TY - JOUR

T1 - A complex-valued gradient flow for the entangled bipartite low rank approximation

AU - Chu, Moody T.

AU - Lin, Matthew M.

N1 - Funding Information:
The first author's research is supported in part by the National Science Foundation under grant DMS-1912816 , and the corresponding author's research is supported in part by the National Center for Theoretical Sciences of Taiwan and by the Ministry of Science and Technology of Taiwan under grant 110-2636-M-006-006 .
Publisher Copyright:
© 2021 Elsevier B.V.

PY - 2022/2

Y1 - 2022/2

N2 - 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.

AB - 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.

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U2 - 10.1016/j.cpc.2021.108185

DO - 10.1016/j.cpc.2021.108185

M3 - Article

AN - SCOPUS:85116536880

SN - 0010-4655

VL - 271

JO - Computer Physics Communications

JF - Computer Physics Communications

M1 - 108185

ER -