bibkey: wolk2016entanglementwidth
authors: Sabine Wölk; Otfried Gühne
year: 2016
title: "Characterizing the width of entanglement"
doi: 10.1088/1367-2630/aa5015
url: https://arxiv.org/abs/1507.07226v2
claim: "The paper introduces the width of entanglement in spin chains, derives criteria from the collective spin and from a magnetic-gradient operator, and conjectures (Appendix D) that every state with entanglement only within the two halves of the chain satisfies a linear upper bound on the gradient second moment in terms of the squared total spin, attained by mixtures of adjacent singlets and the classical state."
strata_touched:
  - D5/S3/QuantumBounds/WidthGradientBalancedCutRefutation
license: citation-only
triage: anchor

Characterizing the width of entanglement

Sabine Wölk and Otfried Gühne, arXiv:1507.07226v2 (quant-ph), 2016-09-06; published in New J. Phys. 18 (2016) 123024. Quotations are from the v2 TeX source.

The gradient operator (Appendix D):

\BE B^{(\ell)}{k}=(2k-N-1) \sigma^{(\ell)}{k}, \EE with $\ell =x,y,z$. Similar to the angular momentum we define $B^{(\ell )}={\sum }_k{B}_k^{(\ell )}$ and $B^2=(B^x{)}^2+(B^y{)}^2+(B^z{)}^2$. Whereas small values of $⟨J^2⟩$ are indicators for entanglement, large values of $⟨B^2⟩$ are indicators for long range correlations, both classical and quantum. As a consequence, all four-qubit states which are separable under the partition $1,2|3,4$ obey the inequality

The reference values (Appendix D):

The only states with $⟨J^2⟩=0$ and only next-neighbor entanglement are given by \mbox{$|{\Psi }^{-}⟩=⨂{|{\psi }^{-}⟩}_{2j-1,2j}$} which leads to $⟨B^2{⟩}_{-}=6N$. The state \mbox{$|{\Psi }^{\text{cl}}⟩={|\uparrow ,\dots ,\uparrow ⟩}_{1,\dots ,N/2}{|\downarrow ,\dots ,\downarrow ⟩}_{N/2+1,\dots ,N}$} possess the highest possible value of $⟨B^2⟩$ for states with only entanglement within the subgroups $\{1,\dots ,N/2\}$ and $\{N/2+1,\dots ,N\}$. This state is characterized by the expectation values $⟨J^2{⟩}_{\text{cl}}=2N$ and \mbox{$⟨B^2{⟩}_{\text{cl}}=(N^4)/4+2(N-1)N(N+1)/3$}. Therefore, all states with expectation values $⟨B^2⟩$ exceeding this bound must posses entanglement between these two subgroups. If we allow entanglement only between nearest neighbors, then compared to $⟨B^2{⟩}_{\text{cl}}$ only the correlation between the two middle particles within the chain are allowed to change from two to six. As a consequence, all states with $⟨B^2⟩>⟨B^2{⟩}_{\text{cl}}+4$ posses entanglement beyond nearest neighbors.

The conjecture (Appendix D):

Furthermore, the behavior of $⟨B^2⟩$ for $N=4$ leads us to the conjecture, that the largest possible value of $⟨B^2⟩$ for a given $⟨J^2⟩$ with only entanglement within within the subgroups $\{1,\dots ,N/2\}$ and $\{N/2+1,\dots ,N\}$ is reach by the state \mbox{$\varrho =p|{\Psi }^{-}⟩⟨{\Psi }^{-}|+(1-p)|{\Psi }^{\text{cl}}⟩⟨{\Psi }^{\text{cl}}|$}. % As a result, we obtain the conjecture that all states with entanglement only within the subgroups $\{1,\dots ,N/2\}$ and $\{N/2+1,\dots ,N\}$ satisfy the inequality \BE \langle B^2 \rangle \leq \langle B^2\rangle_- + \frac{\langle B\rangle_\textrm {cl}-\langle B^2\rangle_-}{2N},\langle J^2\rangle. \EE For $N=4$ the conjecture coincides with the inequality given in \eq{theo:mag_grad}. Furthermore, first tests with random states confirm this bound.

Verified locator

• URL: https://arxiv.org/abs/1507.07226v2 (v2, 2016-09-06; source ent-distance6_9_16.tex, md5 017195d6078fd4bf499d803b9b3c5ac1): the gradient operator (l. 462–466), the reference values (l. 486–491) and the conjecture (l. 493–500).
• DOI: 10.1088/1367-2630/aa5015 (New J. Phys. 18, 123024). The published text was not read.