Mutually Unbiased Bases and Finite Projective Planes
Metod Saniga, Michel Planat, Haret Rosu
证明了在d维Hilbert空间中，如果d不等于素数的幂，那么是否存在一组d+1个互无偏基，这个问题与射影平面的存在性问题密切相关。
It is conjectured that the question of the existence of projective planes whose order is not a power of prime is intimately linked with the problem whether there exists a set of d+1 mutually unbiased bases in a d-dimensional Hilbert space if d differs from a power of prime.
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Circulant matrices, gauss sums and mutually unbiased I. The prime number case
本文研究了素数维$d$中的互无偏基问题。已知精确地提供$d+1$相互无偏的碱基。我们使用一类循环$d \times d$矩阵重新讨论这个问题。建设性的证明一组$d+1$相互无偏基地如下，连同一组性质的高斯和，双幺模序列。
Mutually unbiased bases are an important tool in many applications of quantum information theory. We present a new algorithm for finding the mutually unbiased bases for two-qubit systems. We derive a system of four equations in the Galois field GF(4) and show that the solutions of this system are sufficient for obtaining the most general set of mutually unbiased bases. Further, our algorithm is applied to an example and we show that there are three possible solutions of the system of four equations, each solution leading to a different set of mutually unbiased bases.
A new family of positive, trace-preserving maps is introduced. It is defined using the mutually unbiased measurements, which generalize the notion of mutual unbiasedness of orthonormal bases. This family allows one to define entanglement witnesses whose indecomposability depends on the characteristics of the associated measurement operators.
It has been conjectured that a complete set of mutually unbiased bases in a space of dimension d exists if and only if there is an affine plane of order d. We introduce affine constellations and compare their existence properties with those of mutually unbiased constellations, mostly in dimension six. The observed discrepancies make a deeper relation between the two existence problems unlikely.
Complete sets of mutually unbiased bases are only known to exist in prime-power dimensions. We will describe a few approaches to the problem proving the (non)-existence of four mutually unbiased bases in dimension 6. These will include the notions of Grassmannian distance, quadratic matrix programming, semidefinite relaxations to polynomial programming, as well as various tools from algebraic geometry.
We show how to transform the problem of finding d+1 mutually unbiased bases in the d-dimensional Hilbert space into the one of finding d(d+1) vectors in the N-dimensional Hilbert space with N=d**2. The transformation formulas admit a solution when d is a prime number.
Suppose that for some unit vectors $b_1,\ldots b_n$ in $\mathbb C^d$ we have that for any $j\neq k$ $b_j$ is either orthogonal to $b_k$ or $|\langle b_j,b_k\rangle|^2 = 1/d$ (i.e. $b_j$ and $b_k$ are unbiased). We prove that if $n=d(d+1)$, then these vectors necessarily form a complete system of mutually unbiased bases, that is, they can be arranged into $d+1$ orthonormal bases, all being mutually unbiased with respect to each other.
Two equivalent ways of looking for mutually unbiased bases are discussed in this note. The passage from the search for d+1 mutually unbiased bases in C(d) to the search for d(d+1) vectors in C(d*d) satisfying constraint relations is clarified. Symmetric informationally complete positive-operator-valued measures are briefly discussed in a similar vein.
We introduce a new generalization of the Pauli channels using the mutually unbiased measurement operators. The resulting channels are bistochastic but their eigenvectors are not unitary. We analyze the channel properties, such as complete positivity, entanglement breaking, and multiplicativity of maximal output purity. We illustrate our results with the maps constructed from the Gell-Mann matrices and the Heisenberg-Weyl observables.
We propose a new method of finding the mutually unbiased bases for three qubits. The key element is the construction of the table of striation-generating curves in the discrete phase space. We derive a system of equations in the Galois field GF(8) and show that the solutions of these equations are sufficient for the construction of the general sets of complete mutually unbiased bases. A few examples are presented in order to show how our algorithm works in the cases: striation table with three, two axes, and one and no axis in the discrete phase space.
A new construction of complex Hadamard matrices of composite order d=pq, with primes p,q, is presented which is based on pairs of mutually unbiased bases containing only product states. For product dimensions d < 100, we illustrate the method by deriving many previously unknown complex Hadamard matrices. We obtain at least 12 new isolated matrices of Butson type, with orders ranging from 9 to 91.
