The Limited Role of Mutually Unbiased Product Bases in Dimension Six
Daniel McNulty, Stefan Weigert
我们表明，一个完整的一组七个相互无偏基地在第六维，如果它存在，不能包含一个以上的产品基础。
We show that a complete set of seven mutually unbiased bases in dimension six, if it exists, cannot contain more than one product basis.
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We investigate the interplay between mutual unbiasedness and product bases for multiple qudits of possibly different dimensions. A product state of such a system is shown to be mutually unbiased to a product basis only if each of its factors is mutually unbiased to all the states which occur in the corresponding factors of the product basis. This result implies both a tight limit on the number of mutually unbiased product bases which the system can support and a complete classification of mutually unbiased product bases for multiple qubits or qutrits. In addition, only maximally entangled stat…
All Mutually Unbiased Product Bases in Dimension Six
构造了仅由乘积状态组成的六维中的所有互无偏基。几个连续的家庭对和两个三元组的相互无偏的产品基地被发现存在，但没有四元组。穷举分类导致一个证明，一个完整的一组七个相互无偏基地，如果它存在，不能包含一个三重的相互无偏产品基地。
Orbits of mutually unbiased bases
我们表示Alltop的建设相互无偏基地的轨道下的Weyl-海森堡群在素数维度，并找到一个相关的建设在2和4维。我们重现Alltop的相互无偏基地使用阿贝尔子群的克利福德组在总理层面上，直接类比著名的建设相互无偏基地使用阿贝尔子群的Weyl-海森堡组。最后，我们证明了三个定理有关的距离和线性依赖之间的相互无偏基的不同集合。
A new proof for the existence of mutually unbiased bases
我们建立了正交酉矩阵的极大可换基与互无偏基之间的强联系。对任意有限维，得到了互无偏基存在的一个必要条件。然后构造性地证明了素数的幂维数的互无偏基的存在性。还证明了在任意维数d上，互无偏基的个数至多为d+1。对于d=2^m，给出了用泡利矩阵表示的相互无偏观测量的显式表示。
A geometrical relation between symmetric operators and mutually unbiased operators
本文从有限平面几何的角度研究了对称算子集与互无偏算子集之间的关系。这里对称算子是对称信息完备概率算子测度（SIC POM）的推广，而互无偏算子是互无偏基（MUB）的算子推广。我们还讨论了这种关系的秩-1 SIC POM和MUB的特殊情况下的含义。
Mutually Unbiased Bases for Continuous Variables
研究了N对连续变量的互无偏基的概念。对于与海森堡-外尔群相关的特定状态，要找到相互无偏的基，就归结为辛几何问题。给定一对连续变量，确定三个相互无偏的基地，而五个这样的基地表现出两对连续变量。对于N = 2，黄金比例出现在这些相互无偏的基础的定义中，这表明数论不仅在有限维环境中具有相关性。
Robustness of a quantum key distribution with two and three mutually unbiased bases
研究了各种量子密钥分发协议的鲁棒性。我们首先考虑的情况下qutrits和研究量子协议，采用两个和三个相互无偏基地。然后，我们推导出最佳窃听策略的两个相互无偏基地在第四维和推广的结果，量子密钥分发协议，使用两个相互无偏基地在任意有限维。
We show that maximal families of mutually unbiased bases are characterized in all dimensions by partitioned unitary error bases, up to a choice of a family of Hadamards. Furthermore, we give a new construction of partitioned unitary error bases, and thus maximal families of mutually unbiased bases, from a finite field, which is simpler and more direct than previous proposals. We introduce new tensor diagrammatic characterizations of maximal families of mutually unbiased bases, partitioned unitary error bases, and finite fields as algebraic structures defined over Hilbert spaces.
