The Minimum Hartree Value for the Quantum Entanglement Problem

A general $n$-partite state $| Ψ>$ 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 $| ϕ>$ can be described by $| ϕ> = \otimes_{k=1}^n | ϕ^{(k)}>$ with $| ϕ^{(k)}> \in \HH_k$. We show that $σ:= \min \{< Ψ| ϕ_Ψ> : | Ψ> \in \HH,.$ $. < Ψ| Ψ> = 1\}$ is a positive number, where $| ϕ_Ψ>$ is the nearest separable state to $| Ψ>$. We call $σ$ the minimum Hartree value of $\HH$. We further show that $σ\ge 1/{\sqrt{d_1... d_{n-1}}}$. Thus, the geometric measure of the entanglement content of $Ψ$, $\| | Ψ> - | ϕ_Ψ> \| \le \sqrt{2-2σ} \le \sqrt{2-2(1/{\sqrt{d_1...d_{n-1}}})}$.

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