The Minimum Hartree Value for the Quantum Entanglement Problem

Abstract

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 $|\varphi >$ can be described by $|\varphi >={\otimes }_{k=1}^n|{\varphi }^{(k)}>$ with $| ϕ^{(k)}> \in \HH_k$. We show that $σ:= \min \{< Ψ| ϕ_Ψ> : | Ψ> \in \HH,.$ $.<\Psi |\Psi >=1\}$ is a positive number, where $|{\varphi }_{\Psi }>$ 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 $\Psi$, $\Vert |\Psi >-|{\varphi }_{\Psi }>\Vert \le \sqrt{2-2\sigma }\le \sqrt{2-2(1/\sqrt{d_1...d_{n-1}})}$.

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