Entanglement detection using mutually unbiased measurements
我们研究了利用相互无偏测量的纠缠检测，并给出了一个可用于任意$d$维二体系统的量子可分性判据.我们表明，这个标准是更有效的比基于相互无偏基地的标准。对于各向同性状态，我们的准则是必要的和充分的。
Information-Disturbance Theorem for Mutually Unbiased Observables
我们得到一个新版本的信息扰动定理相互无偏观测。我们表明，Eve的信息增益不可避免地使Bob在共轭基上的结果不仅是错误的，而且是随机的。
Cyclic mutually unbiased bases, Fibonacci polynomials and Wiedemann's conjecture
本文讨论了循环互无偏基的完备集的构造问题，即循环互无偏基的构造问题.例如，在二维幂空间中，将由单一酉算子生成的互无偏基推广到求F_2上具有给定Fibonacci指标的不可约特征多项式的对称矩阵的问题。对于形式为2^（2^k）的维数，我们给出了一个解，它类似于有限场论中的一个公开的Wiedemann猜想。最后，我们讨论了互无偏基的等价性。
Entropic uncertainty relation for mutually unbiased bases
本文导出了量子系统在互无偏基上投影测度的概率的新不等式。这些不等式导致更广泛的有效性和更严格的界限熵不确定性不等式以前在文献中推导。
We systematically study the construction of mutually unbiased bases in $\mathbb{C}^{2}\bigotimes\mathbb{C}^{3}$, such that all the bases are unextendible maximally entangled ones. Necessary conditions of constructing a pair of mutually unbiased unextendible maximally entangled bases in $\mathbb{C}^{2}\bigotimes\mathbb{C}^{3}$ are derived. Explicit examples are presented.
We formulate some properties of a set of several mutually unbiased measurements. These properties are used for deriving entropic uncertainty relations. Applications of mutually unbiased measurements in entanglement detection are also revisited. First, we estimate from above the sum of the indices of coincidence for several mutually unbiased measurements. Further, we derive entropic uncertainty relations in terms of the Rényi and Tsallis entropies. Both the state-dependent and state-independent formulations are obtained. Using the two sets of local mutually unbiased measurements, a method of en…
Entropic uncertainty relations and locking: tight bounds for mutually unbiased bases
我们证明了严格的熵不确定性关系，大量的相互无偏测量。特别是，我们表明，从Maassen和Uffink为2个这样的测量结果得出的界限实际上可以是紧的，在相互无偏的基础上的sqrt{d}测量。然后我们表明，使用更多相互无偏的碱基并不总是会产生更好的锁定效果。我们证明了最佳界的可访问的信息使用sqrt{d}特定的相互无偏基地是log d/2，这是相同的，可以通过使用只有两个基地。我们的结果表明，仅仅使用相互无偏的碱基是不足以实现强锁定效果，我们需要寻找更多的属性。
We present a systematic method to introduce free parameters in sets of mutually unbiased bases. In particular, we demonstrate that any set of m real mutually unbiased bases in dimension N>2 admits the introduction of (m-1)N/2 free parameters which cannot be absorbed by a global unitary operation. As consequence, there are m=k+1 mutually unbiased bases in every dimension N=k^2 with k^3/2 free parameters, where k is even. We construct the maximal set of triplets of mutually unbiased bases for two-qubits systems and triplets, quadruplets and quintuplets of mutually unbiased bases with free parame…
We investigate the $l_{1}$ norm of coherence of quantum states in mutually unbiased bases. We find that the sum of squared $l_{1}$ norm of coherence of the mixed state single qubit is less than two. We derive the $l_{1}$ norm of coherence of three classes of $X$ states in nontrivial mutually unbiased bases for $4$-dimensional Hilbert space is equal. We proposed "autotensor of mutually unbiased basis(AMUB)" by the tensor of mutually unbiased bases, and depict the level surface of constant the sum of the $l_{1}$ norm of coherence of Bell-diagonal states in AMUB. We find the $l_{1}$ norm of coher…
