Horn问题与Littlewood-Richardson系数关系的量子信息理论证明
A quantum information-theoretic proof of the relation between Horn's problem and the Littlewood-Richardson coefficients
摘要：
霍恩问题要求的条件集的整数μ，nu和λ，确保存在的埃尔米特算子A，B和A+B的频谱μ，nu和λ，分别。已经证明，这个问题等价于决定GL（d）的具有最高权lambda的不可约表示是否包含在具有最高权mu和nu的不可约表示的张量积中。在本文中，我们提出了一个量子信息理论证明这两个问题之间的关系是渐近的一个方向。这个结果以前已经由Klyachko使用几何不变理论得到。然而，本文中提出的工作并没有触及这两个问题之间的非渐近等价性，这一结果依赖于最近证明的GL（d）饱和猜想。
英文摘要：
Horn's problem asks for the conditions on sets of integers mu, nu and lambda that ensure the existence of Hermitian operators A, B and A+B with spectra mu, nu and lambda, respectively. It has been shown that this problem is equivalent to deciding whether the irreducible representation of GL(d) with highest weight lambda is contained in the tensor product of irreducible representations with highest weight mu and nu. In this paper we present a quantum information-theoretic proof of the relation between the two problems that is asymptotic in one direction. This result has previously been obtained by Klyachko using geometric invariant theory. The work presented in this paper does not, however, touch upon the non-asymptotic equivalence between the two problems, a result that rests on the recently proven saturation conjecture for GL(d).
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