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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