Aspects of mutually unbiased bases in odd prime power dimensions
我们重新表述伍特斯-菲尔兹结构[安。物理，{ \bf 191}，363（1989）]中的一个例子，其中$N=p^r$维的复向量空间中的互无偏基的全集，其中$p$是奇素数，关于$p$阶循环群$G$的特征向量。这种形式在明确地写下$N=p^r$的相互无偏碱基时可能是有用的。
Constructing Mutually Unbiased Bases in Dimension Six
如果已知一组特定的可观测量的期望值，则可以以最佳效率重构量子点的密度矩阵。在六维中，所需的可观测量只存在于可以识别六个相互无偏的复6x6阿达玛矩阵的情况下。规定第一阿达玛矩阵，我们构造所有其他相互无偏的，使用代数计算进行计算机程序。我们重复这个计算很多次，对所有已知的复阿达玛矩阵进行采样，我们从来没有找到两个以上的相互无偏矩阵。这一结果增加了相当大的支持猜想，没有七个相互无偏基地存在于第六维。
Nonexistence of Mutually Unbiased Bases for Hilbert spaces with non-power-of-a-prime dimensions
这篇论文已经被Reinaldo O.薇安娜
Classifying all mutually unbiased bases in Rel
在不同的维度上寻找所有的互无偏基是量子信息理论和纯数学中的一个基本问题。有限维希尔伯特空间中的一般问题是开放的。在量子力学的范畴方法中，人们可以找到一些范畴的例子，它们在不同的方面表现得“像”有限维希尔伯特空间的范畴，但又有微妙的不同。一个这样的范畴是集合和关系的范畴，$\mathbf{Rel}$。我们也可以在这里用公式表示互无偏基的概念。在本说明中，我们通过将其与组合学中的标准问题联系起来，对该类别中的所有互无偏基进行分类。
Semi-quantum key distribution is traditionally based on two-level quantum systems. In this paper, an unconditional security of a semi quantum key distribution protocol based on higher-dimensional systems using various mutually unbiased bases is presented. We first consider the three dimensional case using three and four mutually unbiased bases and derive a lower bound for the key rate as a function of the quantum channel's noise. We then generalize the result to a semi-quantum key distribution protocol that employs different number of mutually unbiased bases for four-dimensional states. It is …
Weak mutually unbiased bases
考虑具有${\mathbb Z}(d)$中变量的量子系统。研究了这些体系的${\mathbb Z}(d)\times {\mathbb Z}(d)$相空间中的谱线性质。这些系统中的弱相互无偏碱基被定义为两个不同碱基中的任何两个向量的重叠等于$d^{-1/2}$或可替换地等于$d_i^{-1/2},0$之一的碱基（其中$d_i$是$d$的除$d,1$之外的除数）。它们是为${\mathbb Z}(d)\times {\mathbb Z}(d)$相空间的几何而设计的，在这个意义上，弱相互无偏基和通过原点的最大线之间存在对偶性。在素数$d$的特殊情况下，除了$1,d$之外，没有$d$的因子，并且弱互无偏基是互无偏基。
The thesis is mainly about the construction and implementation of cyclic mutually unbiased bases, dealing with different entanglement structures by discussing the related group structures. A recursive construction for Fermat number dimensions is given and related to Wiedemann's conjecture for systems with more than 2048 qubits. The second part of the thesis analyses a quantum public-key encryption scheme.
Quantum process reconstruction based on mutually unbiased basis
我们研究了一个量子过程的重建的基础上使用的相互无偏投影仪（MUB投影仪）作为输入状态的D维量子系统，与D是一个素数的幂。这种方法将使用互无偏基（MUB）的量子态层析成像的结果与量子过程的系数联系起来，并根据MUB投影器进行扩展。我们还研究了在MUB投影仪测量概率时，重建方案对随机误差的性能。
Numerical evidence for the maximum number of mutually unbiased bases in dimension six
自引入量子信息理论以来，确定六维空间中最大互无偏基个数的问题一直受到人们的关注，但至今仍没有一个确定的答案.本文摒弃传统的解析方法，采用数值方法来确定这个个数.我们在数值上最小化一组N+1个正交基的非负函数在D维，只有评估为零，如果基地是相互无偏的。结果，我们发现了强有力的证据，（正如在其他地方所证明的那样）在第六维中不超过三个相互无偏的基础。
New Construction of Mutually Unbiased Bases in Square Dimensions
我们证明了k=w+2互无偏基可以在任意维数d=s^2的正方形中构造，只要有w个相互正交的s阶拉丁方。该构造结合了设计理论对象（k，s）-网（可以从w个相互正交的s阶拉丁方构造，反之亦然）和大小为s的广义Hadamard矩阵。使用已知的相互正交拉丁方数量渐进增长的下界（基于数论筛选技术），我们得到对于所有s（除了有限多个例外），维度d=s^2中相互无偏基的数量大于s^1/14.8。此外，我们的建设提供了更多的相互正交的基础，在许多非素数幂维的建设，减少了问题的素数幂维。
