The Minimum Hartree Value for the Quantum Entanglement Problem

A general $n$-partite state $| \Psi>$ of a composite quantum system can be regarded as an element in a Hilbert tensor product space $\HH = \otimes_{k=1}^n \HH_k$, where the dimension of $\HH_k$ is $d_k$ for $k = 1,..., n$. Without loss of generality we may assume that $d_1 \le...\le d_n$. A separable (Hartree) $n$-partite state $| \phi>$ can be described by $| \phi> = \otimes_{k=1}^n | \phi^{(k)}>$ with $| \phi^{(k)}> \in \HH_k$. We show that $\sigma := \min \{< \Psi | \phi_\Psi> : | \Psi> \in \HH,.$ $. < \Psi | \Psi > = 1\}$ is a positive number, where $| \phi_\Psi >$ is the nearest separable state to $| \Psi >$. We call $\sigma$ the minimum Hartree value of $\HH$. We further show that $\sigma \ge 1/{\sqrt{d_1... d_{n-1}}}$. Thus, the geometric measure of the entanglement content of $\Psi$, $\| | \Psi > - | \phi_\Psi > \| \le \sqrt{2-2\sigma} \le \sqrt{2-2(1/{\sqrt{d_1...d_{n-1}}})}$.