The addition of angular momenta can be reduced to elementary coupling processes of spin-$\frac{1}{2}$-particles. In this way, a method is developed which allows for a non-recursive, simultaneous computation of all Clebsch-Gordan coefficients concerning the addition of two angular momenta. The relevant equations can be interpreted easily, analogously to simple probabilistic considerations. They provide an improved understanding of the addition of angular momenta as well as a practicable evaluation of Clebsch-Gordan coefficients in an easier way than within the well-known methods.
The Clebsch-Gordan coefficients of the group SU(2) are shown to satisfy new inequalities. They are obtained using the properties of Shannon and Tsallis entropies. The inequalities associated with the Wigner 3-j symbols are obtained using the relation of Clebsch-Gordan coefficients with probability distributions interpreted either as distributions for composite systems or distributions for noncomposite systems. The new inequalities were found for Hahn polynomials and hypergeometric functions
Efficient Quantum Circuits for Schur and Clebsch-Gordan Transforms
n d维量子系统的舒尔基是总角动量基的推广，它对于利用排列或集体酉旋转下的对称性是有用的。我们提出了有效的（大小poly（n，d，log（1/ε））的精度ε）量子线路的舒尔变换，这是计算和舒尔基地之间的基础的变化。这些电路基于用于Clebsch-Gordan变换的有效电路。我们还提出了一个有效的电路有限版本的舒尔变换，其中只需要投影到不同的舒尔子空间。这第二个电路是基于任何nonabelian有限群，其中存在一个快速量子傅里叶变换的相位估计的推广。
We discuss the procedure of different partitions in the finite set of $N$ integer numbers and construct generic formulas for a bijective map of real numbers $s_y$, where $y=1,2,\ldots,N$, $N=\prod \limits_{k=1}^{n} X_k$, and $X_k$ are positive integers, onto the set of numbers $s(y(x_1,x_2,\ldots,x_n))$. We give the functions used to present the bijective map, namely, $y(x_1,x_2,...,x_n)$ and $x_k(y)$ in an explicit form and call them the functions detecting the hidden correlations in the system. The idea to introduce and employ the notion of "hidden gates" for a single qudit is proposed. We o…
Quantum Simulation of Random Unitaries from Clebsch-Gordan Transforms
QuSoft, Amsterdam, The Netherlands（QuSoft）；Institute for Logic, Language and Computation, University of Amsterdam, The Netherlands（阿姆斯特丹大学逻辑、语言与计算研究所）；Korteweg-de Vries Institute for Mathematics, University of Amsterdam, The Netherlands（阿姆斯特丹大学科特韦格-德弗里斯数学研究所）；Department of Physics, Graduate School of Science, The University of Tokyo, Japan（东京大学理学研究科物理学系）
本文提出基于克莱布施-戈尔丹变换的精确压缩 oracle 构造方法，用于任意有限维紧群的 Haar 随机群作用，展示了其在量子算法和密码学中的应用。
A symmetrising shift employed by Frenkel and Hartnoll in the approximate computation of the elements of matrix spherical harmonics is further explored and shown to be related to the permutation symmetry of 3-j symbols yielding an extension of the Edmonds classical limit. Some graphs are displayed.
We present a new sum rule for Clebsch-Gordan coefficients using generalized characters of irreducible representations of the rotation group. The identity is obtained from an integral involving Gegenbauer ultraspherical polynomials. A similar procedure can be applied for other types of integrals of such polynomials, and may therefore lead to the derivation of further new relations.
It is shown that, in the relativistic hydrogenic approximation, expectation values of the type $\langle n\ell j|r^k|n\ell j\rangle$ can be expressed in terms of Clebsch-Gordan coefficients or $3jm$ symbols. This generalizes the results obtained by Varshalovich and Karpova [Opt. Spectrosc. 118, 1-5 (2015); Opt. Spektrosk. 118, 3-7] in the non-relativistic hydrogenic case.
In a recent paper [J.-C. Pain, Opt. Spectrosc. ${\bf 218}$, 1105-1109 (2020)], we discussed the link between expectation values of powers of $r$ and Clebsch-Gordan coefficients. In this short note we provide additional information, reminding that such a connection is a direct consequence of group theory. The hydrogenic radial wavefunctions form bases for infinite dimensional representations of the algebra of the non-compact group $O(2,1)$ and the expectation values $r^p$ and $r^{-p}$ ($p$ being positive) transform as tensors with respect to this algebra. As shown a long time ago by Armstrong […
Fast, accurate, and stable computation of the Clebsch-Gordan (C-G) coefficients is always desirable, for example, in light scattering simulations, the translation of the multipole fields, quantum physics and chemistry. Current recursive methods for computing the C-G coefficients are often unstable for large quantum numbers due to numerical overflow or underflow. In this paper, we present an improved method, the so-called sign-exponent recurrence, for the recursive computation of C-G coefficients. The result shows that the proposed method can significantly improve the stability of the computati…
We discuss the construction and symmetries of su(3) Clebsch-Gordan coefficients arising from the su(3) basis states constructed as triple tensor products of two-dimensional harmonic oscillator states. Because of the su(2) symmetry of the basis states, matrix elements and recursion relations are easily expressed in terms of su(2) technology. As the Weyl group has a particularly simple action on these states, Weyl symmetries of the su(3) coupling coefficients generalizing the well known m-> -m symmetry of su(2) coupling can be obtained, so that any coefficient can be obtained as a sum of Weyl-re…
Kronecker product in terms of Hubbard operators and the Clebsch-Gordan decomposition of SU(2)xSU(2)
本文利用Hubbard算子讨论了方阵$A \otimes B \otimes C \cdots$的Kronecker（直积或张量积）的性质。在其最简单的形式中，Hubbard算子$X_n^{i,j}$可以表示为$n$ -方阵，其在位置$(i,j)$中具有条目1，并且在所有其它条目中具有零。在这样一个框架中，定义多体量子系统的可观测量的代数和群性质是非常简单的。特别地，我们利用Hubbard记法中的Kronecker积得到了乘积群$SU(2) \times SU(2)$的Clebsch-Gordan分解.最后，$n$维不可约表示，从而获得的Clebsch-Gordan系数的规则，除了角动量的封闭形式。我们的结果可以在许多不同的方向上进一步发展。