Mutually unbiased measurements for high-dimensional time-bin based photonic states
在两个相互无偏的基础上进行测量的任务是许多量子信息协议的核心，也是根本利益。越来越多的实验关注于产生和控制高维光子态。一种方法是使用光子的到达时间，其可以被分成离散的时间仓。与这种状态相关的一个重要问题是在实验上实现相对于到达时间相互无偏的测量的困难。我们提出了一个简单而紧凑的计划来衡量在到达时间的基础上，是近似相互无偏的到达时间的基础。
The limitations of nice mutually unbiased bases
希尔伯特空间的相互无偏基可以通过分割酉误差基来构造。当单位误差基是一个很好的误差基时，我们考虑这种构造。我们表明，所产生的相互无偏基地的数量可以是最多一个加上最小的素数功率包含在维度中，因此，这种建设不能改善以前的方法。我们证明了这一点，通过建立一个良好的相互无偏基地和阿贝尔子群的指标组的一个不错的错误基础之间的对应关系，然后界定这样的子群的数量。这个界限也对某些称为网的组合对象的构造有影响。
Unextendible sets of Mutually Unbiased Bases (MUBs) are examined from the point of view of complementary subalgebras. We show, that the linear span of less than $d+1$ factors of $M_d \otimes M_d$ does not contain pure states, and therefore some complementary decompositions give rise to undextendable sets of MUBs. We provide some new complementary decompositions, and thus prove strong unextendibility of some set of MUBs.
We develop a new technique to construct mutually unbiased tripartite absolutely maximally entangled bases. We first explore the tripartite absolutely maximally entangled bases and mutually unbiased bases in $\mathbb{C}^{d} \otimes \mathbb{C}^{d} \otimes \mathbb{C}^{d}$ based on mutually orthogonal Latin squares. Then we generalize the approach to the case of $\mathbb{C}^{d_{1}} \otimes \mathbb{C}^{d_{2}} \otimes \mathbb{C}^{d_{1}d_{2}}$ by mutually weak orthogonal Latin squares. The concise direct constructions of mutually unbiased tripartite absolutely maximally entangled bases are remarkably…
Entanglement detection via mutually unbiased bases
我们研究互补观测量之间的相关性。特别是，我们展示了如何利用相互无偏基地（MUBs）的有效检测纠缠在任意高维，多体和连续变量量子系统。引入的纠缠标准是相对容易实现的实验，因为它们只需要几个本地测量设置。此外，我们建立了可分性问题和相互无偏基的最大数量之间的联系-在这个长期存在的开放问题中开辟了一条新的途径。
There is no generalization of known formulas for mutually unbiased bases
在具有有限个正交态的$N$的量子系统中，如果所有内积$<a_i|b_j>$具有相同的模$1/\sqrt{N}$，则两个正交基$\{a_i\}$和$\{b_j\}$被称为互无偏基。这个概念出现在几个量子信息问题中。成对相互无偏碱基的数目至多为$N+1$，并且当$N$是素数的幂时，已经发现了$N+1$这种碱基的各种构造。我们利用有限环研究了将这些构造推广到任意维数的公式族，并证明了存在一组由这些公式描述的$N+1$互无偏基当且仅当$N$是素数的幂.
Mutually unbiased bases and generalized Bell states
我们采用了一个直接的相互之间的关系，无偏和贝尔基地，以扩大后者的直接建设前者。我们详细分析了这些新的广义贝尔态的性质，表明它们构成了一个适当的工具，用于测试纠缠的二分多量子系统。
Three ways to look at mutually unbiased bases
这是一个审查问题的相互无偏基地在有限维希尔伯特空间，实和复杂。此外，几何措施的“mubness”的介绍，并适用于最近的一些计算在六个方面（部分由比约克和Grassl）。虽然这还不能解决任何问题，但一些有吸引力的结构出现了。
We study entanglement witness and present a construction of entanglement witnesses in terms of the mutually unbiased measurements (MUMs). These witnesses include the entanglement witnesses constructed from mutually unbiased bases (MUBs) as a special case. Comparing with the dimension dependence of MUBs, the witnesses can be always constructed from a complete set of $d+1$ MUMs for any dimension $d$. We show that our witness can detect entanglement better than previous separability criterion given also by MUMs. And the approach can be experimentally implemented.
