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Low-rank approximation to entangled multipartite quantum systems

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Abstract

Qualifying the entanglement of a mixed multipartite state by gauging its distance to the nearest separable state of a fixed rank is a challenging but critically important task in quantum technologies. Such a task is computationally demanding partly because of the necessity of optimization over the complex field in order to characterize the underlying quantum properties correctly and partly because of the high nonlinearity due to the multipartite interactions. Representing the quantum states as complex density matrices with respect to some suitably selected bases, this work offers two avenues to tackle this problem numerically. For the rank-1 approximation, an iterative scheme solving a nonlinear singular value problem is investigated. For the general low-rank approximation with probabilistic combination coefficients, a projected gradient dynamics is proposed. Both techniques are shown to converge globally to a local solution. Numerical experiments are carried out to demonstrate the effectiveness and the efficiency of these methods.

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Data Availability Statements

The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.

Notes

  1. The very same notation \(\otimes \) has been used for many different meanings in the literature. The distinction between a tensor product and the Kronecker product is necessary for computation and will be explained in Footnote 2. For a general composite system \({\mathscr {H}}_{1} \otimes {\mathscr {H}}_{2}\), we emphasize that \(\otimes \) is merely a bilinear map.

  2. The tensor product of tensors leads to a multi-indexed array. While the way to enumerate its elements is often immaterial in theory, it is essential to enumerate them consistently for numerical calculation. One general rule adopted is that the indices of the leftmost tensor are counted first, e.g., the indices in the tensor product \({\mathbf {a}} \circ {\mathbf {b}}\) of two vectors are enumerated in the same way as the matrix \({\mathbf {a}}{\mathbf {b}}^{\top }\). The relationship (2.3) therefore follows.

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Authors and Affiliations

  1. Department of Mathematics, National Cheng Kung University, Tainan, 701, Taiwan

    Matthew M. Lin

  2. Department of Mathematics, North Carolina State University, Raleigh, NC, 27695-8205, USA

    Moody T. Chu

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M. M. Lin: This research was supported in part by the National Center for Theoretical Sciences of Taiwan and by the Ministry of Science and Technology of Taiwan under Grant 111-2636-M-006-018. M. T. Chu: This research was supported in part by the National Science Foundation under Grant DMS-1912816.

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Lin, M.M., Chu, M.T. Low-rank approximation to entangled multipartite quantum systems. Quantum Inf Process 21, 120 (2022). https://doi.org/10.1007/s11128-022-03467-z

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