We study the connection between mutually unbiased bases and mutually orthogonal extraordinary supersquares, a wider class of squares which does not contain only the Latin squares. We show that there are four types of complete sets of mutually orthogonal extraordinary supersquares for the dimension $d=8$. We introduce the concept of physical striation and show that this is equivalent to the extraordinary supersquare. The general algorithm for obtaining the mutually unbiased bases and the physical striations is constructed and it is shown that the complete set of mutually unbiased physical stria…
Feynman's Integral is About Mutually Unbiased Bases
费曼积分可以被看作是一种尝试，在某些情况下，相互无偏基的量子信息论分离性与测量过程的因果接近性联系起来。
All complex Hadamard matrices in dimensions two to five are known. We use this fact to derive all inequivalent sets of mutually unbiased (MU) bases in low dimensions. We find a three-parameter family of triples of MU bases in dimension four and two inequivalent classes of MU triples in dimension five. We confirm that the complete sets of (d+1) MU bases are unique (up to equivalence) in dimensions below six, using only elementary arguments for d less than five.
A complex Hilbert space of dimension six supports at least three but not more than seven mutually unbiased bases. Two computer-aided analytical methods to tighten these bounds are reviewed, based on a discretization of parameter space and on Grobner bases. A third algorithmic approach is presented: the non-existence of more than three mutually unbiased bases in composite dimensions can be decided by a global optimization method known as semidefinite programming. The method is used to confirm that the spectral matrix cannot be part of a complete set of seven mutually unbiased bases in dimension…
Angular Momentum and Mutually Unbiased Bases
群SU（2）的李代数是由两个变形振子代数构造而成的，其中变形参数是单位根。这导致不寻常的量化方案，{J2，Ur}方案，熟悉的{J2，Jz}量化方案的替代方案，对应于Casimir算子J2和Cartan算子Jz的公共本征向量。建立了交换算子完备集{J2，Ur}的本征向量与常角动量空间中的互无偏基之间的联系。
In this work, the concept of mutually unbiased frames is introduced as the most general notion of unbiasedness for sets composed by linearly independent and normalized vectors. It encompasses the already existing notions of unbiasedness for orthonormal bases, regular simplices, equiangular tight frames, positive operator valued measure, and also includes symmetric informationally complete quantum measurements. After introducing the tool, its power is shown by finding the following results about the last mentioned class of constellations: (i) real fiducial states do not exist in any even dimens…
A SU(2) recipe for mutually unbiased bases
从一个作用在常角动量空间j上的矩阵出发，利用循环群C（2j +1）的不可约特征标，给出了一个生成（2j +1）**e维互无偏基完备集的简单方法，其中2j + 1是素数，e是正整数.作为两个待定结果，该矩阵用于SU（2）的极分解和FFZ代数的推导。
Equivalence relations between conical 2-designs and mutually unbiased generalized equiangular tight frames
本文研究了锥形2设计与互正交广义等角紧框架之间的等价关系，揭示了SIC POVMs和MUBs在任意秩算子下的推广形式及其应用意义。
A Survey of Finite Algebraic Geometrical Structures Underlying Mutually Unbiased Quantum Measurements
讨论了在任意有限维的希尔伯特空间中构造互无偏基的基本方法，并概述了它们之间的一种新的联系。它表明，这些方法采用了广泛的重要的数学概念，如，例如，傅立叶变换，伽罗瓦域和环，有限和相关的射影几何，纠缠，仅举几例。文中还介绍了该理论在量子信息任务中的一些应用。
Block circulant matrices with circulant blocks, weil sums and mutually unbiased bases, II. The prime power case
在我们以前的文件\cite{co1}中，我们已经表明，循环矩阵的理论允许恢复的结果，存在$p+1$相互无偏基的尺寸$p$，$p$是一个任意的素数。如果$\forall b\in \mathcal B, \forall b' \in \mathcal B'$之一具有该$$| b\cdot b'| = \frac{1}{\sqrt d}$$（$\mathbb C^d$中的$b\cdot b'$厄米标量积），则称$\mathbb C^d$的两个正交基$\mathcal B, \mathcal B'$相互无偏。本文证明了具有循环块的块循环矩阵的理论可以非常简单地证明已知的结果：如果$d=p^n$（$p$是素数，$n$是任意整数），则在$\mathbb C^d$中存在$d+1$互无偏基.我们的结果在很大程度上依赖于Klimov，Munoz，Romero \cite{klimuro}的一个想法。作为一个子产品，我们恢复二次韦伊和$p\ge 3$的性质，这推广了这样一个事实，即在素数的情况下，二次高斯和性质遵循我们的结果。
Finding four six-dimensional mutually unbiased bases (MUBs) containing the identity matrix is a long-standing open problem in quantum information. We show that if they exist, then the $H_2$-reducible matrix in the four MUBs has exactly nine $2\times2$ Hadamard submatrices. We apply our result to exclude from the four MUBs some known CHMs, such as symmetric $H_2$-reducible matrix, the Hermitian matrix, Dita family, Bjorck's circulant matrix, and Szollosi family. Our results represent the latest progress on the existence of six-dimensional MUBs.
