Informationally complete measurements on a quantum system allow to estimate the expectation value of any arbitrary operator by just averaging functions of the experimental outcomes. We show that such kind of measurements can be achieved through positive-operator valued measures (POVM's) related to unitary irreducible representations of a group on the Hilbert space of the system. With the help of frame theory we provide a constructive way to evaluate the data-processing function for arbitrary operators.
We propose entanglement criteria for multipartite systems via symmetric informationally complete (SIC) measurement and general symmetric informationally complete (GSIC) measurement. We apply these criteria to detect entanglement of multipartite states, such as the convex of Bell states, entangled states mixed with white noise. It is shown that these criteria are stronger than some existing ones.
A broad class of informationally complete symmetric measurements is introduced. It can be understood as a common generalization of symmetric, informationally complete POVMs and mutually unbiased bases. Additionally, it provides a natural way to define two new families of mutually unbiased symmetric measurement operators in any finite dimension. We show a general method of their construction, together with an example of an optimal measurement. Finally, we analyze the properties of symmetric measurements and provide applications in entropic relations and entanglement detection.
We study entanglement witness and present a construction of entanglement witnesses in terms of the symmetric informationally complete measurements (SIC-POVM). The capability of our witness is shown by some examples and it can be found this witness detects more entanglement than previous separability method given also by SIC-POVM. What's more, comparing with the dimension dependence of SIC-POVM, we also give the entanglement witnesses can be always constructed from general symmetric informationally complete measurements (GSIC-POVM).
Wigner functions provide a way to do quantum physics using quasiprobabilities, that is, "probability" distributions that can go negative. Informationally complete POVMs, a much younger subject than phase space formulations of quantum mechanics, are less familiar but provide wholly probabilistic representations of quantum theory. In this paper, we show that the Born Rule links these two classes of structure and discuss the art of interconverting between them. In particular, we demonstrate that the operator bases corresponding to minimal discrete Wigner functions (Wigner bases) are orthogonaliza…
The Poincaré half-plane for informationally complete POVMs
已经表明，可以使用作用于适当的基准状态的多粒子泡利群来构建d维中的（最小不对称）信息完备POVM类[M。Planat和Z.盖迪克河Soc. open sci. 4，170387（2017）]。后者的状态也可以从庞加莱上半平面模型H开始导出。为了做到这一点，将模群的指数为d的全等（或非全等）子群转化为置换门的群，置换门的一些本征态是所寻找的基准。一些IC-POVM的结构被发现是密切相关的Kochen-Specker定理。
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 …
Mutually unbiased bases (MUBs) and symmetric informationally complete projectors (SICs) are crucial to many conceptual and practical aspects of quantum theory. Here, we develop their role in quantum nonlocality by: i) introducing families of Bell inequalities that are maximally violated by $d$-dimensional MUBs and SICs respectively, ii) proving device-independent certification of natural operational notions of MUBs and SICs, and iii) using MUBs and SICs to develop optimal-rate and nearly optimal-rate protocols for device independent quantum key distribution and device-independent quantum rando…
Although quantum entanglement has already been verified experimentally and applied in quantum computing, quantum sensing and quantum networks, most of the existing measures cannot characterize the entanglement faithfully. In this work, by exploiting the Schmidt decomposition of a bipartite state $|ψ\rangle_{AB}$, we first establish a one-to-one correspondence relation between the characteristic polynomial of the reduced state $ρ_A$ and the polynomials its trace. Then we introduce a family of entanglement measures which are given by the complete eigenvalues of the reduced density matrices of th…
Symmetric informationally complete measurements (SICs) are elegant, celebrated and broadly useful discrete structures in Hilbert space. We introduce a more sophisticated discrete structure compounded by several SICs. A SIC-compound is defined to be a collection of $d^3$ vectors in $d$-dimensional Hilbert space that can be partitioned in two different ways: into $d$ SICs and into $d^2$ orthonormal bases. While a priori their existence may appear unlikely when $d>2$, we surprisingly answer it in the positive through an explicit construction for $d=4$. Remarkably this SIC-compound admits a close …
Many prominent quantum computing algorithms with applications in fields such as chemistry and materials science require a large number of measurements, which represents an important roadblock for future real-world use cases. We introduce a novel approach to tackle this problem through an adaptive measurement scheme. We present an algorithm that optimizes informationally complete positive operator-valued measurements (POVMs) on the fly in order to minimize the statistical fluctuations in the estimation of relevant cost functions. We show its advantage by improving the efficiency of the variatio…
In the present work, we suggest an approach for describing dynamics of finite-dimensional quantum systems in terms of pseudostochastic maps acting on probability distributions, which are obtained via minimal informationally complete quantum measurements. The suggested method for probability representation of quantum dynamics preserves the tensor product structure, which makes it favourable for the analysis of multi-qubit systems. A key advantage of the suggested approach is that minimal informationally complete positive operator-valued measures (MIC-POVMs) are easier to construct in comparison…
Quamtumness, Generalized Spherical 2-Design and Symmetric Informationally Complete POVM
C. A. Fuchs和M. Sasaki在\cite{Quantumness}中定义了一组量子态的量子性，这与量子态通过经典信道传输时的保真度损失密切相关。在\cite{Fuchs}，富克斯表明，在$d$维希尔伯特空间，最小量子是$\frac{2}{d+1}$，这可以实现的所有射线的空间。他留下了一个悬而未决的问题，即少于$d^2$的状态是否可以达到这个界限。最近，在一个不同的背景下，A. Scott在\cite{GenSphet}中引入了广义$t$设计的概念，它是球形$t$设计的自然推广.在本文中，我们证明了当且仅当态形成广义2-设计时，量子度的下界才能达到。作为一个推论，我们表明，只有当状态的数量大于或等于$d^2$，回答开放的问题，才能达到这个界限。此外，我们还证明了这种集成的最小集是对称信息完全POVM（SIC-POVM）。这导致了SIC-POVM和实现最小量子性的系综的最小集合之间的等价关系。
State of a $d$-dimensional quantum system can only be inferred by performing an informationally complete measurement with $m\geqslant d^2$ outcomes. However, an experimentally accessible measurement can be informationally incomplete. Here we show that a single informationally incomplete measuring apparatus is still able to provide all the information about the quantum system if applied several times in a row. We derive a necessary and sufficient condition for such a measuring apparatus and give illustrative examples for qubits, qutrits, general $d$-level systems, and composite systems of $n$ q…
The article establishes a framework for dynamic generation of informationally complete POVMs in quantum state tomography. Assuming that the evolution of a quantum system is given by a dynamical map in the Kraus representation, one can switch to the Heisenberg picture and define the measurements in the time domain. Consequently, starting with an incomplete set of positive operators, one can obtain sufficient information for quantum state reconstruction by multiple measurements. The framework has been demonstrated on qubits and qutrits. For some types of dynamical maps, it suffices to initially …
Informationally Complete Distributed Metrology Without a Shared Reference Frame
University of Science and Technology of China（中国科学技术大学）；Anhui Province Key Laboratory of Quantum Network（安徽省量子网络重点实验室）；CAS Center For Excellence in Quantum Information and Quantum Physics（中国科学院量子信息与量子物理学前沿卓越创新中心）；Hefei National Laboratory（合肥国家实验室）
Detecting $k$-nonstretchability via a class of informationally complete symmetric measurements
Hebei GEO University（河北地质大学）；Intelligent Sensor Network Engineering Research Center of Hebei Province, Hebei GEO University（河北省智能传感网络工程研究中心）；Hengshui University（衡水学院）
