bibkey: holevo2026logsobolevformation
authors: A. S. Holevo; M. E. Shirokov
year: 2026
title: "Log-Sobolev inequality, von Neumann entropy and Entanglement of Formation"
doi: null
url: https://arxiv.org/abs/2609.12667v3
claim: "With E_F the convex roof of the marginal entropy over finite pure ensembles, the paper states that neither a proof nor a counterexample is known for the existence of a supporting affine functional for the entanglement of formation at every pure state of a finite-dimensional bipartite system."
strata_touched:
  - D5/S3/Quantum/PureState/PureStateFormationSupportingFunctional
license: citation-only
triage: anchor

Log-Sobolev inequality, von Neumann entropy and Entanglement of Formation

A. S. Holevo and M. E. Shirokov, arXiv:2609.12667v3 (quant-ph), 2026-09-15; v1 2026-09-11. Quotations are from the v3 TeX source.

The entropy (Section 1):

The von Neumann entropy $S$ (the entropy, in what follows) is the main characteristic of a quantum state $\rho$ \cite{H-SCI,W,Wilde}. It is defined by the formula \begin{equation}\label{ent-def} S(\rho)=\operatorname{Tr}\eta(\rho), \end{equation} where $\eta (x)=-x\ln x$ if $x>0$ and $\eta (0)=0$.

The entanglement of formation:

\begin{equation}\label{EF-d} E_F(\rho)=\inf_{\sum_k!p_k\rho_k=\rho}\sum_kp_kS([\rho_k]_A), \end{equation} where the infimum is over all finite ensembles $\{p_k,{\rho }_k\}$ of pure states in $\text{St}({\text{HH}}_{AB})$ with the average state $\rho$ \cite{Bennett}.

The question (Introduction):

It is somewhat surprising that the above problem is nontrivial even in the case when the state $\rho$ is pure. We and GPT-5.6 have found neither a general proof nor counterexamples which would confirm or refute the hypothesis of the existence of a supporting affine functional for the EoF at any pure bipartite state.

The question restated (Section 3.1):

One of the questions which are not resolved in \cite{H&Sh-5} concerns the existence of supporting affine functional at any pure state of a finite-dimensional composite quantum system.

Verified locator

• URL: https://arxiv.org/abs/2609.12667v3 (v3, 2026-09-15; v1, 2026-09-11; source EFD-1409.tex, md5 8efd5ec39f0985ff58db8024fa351b70): the entropy (l. 127–132), E_F (l. 170–173), the question (l. 202–205) and its restatement (l. 295–298).