Optimal arrangements of classical and quantum states with limited purity
Optimal arrangements of classical and quantum states with limited purity
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DOI：
10.1112/jlms.12276
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发表时间：
2018-11
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影响因子：
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通讯作者：
B. Bodmann;Emily J. King
中科院分区：
文献类型：
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作者：
B. Bodmann;Emily J. King
我们考虑Hilbert-Schmidt球中迹归一化非负算子的集合，使它们的相互Hilbert-Schmidt距离最大化；这些是在有限维希尔伯特空间中纯度有限的经典态或量子态集合中的最佳排列。经典状态被理解为由对角矩阵表示，对角条目形成一个概率向量。我们还引入了光谱面体排列的概念，它为经典排列和量子排列提供了统一的框架，并提供了定义新型最优填料的灵活性。继续先前的工作，我们将组合结构和与帧相关的线包装结合起来，以达到高阶量子态的最佳排列。提出了一种新的构造，它涉及到生成一个最优排列，我们称之为gaber - steiner等角紧框架，作为Weyl-Heisenberg群在任何有限阿贝尔群上的射影表示的轨道。然后描述了Gabor-Steiner等角紧框架的线性相关向量的最小集，即所谓的结合部；在一定条件下，它们形成组合块设计，在一种情况下产生一类新的块设计。然后利用在gaborr - steiner等角紧框架中的最小线性相关集上的投影来生成进一步的最优谱面体排列。
We consider sets of trace‐normalized non‐negative operators in Hilbert–Schmidt balls that maximize their mutual Hilbert–Schmidt distance; these are optimal arrangements in the sets of purity‐limited classical or quantum states on a finite‐dimensional Hilbert space. Classical states are understood to be represented by diagonal matrices, with the diagonal entries forming a probability vector. We also introduce the concept of spectrahedron arrangements which provides a unified framework for classical and quantum arrangements and the flexibility to define new types of optimal packings. Continuing a prior work, we combine combinatorial structures and line packings associated with frames to arrive at optimal arrangements of higher rank quantum states. One new construction that is presented involves generating an optimal arrangement we call a Gabor–Steiner equiangular tight frame as the orbit of a projective representation of the Weyl–Heisenberg group over any finite abelian group. The minimal sets of linearly dependent vectors, the so‐called binder, of the Gabor–Steiner equiangular tight frames are then characterized; under certain conditions, these form combinatorial block designs and in one case generate a new class of block designs. The projections onto the span of minimal linearly dependent sets in the Gabor–Steiner equiangular tight frame are then used to generate further optimal spectrahedron arrangements.