We present a new quantum communication complexity protocol, the promise--Quantum Random Access Code, which allows us to introduce a new measure of unbiasedness for bases of Hilbert spaces. The proposed measure possesses a clear operational meaning and can be used to investigate whether a specific number of mutually unbiased bases exist in a given dimension by employing Semi--Definite Programming techniques.
We generalize the notion of unextendible maximally entangled basis from bipartite systems to multipartite quantum systems. It is proved that there do not exist unextendible maximally entangled bases in three-qubit systems. Moreover,two types of unextendible maximally entangled bases are constructed in tripartite quantum systems and proved to be not mutually unbiased.
In order to describe the right setting to handle Zauner's conjecture on mutually unbiased bases (MUBs) (saying that in $\mathbb{C}^d$, a set of MUBs of the theoretical maximal size $d + 1$ exists only if $d$ is a prime power), we pose some fundamental questions which naturally arise. Some of these questions have important consequences for the construction theory of (new) sets of maximal MUBs.
Real Mutually Unbiased Bases
我们列出了R^d中相互无偏基的最佳数目的界。对于大多数维数d，可以用相对简单的方法证明，要么不存在相互无偏的实正交基，要么最优数至多为2或3。我们讨论了这些方法在应用于所有维数时的局限性，揭示了获得具有d=16n^2形式的其余维数的紧界的困难，其中n可以是任何数。我们还给出了一个更简单的，替代的证明，可以有最多d/2+1真正的相互无偏基地在D维，而不是调用已知的结果极值欧几里德线集卡梅伦和赛德尔，Delsarte，和Calderbank等人。
Uncertainty relations based on mutually unbiased measurements
根据互无偏测量，导出了不确定关系不等式。通过计算$d+1$ MUM对$\mathbb{C}^{d}$中任意密度算符$ρ$给出的概率分布重合指数，给出了状态相关和状态无关形式的熵下界.此外，我们制定的不确定性关系的MUM的Rényi和Tsallis熵。
Mutually unbiased measurements (MUMs) are generalized from the concept of mutually unbiased bases (MUBs) and include the complete set of MUBs as a special case, but they are superior to MUBs as they do not need to be rank one projectors. We investigate entanglement detection using sets of MUMs and derived separability criteria for $d$-dimensional multipartite systems, and arbitrary high-dimensional bipartitie and multipartite systems. These criteria provide experimental implementation in detecting entanglement of unknown quantum states.
Unextendible maximally entangled bases and mutually unbiased bases
研究了任意二分空间中不可扩张的最大纠缠基。给出了在$\mathbb{C}^{d}\bigotimes\mathbb{C}^{d'}(\frac{d'}{2}<d<d')$中构造一组$d^{2}$正交最大纠缠态的系统方法。这些$d^{2}$正交最大纠缠态的集合的互补空间不包含与它们正交的最大纠缠态。此外，我们还研究了所有基都是不可扩展的最大纠缠基的互无偏基。我们在$\mathbb{C}^{2}\bigotimes\mathbb{C}^{3}$中提出了两个不可扩展的最大纠缠基，它们是相互无偏的。
Mutually unbiased measurements in finite dimensions
我们推广的概念，相互无偏基地（MUB）的测量，不一定是由秩一投影仪描述。因此，这些测量可以是一个有用的工具来研究长期存在的问题MUB的存在。我们推导出它们的一般形式，并表明，在一个有限的，$d$维希尔伯特空间，可以构建一个完整的一组$d+1$相互无偏测量。除了它们与MUB的内在联系之外，我们还表明，这些测量的统计数据提供了有关系统状态的完整信息。此外，它们捕捉了无偏性的物理本质，特别是，它们满足类似于$d+1$ MUB的非平凡熵不确定关系。
Entanglement in mutually unbiased bases
量子力学的本质特征之一是大多数对可观测量不能同时测量。当可观测量与相互无偏的基础相关时，这种现象表现得最为强烈。在本文中，我们揭示了相互无偏基和量子力学的另一个基本特征，量子纠缠之间的联系。它表明，一个完整的二分系统的相互无偏基地的集合包含一个固定量的纠缠，独立的选择集。这对完整集合的状态之间的纠缠分布有影响。在素数平方维，我们提出了一个明确的实验友好的建设一个完整的一套特别简单的纠缠分布。最后，我们描述的基本性质的相互无偏基地组成的产品状态。在低维的结构与明确的例子说明。我们相信，在相互无偏基中的纠缠性质可能是解决完备集存在性问题的一个因素。我们还期望它们将与在高维希尔伯特空间中量子协议的实验实现中的基的应用有关。
