experiment produces sufficiently low error to allow the saturation of the theoretical limits, given by the Nagaoka--Hayashi bound. We also present simulations of adaptive measurement schemes utilizing the proposed method. The results of the simulations show the robustness of the method in characterizing arbitrary qubit states with different amounts of prior knowledge

The SIC Question: History and State of Play

Recent years have seen significant advances in the study of symmetric informationally complete (SIC) quantum measurements, also known as maximal sets of complex equiangular lines. Previously, the published record contained solutions up to dimension 67, and was with high confidence complete up through dimension 50. Computer calculations have now furnished solutions in all dimensions up to 151, and in several cases beyond that, as large as dimension 844. These new solutions exhibit an additional type of symmetry beyond the basic definition of a SIC, and so verify a conjecture of Zauner in many new cases. The solutions in dimensions 68 through 121 were obtained by Andrew Scott, and his catalogue of distinct solutions is, with high confidence, complete up to dimension 90. Additional results in dimensions 122 through 151 were calculated by the authors using Scott's code. We recap the history of the problem, outline how the numerical searches were done, and pose some conjectures on how the search technique could be improved. In order to facilitate communication across disciplinary boundaries, we also present a comprehensive bibliography of SIC research.Comment: 16 pages, 1 figure, many references; v3: updating bibliography, dimension eight hundred forty fou

Construction of ASIC-POVMs via 2-to-1 PN functions and the Li bound

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. [33] introduced an approximate version of SIC-POVMs called approximately symmetric informationally complete POVMs (ASIC-POVMs). In this paper, we present two new constructions of ASIC-POVMs in dimensions $q$ and $q+1$ by $2$-to-$1$ PN functions and the Li bound, respectively, where $q$ is a prime power. In the first construction, we show that all $2$-to-$1$ PN functions can be used for constructing ASIC-POVMs of dimension $q$, which not only generalizes the construction in [33, Theorem 5], but also generalizes the general construction in [11, Theorem III.3]. We show that some $2$-to-$1$ PN functions that do not satisfy the condition in [11, Theorem III.3] can be also utilized for constructing ASIC-POVMs of dimension $q$. We also give a class of biangular frames related to our ASIC-POVMs. The second construction gives a new method to obtain ASIC-POVMs in dimension $q+1$ via a multiplicative character sum estimate called the Li bound