bibkey: paltothbenevertesi2021singlet
authors: Pál, Károly F.; Tóth, Géza; Bene, Erika; Vértesi, Tamás
year: 2021
title: "Bound entangled singlet-like states for quantum metrology"
doi: 10.1103/PhysRevResearch.3.023101
url: "https://arxiv.org/abs/2002.12409v4"
claim: "The paper constructs two families of PPT entangled states on (2d x 2d)-dimensional systems whose quantum Fisher information 16 sqrt(d)/(1+sqrt(d)) exceeds every separable state; the second family is defined through orthogonal matrices with prescribed properties, constructed for d = 3 and d = 2^n, and the authors believe it exists for all d."
strata_touched:
  - D5/S3/Quantum/Entanglement/BoundEntanglement/SecondFamilyAllDimensions
license: citation-only
triage: anchor

Pál, Tóth, Bene and Vértesi, Bound entangled singlet-like states for quantum metrology

K. F. Pál, G. Tóth, E. Bene and T. Vértesi, Phys. Rev. Research 3, 023101 (2021), DOI 10.1103/PhysRevResearch.3.023101; arXiv:2002.12409v4. Quotations are from the arXiv v4 TeX source metroboundent53.tex.

Definition 2 gives the second family on ${\mathbb{C}}_A^2\otimes {\mathbb{C}}_B^2\otimes {\mathbb{C}}_{A^{\prime }}^d\otimes {\mathbb{C}}_{B^{\prime }}^d$ as $\varrho =\frac{p_1}{d^2}{\sum }_{i,j}|z_{ij}⟩⟨z_{ij}|+\frac{p_2}{2d}{\sum }_i|s_i⟩⟨s_i|+\frac{p_2}{2d}{\sum }_i|10ii⟩⟨10ii|$ with $|z_{ij}⟩=\frac{1}{\sqrt{2}}(|00ij⟩+{\sum }_k{Q}_{ik}^j|11jk⟩)$, $p_1=\sqrt{d}/(1+\sqrt{d})$ and $p_2=1/(1+\sqrt{d})$.

It requires:

• every $Q^j$ orthogonal;
• each $|q_{ij}⟩={\sum }_k{Q}_{ij}^k|10kk⟩$ either zero or $\pm \sqrt{d/N_m}$ times one of $d$ orthonormal vectors $|{\bar{q}}_m⟩$, where $N_m$ counts the pairs assigned to $m$;
• orthonormal $|s_i⟩$ in the $|01{⟩}_{AB}$ sector with $({\sum }_i|s_i⟩⟨s_i|{)}^{\Gamma }={\sum }_m|t_m⟩⟨t_m|$, where $|t_m⟩=d^{-1/2}{\sum }_{(i,j)\in {\Xi }_m}{S}_{ij}|01ij⟩$.

Under these requirements the paper shows that ${\varrho }^{\Gamma }⪰0$ and that the quantum Fisher information is $16\sqrt{d}/(1+\sqrt{d})$. The paper states:

For this family of states we have constructed analytical examples only for $d=3$ and $2^n.$ However, we believe that such states exist for all $d$.

Verified locator

• DOI: https://doi.org/10.1103/PhysRevResearch.3.023101
• Source: https://arxiv.org/abs/2002.12409v4