Discrete phase-space structure of $n$-qubit mutually unbiased bases
我们计算了一个$n$量子比特系统的相空间结构。我们用有限域$\Gal{2^n}$代替连续相空间的实数域，并研究与无偏性概念相容的几何结构。这些由仅在原点相交并满足某些附加性质的离散曲线束组成。我们提供了一个简单的分类，这样的曲线和研究详细的四维和八维的情况下，还分析了局部变换的效果。通过这种方式，我们提供了一个全面的相空间方法来构建$n$量子位的相互无偏基。
The Mutually Unbiased Bases Revisited
互无偏基的研究在量子信息理论中得到了蓬勃的发展，并提出了一些挑战。如果$\forall b\in B, b'\in B'$，标量积$b\cdot b'$的模为$d^{-1/2}$，则称$\mathbb C^d, B {and} B'$中的两个正交基相互无偏。特别是这个属性已被引入，以便允许任何状态$ψ\in \mathbb C^d$的测量驱动的量子演化过程的优化时，在相互无偏基地$B\_{j} {of} \mathbb C^d$测量。当$d$不是素数的幂时，如何求最大互无偏基数是一个尚未解决的问题。\noindent在这篇文章中，我们重新讨论了在任何维度$d$中寻找互无偏基（MUB）的问题。该方法是非常初级的，使用简单的酉矩阵介绍了Schwinger在1960年，连同他们的对角化。基于单位根的$d$的范德蒙矩阵起着重要作用。这使我们能够证明在任何维度上存在一组3个MUB，给出$d$偶数或奇数存在3个以上MUB的条件，并恢复对于$d$为素数的$d+1$ MUB存在的已知结果。此外，这些MUB的构造非常明确。作为一个副产品，我们恢复高斯和的结果，在数论中已知的，但显然没有以前来自MUB属性。
An analytical method for getting new complex Hadamard matrices by using mutually unbiased bases and a nonlinear doubling formula is provided. The method is illustrated with the n=4 case that leads to a rich family of eight-dimensional Hadamard matrices that depend on five arbitrary parameters whose modulus is equal to unity.
