bibkey: yu2013skewinformation
authors: Sixia Yu; C. H. Oh
year: 2013
title: "Robertson-Schrödinger Uncertainty Relation Refined by Skew Information"
doi: null
url: https://arxiv.org/abs/1303.6404v1
claim: "For regular symmetric operator monotone functions f, the paper refines the Robertson-Schrödinger uncertainty relation by f-skew information with a constant λ_f, proves 1 − f(0) ≤ λ_f ≤ min{1, 1/[4f(0)]}, and conjectures that λ_f = min{1, 1/[4f(0)]} always holds."
strata_touched:
  - D5/S3/Quantum/Information/SkewInformationLambdaRefutation
license: citation-only
triage: anchor

Robertson-Schrödinger Uncertainty Relation Refined by Skew Information

Sixia Yu and C. H. Oh, arXiv:1303.6404v1 (quant-ph), 2013-03-26. Quotations are from the v1 TeX source.

The class of functions:

A nonnegative function $f(x)$ for $x\ge 0$ is operator monotone if $f(A)\le f(B)$ for any two Hermitian matrices satisfying $0\le A\le B$, symmetric if $xf(1/x)=f(x)$, normalized if $f(1)=1$, and regular if $f(0)>0$ \cite{petz96}.

The constant (Appendix 2):

Here we shall determine the range of ${\lambda }_f={\min }_{x\ge 0}F(x)$ with

F(x)=\frac1{2f(x)}\left(1+x-\frac{f(0)(1-x)^2}{2f(x)}\right)

The conjecture (Appendix 2):

${\lambda }_f^{\ast }\ge f(0)$ as $f(0)\le 1/2$. In general we have $F^{\prime }(1)=0$, i.e., $F(1)$ is always a local extremal point. Numerical evidences show that ${\lambda }_f=1$ whenever $4f(0)\le 1$ and we conjecture that ${\lambda }_f=\min \{1,1/[4f(0)]\}$ always holds.

Verified locator

• URL: https://arxiv.org/abs/1303.6404v1 (v1, 2013-03-26; source RRSUnet.tex, md5 f03729692cc84121ca4387b67f083b18): the class of functions (l. 204), the constant λ_f and F (l. 281–285), the bounds (l. 286–292) and the conjecture (l. 293).
• The arXiv record has only v1 and no journal reference.