bibkey: derksenfriedlandlimwang2017entanglement
authors: Harm Derksen, Shmuel Friedland, Lek-Heng Lim and Li Wang
year: 2017
title: "Theoretical and computational aspects of entanglement"
doi: null
url: "https://arxiv.org/abs/1705.07160v1"
claim: "For a d-partite density tensor A, log₂‖A‖₁ ≤ 2 log₂ α(n,ℂ), with equality for the pure density tensor of a maximally entangled state; the paper conjectures that equality implies that A is such a pure state."
strata_touched:
  - D5/S3/Quantum/Entanglement/Witness/DensityTensorNuclearNormEqualityRefutation
license: citation-only
triage: anchor

Derksen–Friedland–Lim–Wang, Theoretical and computational aspects of entanglement

H. Derksen, S. Friedland, L.-H. Lim and L. Wang, arXiv:1705.07160v1 (the only version). Quotations are from the v1 TeX source.

For a tensor $\mathcal{T}\in {\mathbb{F}}^{\mathbf{n}}$ the nuclear norm $\Vert \mathcal{T}{\Vert }_{1,\mathbb{F}}$ is the minimum of ${\sum }_i{\prod }_j\Vert {\mathbf{x}}_{i,j}\Vert$ over decompositions $\mathcal{T}={\sum }_i{\otimes }_j{\mathbf{x}}_{i,j}$, and $\alpha (\mathbf{n},\mathbb{F})$ is its maximum over unit tensors. A density tensor is the $2d$-mode tensor of a positive semidefinite trace-one matrix on ${\mathbb{C}}^{\mathbf{n}}$. Lemma maxinsep bounds ${\log }_2\Vert \mathcal{A}{\Vert }_1\le 2{\log }_2\alpha (\mathbf{n},\mathbb{C})$ for every density tensor, with equality for $\mathcal{T}\otimes \overline{\mathcal{T}}$ when $\mathcal{T}$ is maximally entangled, and the paper states:

We conjecture that equality in (6.4) implies that $\mathcal{A}$ is a pure density state corresponding to maximum entangled state.

Verified locator

Source: https://arxiv.org/abs/1705.07160v1

• URL: https://arxiv.org/abs/1705.07160v1 (no journal version located). The e-print member dflw19May17.tex has 2027 lines and md5 1a948baafea109a303b169459eff9371. Locations:
  • the nuclear norm, l. 494–497; $\alpha (\mathbf{n},\mathbb{F})$, l. 525–526;
  • density tensors and pure states, l. 680–695; Lemma maxinsep, l. 860–876; the conjecture, l. 878.