量子纠缠与行列式簇的几何
Quantum entanglement and geometry of determinantal varieties
摘要：
量子纠缠在爱因斯坦、波多尔斯基和罗森的著名论文中首次被认为是量子力学的一个特征[18]。近年来人们认识到，量子纠缠是量子计算、量子通信和量子密码学的重要组成部分（[16]，[17]，[6]）。本文引入了二分或多分量子系统中混合态的代数集作为它们在局域幺正变换下的不变量，代数集是复射影空间或复射影空间的乘积中的行列式簇。这些不变量自然产生于用可分离的纯态测量混合态的物理考虑。这样，这些代数集合的代数几何和复微分几何就成为理解量子对合的有力工具。我们的构造在量子信息理论中的下列重要问题中有应用：1）可分性判据，证明了如果混合态是可分的，则代数集必须是线性子空间的和：2）Schmidt数的下界，即一般的低秩二分混合态在多个自由度上纠缠; 3）Hamilton算子的模拟，证明了同秩半正Hamilton算子的模拟蕴涵了相应代数集的投射同构;（4）束缚排列的构建，利用我们提出的新的可分性判据，系统地构造了在部分转置下不变的纠缠混合态的例子（即PPT束缚纠缠）。另一方面，从这一观点出发，构造和研究了许多纠缠混合态的例子，这些纠缠混合态在其相关的行列式簇中具有丰富的代数几何结构。
英文摘要：
Quantum entanglement was first recognized as a feature of quantum mechanics in the famous paper of Einstein, Podolsky and Rosen [18]. Recently it has been realized that quantum entanglement is a key ingredient in quantum computation, quantum communication and quantum cryptography ([16],[17],[6]). In this paper, we introduce algebraic sets, which are determinantal varieties in the complex projective spaces or the products of complex projective spaces, for the mixed states in bipartite or multipartite quantum systems as their invariants under local unitary transformations. These invariants are naturally arised from the physical consideration of measuring mixed states by separable pure states. In this way algebraic geometry and complex differential geometry of these algebraic sets turn to be powerful tools for the understanding of quantum enatanglement. Our construction has applications in the following important topics in quantum information theory: 1) separability criterion, it is proved the algebraic sets have to be the sum of the linear subspaces if the mixed states are separable; 2) lower bound of Schmidt numbers, that is, generic low rank bipartite mixed states are entangled in many degrees of freedom; 3) simulation of Hamiltonians, it is proved the simulation of semi-positive Hamiltonians of the same rank implies the projective isomorphisms of the corresponding algebraic sets; 4) construction of bound enatanglement, examples of the entangled mixed states which are invariant under partial transpositions (thus PPT bound entanglement) are constructed systematically from our new separability criterion. On the other hand many examples of entangled mixed states with rich algebraic-geometric structure in their associated determinantal varieties are constructed and studied from this point of view.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
