Wei and Goldbart (Phys Rev A 68:042307, 2003) extend pure geometric measures \(E_\textrm{pure}\) to mixed states by using convex combinations. But, it is hard to calculate this geometric measure. So, Yang et al. (Phys Rev A 108:052217, 2003) define a new geometric measure \(E_\textrm{mix}\) for quantum mixed states as the minimum Frobenius distance between the quantum state and all separable mixed states. However, there is an important theoretical problem of whether this measure is a well-defined geometric measure. In this paper, we first theoretically prove that geometric measure \(E_\textrm{mix}\) of mixed states is well defined from the following three aspects: \(E_\textrm{mix}\) satisfies the criteria (C1)–(C4); separability is consistent under two different geometric measures for pure states; geometric measure \(E_\textrm{mix}\) exists. Further, we prove the existence of the optimal solution for the unconstrained best low-rank positive Hermitian approximation problem. Finally, through numerical experiments, we find that, for the m-partite n-dimensional isotropic state \(\rho _\textrm{iso}(F)\) , there is a linear relationship between F and mixed geometric measure \(E_\textrm{mix}(\rho _\textrm{iso}(F))\) if it is an entangled state, and provide a novel method for calculating the separable critical point \(F_c\) .
