The Minimum Hartree Value for the Quantum Entanglement Problem

2012/02/14 by Liqun Qi, Qi, Liqun · 2 citations
Computer Science · Physics and Astronomy · #FOS: Physical sciences #Quantum Computing Algorithms and Architecture #Quantum Information and Cryptography #Quantum Mechanics and Applications #Quantum Physics (quant-ph) #quant-ph

paper · pdf · doi:10.48550/arxiv.1202.2983

arxiv created 2012/02/14 · openalex publication_date 2012/02/14 · arxiv updated 2012/02/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A general n-partite state | Ψ> of a composite quantum system can be regarded as an element in a Hilbert tensor product space \HH = ⊗_k=1ⁿ \HH_k, where the dimension of \HH_k is d_k for k = 1,..., n. Without loss of generality we may assume that d₁ ≤...≤ d_n. A separable (Hartree) n-partite state | ϕ> can be described by | ϕ> = ⊗_k=1ⁿ | ϕ^(k)> with | ϕ^(k)> ∈ \HH_k. We show that σ:= min \< Ψ| ϕ_Ψ> : | Ψ> ∈ \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≥ 1/√d₁... d_n-1. Thus, the geometric measure of the entanglement content of Ψ, ‖ | Ψ> - | ϕ_Ψ> ‖ ≤ √(2-2σ) ≤ √2-2(1/√d₁...d_n-1).

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• Geometric measure of entanglement of multipartite mixed states
• The geometric measure of entanglement of pure states with nonnegative amplitudes and the spectral theory of nonnegative tensors

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