论文标题
量子状态的平均二元性在任意对称群体上揭示了schur-weyl双重性
Duality of averaging of quantum states over arbitrary symmetry groups revealing Schur-Weyl duality
论文作者
论文摘要
在量子信息理论中，这是一个公认的事实，它平均对单一群体在多部分量子状态的集体行动上平均将状态投射到等同于子系统置换操作员的形式。因此，相当于置换操作员的状态不会被集体统一的噪声所触及。一个微不足道的观察表明，统一的对排列操作员的平均均匀构图将状态投射为一种形式，其块对基结构的结构等于统一群体的集体行动之一。我们介绍了此属性的名称：平均二元性。这种二元性背后的数学原因是，统一组在多部分量子系统的张量产物状态空间上的集体作用，而置换操作的作用是相互交换者，当将其视为矩阵代数时。这样的基质代数对被称为双还原对。在这项工作中，我们表明，在有限的尺寸量子系统的情况下，这种平均二元性对于任何对称组是双重还原对，无论它们是否紧凑，只要平均操作是通过迭代的集成在集体动作的cartan demotions上定义的。尽管我们的结果非常笼统，但我们将非常关注的重点集中在双重还原对的具体示例上，该双重还原对由特殊的线性矩阵和置换操作的集体作用组成，这些矩阵和置换操作的集体作用实际上对应于非单独的SLOCC-type（随机局部操作和经典通信）操作的多个分数量子状态。在这种情况下，我们表明，在SLOCC以条件方式平均的情况下，从集体单位平均持续存在的无噪声子系统：按照选择后，对特定的不变子空间。
It is a well-established fact in quantum information theory, that uniform averaging over the collective action of a unitary group on a multipartite quantum state projects the state to a form equivalent to a permutation operator of the subsystems. Hence states equivalent to permutation operators are untouched by collective unitary noise. A trivial observation shows that uniform averaging over permutation operators projects the state into a form with block-diagonal structure equivalent to the one of the collective action of the unitary group. We introduce a name for this property: duality of averaging. The mathematical reason behind this duality is the fact that the collective action of the unitary group on the tensor product state space of a multipartite quantum system and the action of the permutation operations are mutual commutants when treated as matrix algebras. Such pairs of matrix algebras are known as dual reductive pairs. In this work we show, that in the case of finite dimensional quantum systems such duality of averaging holds for any pairs of symmetry groups being dual reductive pairs, regardless of whether they are compact or not, as long as the averaging operation is defined via iterated integral over the Cartan decomposition of the group action. Although our result is very general, we focus much attention on the concrete example of a dual reductive pair consisting of collective action of special linear matrices and permutation operations, which physically corresponds to averaging multipartite quantum states over non-unitary SLOCC-type (Stochastic Local Operations and Classical Communication) operations. In this context we show, that noiseless subsystems known from collective unitary averaging persist in the case of SLOCC averaging in a conditional way: upon postselection to specific invariant subspaces.