Mutual unbiasedness of the eigenstates of phase-space operators-such as position and momentum, or their standard coarse grained versions-exists only in the limiting case of infinite squeezing. In [Phys. Rev. Lett. 120, 040403 (2018)] it was shown that mutual unbiasedness can be recovered for periodic coarse grainings of these two operators. Here we investigate mutual unbiasedness of coarse-grained measurements for more than two phase-space variables. We show that mutual unbiasedness can be recovered between periodic coarse graining of any two non-parallel phase-space operators. We illustrate t…
Sizing up entanglement in mutually unbiased bases with Fisher information
提出了一种评价量子态层析成像质量的有效方法。特别注意的是支付给多体系统的无偏测量方面的层析成像。虽然不同组的相互无偏基的总体重构误差是相同的，但是当考虑被测系统的特定方面时出现差异。这一点是通过估计真正的三方纠缠态的量子数来说明的。
An uncertainty relation is introduced for a symmetric arrangement of three mutually unbiased bases in continuous variable phase space, and then used to derive a bipartite entanglement criterion based on the variance of global operators composed of these three phase space variables. We test this criterion using spatial variables of photon pairs, and show that the entangled photons are correlated in three pairs of bases.
Encryption via Entangled states belonging to Mutually Unbiased Bases
我们考虑两个粒子的特殊纠缠，它们的状态矢量在相互无偏的基（MUB）中，即“表现出最大程度的不相容性”（J.Schwinger，Nat. Ac. Sci. (USA)，1960））。我们使用这种纠缠和MUB之间的联系，概述了一个协议，共享这些纠缠态的各方之间的安全密钥分发。分析导致了纠缠态和MUB集合中的状态的关联：两者都带有相同的标签。
We propose a new method for implementing mutually unbiased generation and measurement of time-bin qudits using a cascade of electro-optic phase modulator -- coded fiber Bragg grating pairs. Our approach requires only a single spatial mode and can switch rapidly between basis choices. We obtain explicit solutions for dimensions $d=2,3,4$ that realize all $d+1$ possible mutually unbiased bases and analyze the performance of our approach in quantum key distribution. Given its practicality and compatibility with current technology, our approach provides a promising springboard for scalable process…
The number of measurements necessary to perform the quantum state reconstruction of a system of qubits grows exponentially with the number of constituents, creating a major obstacle for the design of scalable tomographic schemes. We work out a simple and efficient method based on cyclic generation of mutually unbiased bases. The basic generator requires only Hadamard and controlled-phase gates, which are available in most practical realizations of these systems. We show how complete sets of mutually unbiased bases with different entanglement structures can be realized for three and four qubits…
We provide a class of entanglement witnesses constructed in terms of Mutually Unbiased Bases (MUBs). This construction reproduces many well-known examples like the celebrated reduction map and Choi map together with its generalizations. We illustrated our construction by the detailed analysis of the 3-dimensional case: in this case, one obtains a family of entanglement witnesses parameterized by an L-dimensional torus (L = 2,3,4 being a number of MUBs used in the construction).
