The purpose of this paper is to introduce techniques of obtaining optimal ways to determine a d-level quantum state or distinguish such states. It entails designing constrained elementary measurements extracted from maximal abelian subsets of a unitary basis U for the operator algebra ℬ(ℋ) of a Hilbert space ℋ of finite dimension d > 3 or, after choosing an orthonormal basis for ℋ, for the ⋆-algebra Md of complex matrices of order d > 3. Illustrations are given for the techniques. It is shown that the Schwinger basis U of unitary operators can give for d, a product of primes p and a, the ideal number d2 of rank one projectors that have a few quantum mechanical overlaps (or, for that matter, a few angles between the corresponding unit vectors). Finally, we give a combination of the tensor product and constrained elementary measurement techniques to deal with all d, though with more overlaps or angles depending on the factorization of d as a product of primes or their powers like d =∏j=1kd j with dj = pjsj,p1 < p2 < ⋯ < pk, all primes, sj ≥ 1 for 1 ≤ j ≤ k, or other types. A comparison is drawn for different forms of unitary bases for the Hilbert space factors of the tensor product like L2(𝔽 t) or L2(ℤ u), where 𝔽t is the Galois field of size t = ps and ℤu is the ring of integers modulo u. Even though as Hilbert spaces they are isomorphic, but quantum mechanical system-wise, these tensor products are different. In the process, we also study the equivalence relation on unitary bases defined by R. F. Werner [J. Phys. A: Math. Gen. 34 (2001) 7081–7094], connect it to local operations on maximally entangled vectors bases, find an invariant for equivalence classes in terms of certain commuting systems, called fan representations, and, relate it to mutually unbiased bases and Hadamard matrices. Illustrations are given in the context of Latin squares and projective representations as well.

中文翻译：

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具有由酉基产生的约束测量的最佳量子断层扫描

本文的目的是介绍获得最佳方法的技术，以确定d级量子态或区分这些状态。它需要设计从酉基的最大阿贝尔子集中提取的约束基本测量ü对于算子代数ℬ(ℋ)希尔伯特空间ℋ有限维d > 3或者，在为ℋ, 对于 ⋆-代数米d复杂的阶矩阵d > 3. 为这些技术提供了插图。表明 Schwinger 基ü的酉算子可以给出d, 素数的乘积p和一种, 理想数d2具有一些量子力学重叠（或者，就此而言，相应的单位向量之间的几个角度）的第一级投影仪。最后，我们给出了张量积和约束基本测量技术的组合来处理所有d, 虽然有更多的重叠或角度取决于分解d作为素数或它们的幂的乘积d =∏j=1ķd j和dj = pjsj,p1 < p2 < ⋯ < pķ, 所有素数,sj ≥ 1为了1 ≤ j ≤ ķ，或其他类型。比较张量积的希尔伯特空间因子的不同形式的酉基，如大号2(𝔽 吨)要么大号2(ℤ 你)， 在哪里𝔽吨是大小的伽罗瓦域吨 = ps和ℤ你是整数环模你. 尽管作为希尔伯特空间，它们是同构的，但在量子力学系统方面，这些张量积是不同的。在这个过程中，我们还研究了 RF Werner 定义的酉基上的等价关系[J.物理。答：数学。将军 34(2001) 7081–7094]，将其连接到最大纠缠向量基上的局部操作，根据某些通勤系统找到等价类的不变量，称为扇形表示，并将其与相互无偏的基和 Hadamard 矩阵相关联。插图也是在拉丁方格和投影表示的上下文中给出的。