A complex-valued gradient flow for the entangled bipartite low rank approximation
A complex-valued gradient flow for the entangled bipartite low rank approximation
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DOI：
10.1016/j.cpc.2021.108185
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发表时间：
2021-10
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影响因子：
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通讯作者：
M. Chu;Matthew M. Lin
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文献类型：
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作者：
M. Chu;Matthew M. Lin
复合系统中量子态的纠缠在许多应用中具有重要意义。对于某些适当选择的基，纠缠可以通过复值密度矩阵的Kronecker积来数学描述。对混合态的近似可以看作是在计算其最近的可分离态。这样的任务在计算方面遇到了几个挑战。首先，通过Kronecker积产生的纠缠所增加的扭曲破坏了多线性。常用的交替最小二乘张量逼近技术很难应用。其次，没有明确的策略来为近似选择一个先验的适当的低排名。第三，传统的微积分不足以解决复变量上实值函数的最优化问题。本文提出了一种解决纠缠二体系统低阶近似的动力学系统方法，该方法具有以下优点：1)可以用相当简洁的方式描述复杂空间中的梯度动力学；2)保证从任意起点到局部解的全局收敛；3)可以保证纯态的组合系数必须是概率分布的要求；4)可以动态地调整秩值。本文讨论了该方法的原理、算法，并给出了一些数值实验。
Entanglement of quantum states in a composite system is of profound importance in many applications. With respect to some suitably selected basis, the entanglement can be mathematically characterized via the Kronecker product of complex-valued density matrices. An approximation to a mixed state can be thought of as calculating its nearest separable state. Such a task encounters several challenges in computation. First, the added twist by the entanglement via the Kronecker product destroys the multi-linearity. The popular alternating least squares techniques for tensor approximation can hardly be applied. Second, there is no clear strategy for selecting a priori a proper low rank for the approximation. Third, the conventional calculus is not enough to address the optimization of real-valued functions over complex variables. This paper proposes a dynamical system approach to tackle low rank approximation of entangled bipartite systems, which has several advantages, including 1) A gradient dynamics in the complex space can be described in a fairly concise way; 2) The global convergence from any starting point to a local solution is guaranteed; 3) The requirement that the combination coefficients of pure states must be a probability distribution can be ensured; 4) The rank can be dynamically adjusted. This paper discusses the theory, algorithms, and presents some numerical experiments.