Orthogonal Polynomials on the Unit Circle, Mutually Unbiased Bases, and Balanced States
本研究采用单位圆上的正交多项式构造方法，探究量子态构建中的互无偏基与平衡态现象，证明该方法无法生成互无偏基，给出平衡态相关示例，并探讨了无穷维拓展方向。
Based on maximally entangled states, we explore the constructions of mutually unbiased bases in bipartite quantum systems. We present a new way to construct mutually unbiased bases by difference matrices in the theory of combinatorial designs. In particular, we establish $q$ mutually unbiased bases with $q-1$ maximally entangled bases and one product basis in $\mathbb{C}^q\otimes \mathbb{C}^q$ for arbitrary prime power $q$. In addition, we construct maximally entangled bases for dimension of composite numbers of non-prime power, such as five maximally entangled bases in $\mathbb{C}^{12}\otimes…
A set of mutually unbiased bases (MUBs) is said to be unextendible if there does not exist another basis that is unbiased with respect to the given set. Here, we prove the existence of smaller sets of MUBs in prime-squared dimensions ($d=p^{2}$) that cannot be extended to a complete set using the generalized Pauli operators. We further observe an interesting connection between the existence of unextendible sets and the tightness of entropic uncertainty relations (EURs) in these dimensions. In particular, we show that our construction of unextendible sets of MUBs naturally leads to sets of $p+1…
Hjelmslev Geometry of Mutually Unbiased Bases
本文证明了q维Hilbert空间Hq（q = p^r，p为素数，r为正整数）的互无偏基完备集的基本组合性质，可由定义在特征为p ^2秩为r的伽罗瓦环上的射影Hjelmslev平面上的真二次曲线上的点的构型定性地模拟. Hq的基的q个向量对应于（所谓的）相邻类的q个点，并且q+1个MUB对应于圆锥曲线上（成对不相交的）相邻类的总数。
Qudits of composite dimension, mutually unbiased bases and projective ring geometry
附加到维度为$d$的复合量子点的$d^2$泡利算子可以映射到辛模$\mathcal{Z}_d^{2}$（$\mathcal{Z}_d$模环）的向量。因此，垂直向量对应于交换算子，一个自由循环子模对应于一个最大交换集，并且不相交的集合对应于相互无偏基。对于维数$d=6,~10,~15,~12$，和18，我们分别发现了环$\mathcal{Z}_{6}$，$\mathcal{Z}_{10}$，$\mathcal{Z}_{15}$，$\mathcal{Z}_6 \times \mathbf{F}_4$和$\mathcal{Z}_6 \times \mathcal{Z}_3$上的投影线的精细结构和极大交换集之间的关联.
We propose a group-theoretical approach to the generalized oscillator algebra Ak recently investigated in J. Phys. A: Math. Theor. 43 (2010) 115303. The case k > or 0 corresponds to the noncompact group SU(1,1) (as for the harmonic oscillator and the Poeschl-Teller systems) while the case k < 0 is described by the compact group SU(2) (as for the Morse system). We construct the phase operators and the corresponding temporally stable phase eigenstates for Ak in this group-theoretical context. The SU(2) case is exploited for deriving families of mutually unbiased bases used in quantum information…
The present chapter [submitted for publication in "Fourier Transforms, Theory and Applications", G. Nikolic (Ed.), InTech (Open Access Publisher), Vienna, 2011] is concerned with the introduction and study of a quadratic discrete Fourier transform. This Fourier transform can be considered as a two-parameter extension, with a quadratic term, of the usual discrete Fourier transform. In the case where the two parameters are taken to be equal to zero, the quadratic discrete Fourier transform is nothing but the usual discrete Fourier transform. The quantum quadratic discrete Fourier transform plays…
The connection between maximal sets of mutually unbiased bases (MUBs) in a prime-power dimensional Hilbert space and finite phase-space geometries is well known. In this article we classify MUBs according to their degree of covariance with respect to the natural symmetries of a finite phase-space, which are the group of its affine symplectic transformations. We prove that there exist maximal sets of MUBs that are covariant with respect to the full group only in odd prime-power dimensional spaces, and in this case their equivalence class is actually unique. Despite this limitation, we show that…
We investigate the number of real entries of an $n\times n$ complex Hadamard matrix (CHM). We analytically derive the numbers when $n=2,3,4,6$. In particular, the number can be any one of $0-22,24,25,26,30$ for $n=6$. We apply our result to the existence of four mutually unbiased bases (MUBs) in dimension six, which is a long-standing open problem in quantum physics and information. We show that if four MUBs containing the identity matrix exists then the real entries in any one of the remaining three matrices does not exceed $22$.
Mutually unbiased bases: common states, completion, and extension obstructions
该研究探讨相互无偏基的公共态、完备性与扩展障碍，在特定维度下给出完备化方法、不可扩展性证明及相关代数性质，未确定相互无偏基的全局最大数量。
Intrinsic volumes of the quantum state space and mutually unbiased bases
本文针对有限能级量子系统态空间推导两类内蕴体积的显式公式，验证其无法通过体积比较排除完备互无偏基对应的多面体内接可能，但可用于排除部分单位向量构型的存在性。
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