On mutually unbiased bases
量子自由度的相互无偏基础是所有理论研究和互补性质的实际开发的核心。关于相互无偏的基础，我们已经知道得很多，但也有相当数量的重要问题尚未得到充分的回答。特别地，可以在素数幂维${N=p^\m}$的希尔伯特空间中找到${N+1}$互无偏基的最大集合，其中$p$素数和$\m$是正整数，并且对于连续的自由度，例如沿着直线的运动，存在互无偏基的连续体。但如果维数是另一个合数（$N=6,10,12,...$），则没有一个极大集的例子是已知的。在这篇综述中，我们提出了一个统一的方法，在该方法中，基态标记的数字${0,1,2,...,N-1}$，这是一个伽罗瓦域的元素和普通的整数。这种双重性质允许一个紧凑的系统建设的最大集的相互无偏基地时，他们是已知的存在，但在其他情况下，没有光的开放存在问题。我们展示了如何使用这样构建的相互无偏基地在量子信息学的应用，包括密集编码，隐形传态，纠缠交换，协变克隆，和状态层析成像，所有这些都依赖于一个明确的一组最大纠缠态（熟悉的两个-q比特贝尔状态的推广），是相互无偏基地。有一个链接到数学的有限仿射平面。我们还利用了一对一的对应关系之间的无偏基地和复杂的阿达玛矩阵，把基地变成对方。最终的希望，尚未实现，是开放的问题，相互无偏基地可以涉及开放的问题，阿达玛矩阵或仿射飞机，特别是. [rest删除]
Group theoretical construction of mutually unbiased bases in Hilbert spaces of prime dimensions
有限维希尔伯特空间中的互无偏基与量子互补概念密切相关。利用Hilbert空间中的有限Heisenberg群（也称Pauli群）和SL（2，ZN）在有限相空间ZNxZN上由酉算子实现的作用，给出了素数维N的Hilbert空间中存在N+1个互无偏基的极大集合的另一种证明.证明的关键是，对于素数N，Z_N也是有限域。
We derive a framework for quantifying entanglement in multipartite and high dimensional systems using only correlations in two unbiased bases. We furthermore develop such bounds in cases where the second basis is not characterized beyond being unbiased, thus enabling entanglement quantification with minimal assumptions. Furthermore, we show that it is feasible to experimentally implement our method with readily available equipment and even conservative estimates of physical parameters.
Akin to the idea of complete sets of Mutually Unbiased Bases for prime dimensional Hilbert spaces, $\mathcal{H}_d$, we study its analogue for a $d$ dimensional subspace of $M (d,\mathbb{C})$, i.e. Mutually Unbiased Unitary Bases (MUUBs) comprising of unitary operators. We note an obvious isomorphism between the vector spaces and beyond that, we define a relevant monoid structure for $\mathcal{H}_d$ isomorphic to one for the subspace of $M (d,\mathbb{C})$. This provides us not only with the maximal number of such MUUBs, but also a recipe for its construction.
Mutually unbiased bases for the rotor degree of freedom
我们认为存在一组连续的相互无偏基地的连续和周期性的自由度，描述了一个圆（转子自由度）上的运动。通过一个奇异的映射的圆的线，我们找到了第一个，但有点不令人满意的，连续的一套不涉及到一个潜在的海森堡对互补的可观测量。然后，通过一个非奇异映射的离散角动量的基础上的转子到Fock的基础上的线性运动，我们构造这样的海森堡对转子，并使用它来获得第二个，完全满意，一组相互无偏的基础。
We introduce orthogonal quantum Latin squares, which restrict to traditional orthogonal Latin squares, and investigate their application in quantum information science. We use quantum Latin squares to build maximally entangled bases, and show how a pair of mutually unbiased maximally entangled bases can be constructed in square dimension from orthogonal quantum Latin squares. We also compare our construction to an existing construction due to Beth and Wocjan and show that ours is strictly more general.
Mutually Unbiased Bases and Orthogonal Latin Squares -- version 3
本文证明了N维希尔伯特空间中存在完整互不偏正交基与存在完整互正交拉丁方之间的关系，并指出六维空间中不存在完整MUBs。
Improving Quantum State Estimation with Mutually Unbiased Bases
当用于量子态估计时，在相互无偏基上的投影具有使每次测量的信息提取最大化并使冗余最小化的能力。我们提出了第一个实验演示的量子态层析成像的两个量子比特的偏振态，以利用相互无偏基地。我们证明了改进的状态估计相比，标准的测量策略，并讨论如何可以理解我们使用的测量结构。我们通过实验将我们的方法与三种不同状态的标准状态估计方法进行了比较，并观察到使用我们的技术的不保真度比使用标准状态估计方法低1.84+/-0.06倍。
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