bibkey: peledteenicarmicohen2021minor
authors: Bar Y. Peled; Amit Te'eni; Avishy Carmi; Eliahu Cohen
year: 2021
title: "Correlation minor norms, entanglement detection and discord"
doi: 10.1038/s41598-021-82303-3
url: "https://arxiv.org/abs/2005.12079v2"
claim: "The paper defines the correlation minor norms of a bipartite state as Schatten norms of compound matrices of its correlation matrix, bounds them for separable states in filter normal form, constructs saturating states from quantum designs, and conjectures that the bounds hold without the filter-normal-form assumption."
strata_touched:
  - D5/S3/Quantum/Entanglement/Witness/CorrelationMinorNormFNFRefutation
license: citation-only
triage: anchor

Peled, Te’eni, Carmi and Cohen, Correlation minor norms, entanglement detection and discord

B. Y. Peled, A. Te’eni, A. Carmi and E. Cohen, Scientific Reports 11 (2021) 2849, DOI 10.1038/s41598-021-82303-3; arXiv:2005.12079v2. Quotations are from the arXiv v2 TeX source.

For orthonormal bases $\{A_i\}$, $\{B_j\}$ of the Hermitian matrices on ${\mathbb{C}}^{d_A}$ and ${\mathbb{C}}^{d_B}$, the correlation matrix of a state $\rho$ is ${\mathcal{C}}_{ij}=\mathrm{tr}(\rho A_i\otimes B_j)$, and ${\mathcal{M}}_{h,p}$ is the Schatten $p$-norm of its $h$-th compound matrix. With $d=\min (d_A,d_B)$ and $D=\max (d_A,d_B)$, Theorem 3 states: “Let $\rho$ be a separable state in FNF, and $h\ge \sqrt{Dd}$. Then: ${\mathcal{M}}_{h,p=\infty }\le \frac{1}{\sqrt{Dd}}{\left[\frac{D-1}{D\left(h-1\right)}\frac{d-1}{d\left(h-1\right)}\right]}^{\frac{h-1}{2}}$.” The paper adds:

As in Theorem \ref{thm_bound_p_1}, we conjecture this theorem still holds without the assumption that $\rho$ is in FNF.

Verified locator

Source: https://arxiv.org/abs/2005.12079v2

• DOI: https://doi.org/10.1038/s41598-021-82303-3 (journal version; title, authors, journal, volume, issue and article number from Crossref; the journal text was not read).
• URL: https://arxiv.org/abs/2005.12079v2. The e-print member CMN_16_12_20_arXiv.tex has 378 lines and md5 8c626d38169cc53f41f01da6643e7760. Locations:
  • the correlation matrix, l. 177–183;
  • the correlation minor norms and the compound matrix, l. 214–233;
  • filter normal form, l. 248; Theorem 3 and the conjecture, l. 274–281.