A complex-valued gradient flow for the entangled bipartite low rank approximation

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.