{"aixiv_id":"2608.00014","url":"https://aixiv.online/abs/2608.00014","pdf_url":"https://arxiv.org/pdf/2608.27363","title":"On supporting affine functionals for Entanglement of Formation","abstract":"In several articles, the authors assume that the convex roof structure of the EoF and finite-dimensionality of subsystems $A$ and $B$ guarantee the existence of the (global) supporting affine functional for the EoF at any state of the system $AB$. This means that for any state $ρ$ of $AB$ there is a Hermitian operator $Λ_ρ$ on $\\mathcal{H}_{AB}=\\mathcal{H}_A\\otimes\\mathcal{H}_B$ such that $E_F(ρ)=\\mathrm{Tr}Λ_ρρ$ and $E_F(σ)\\geq\\mathrm{Tr}Λ_ρσ$ for any state $σ$ of $AB$. We present an explicit example showing that, when $ρ$ is degenerate, this is not true even in the simplest case when $A$ and $B$ are qubit systems. The construction is based on the fact that the existence of a supporting affine functional for the EoF at a state $ρ$ is equivalent to the Lipschitz lower semicontinuity of the EoF at this state $ρ$. We use Wootters' formula and the help of Claude Fable 5 to find a state $ρ$ of the system $AB$ for which the latter property does not hold. We also describe conditions for the existence the local and global supporting affine functionals for the EoF at a given state of both finite and infinite-dimensional bipartite quantum systems. These conditions allow us to find Lipschitz lower semicontinuity bounds for the EoF at a given finite rank state $ρ$ (i.e. inequalities of the form $\\,E_F(ρ)-E_F(σ)\\leq C_ρ\\|ρ-σ\\|_1$) with and without restrictions on the support of the state $σ$.","authors":["A. S. Holevo","M. E. Shirokov"],"primary_category":"quant-ph","categories":["quant-ph","math-ph","math.OC"],"published":"2026-08-27","updated_at":"2026-08-30T23:08:48.000Z","latest_version":1,"license":"See original source","kind":"imported","withdrawn":false,"listed":true,"withdrawal_reason":null,"fulltext_indexed":false,"claimed_by_author":false,"source":{"name":"arXiv","url":"https://arxiv.org/abs/2608.27363"},"repository":null,"ai_process":{"models":null,"contribution_level":null,"has_process_summary":false,"has_prompts_harness":false,"has_verification_notes":false,"has_negative_results":false,"trace_url":null},"reproduction_status":"unverified","editors_pick":true,"editorial_summary":"An explicit two-qubit counterexample shows Entanglement of Formation need not admit a global supporting affine functional at degenerate states, contradicting an assumption used in prior work. Holevo and Shirokov credit Claude with helping find the counterexample state.","comment_count":0,"priority_record":{"note":"Each version is timestamped at receipt; the hash is immutable evidence of content at that time.","versions":[]},"process_summary":null,"prompts_harness":null,"verification_notes":null,"negative_results":null,"reproduction_reports":[],"bibtex_url":"https://aixiv.online/bibtex/2608.00014"}