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bibkey: vomende2025witnessoptimality authors: Frederik vom Ende and Simon Cichy year: 2025 title: “Simple Sufficient Criteria for Optimality of Entanglement Witnesses” doi: null url: https://arxiv.org/abs/2505.15615v2 claim: “Section IV asks whether the trace criterion of Corollary 2 (saturation of ⟨Ω|W|Ω⟩ ≥ −tr(W)/min{m,n} by a maximally entangled Ω) is equivalent to the kernel criterion of Theorem 2 (a vector of full Schmidt rank in the kernel of W + tr₂(W) ⊗ 1, or of W + 1 ⊗ tr₁(W)).” strata_touched:

• D5/S3/Quantum/Entanglement/WitnessKernelTraceCriteriaRefutation license: citation-only triage: anchor

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Source: https://arxiv.org/abs/2505.15615v2

Version 2 (27 June 2025; v1 21 May 2025), comments “submitted to Phys. Rev. A.”; Crossref lists no journal version (checked 2026-10-09). The quotations are from the v2 TeX source (main.tex); page numbers refer to the v2 PDF. Witnesses and block positivity: Section II B, p. 3. Theorem 2: Section III A, p. 8. Remark 3(i): p. 8. Corollary 2: p. 9. The question: Section IV, p. 12.

Witnesses (p. 3):

Next, a convenient tool for detecting entanglement are entanglement witnesses. These are block-positive operators $W\in {\mathbb{C}}^{m\times m}\otimes {\mathbb{C}}^{n\times n}$ for which there exists $\sigma \ge 0$ such that $\mathrm{tr}(W\sigma )<0$; recall that $W$ is called block-positive if it satisfies $⟨x\otimes y|W|x\otimes y⟩\ge 0$ for all $x,y$.

Theorem 2, the kernel criterion (p. 8):

Given any witness $W\in {\mathbb{C}}^{m\times m}\otimes {\mathbb{C}}^{n\times n}$, the following statements hold. (i) If $m\le n$ and if $\ker (W+{\mathrm{tr}}_2(W)\otimes 1)$ contains a vector of Schmidt rank $m$, then $W$ is optimal. (ii) If $m\ge n$ and if $\ker (W+1\otimes {\mathrm{tr}}_1(W))$ contains a vector of Schmidt rank $n$, then $W$ is optimal.

Remark 3(i) (p. 8) fixes the Schmidt rank of a vector as the rank of its coefficient matrix:

… has the form $(0,a,b,-a,0,c,-b,-c,0{)}^T$ for some $a,b,c\in \mathbb{C}$, the Schmidt rank of which—by definition—is the rank of $\left(\begin{array}{ccc}0& -a& -b\\ a& 0& -c\\ b& c& 0\end{array}\right)$.

Corollary 2, the trace criterion (p. 9):

For all witnesses $W\in {\mathbb{C}}^{m\times m}\otimes {\mathbb{C}}^{n\times n}$ and all maximally entangled states $\Omega$—i.e., $\Omega$ has Schmidt rank $\min \{m,n\}$ and all its Schmidt coefficients equal $(\min \{m,n\}{)}^{-1/2}$—it holds that $⟨\Omega |W|\Omega ⟩\ge -\frac{\mathrm{tr}(W)}{\min \{m,n\}}$. Moreover, if equality holds in (11) for some maximally entangled state $\Omega$, then $W$ is optimal.

Its proof writes $|\Omega ⟩={\sum }_{j=1}^m\frac{1}{\sqrt{m}}|u_j⟩\otimes |v_j⟩$ with orthonormal $\{u_j\}$, $\{v_j\}$ ($m\le n$) and derives the kernel criterion from equality, so the trace criterion implies the kernel criterion.

The question (Section IV, p. 12):

Next: is our trace-based criterion (Coro. 2) actually equivalent to our kernel criterion (Thm. 2)? This is the question mark in Fig. 2, and while the kernel criterion seems stronger we were not able to find a witness for which only the kernel criterion holds.

After the proof of Theorem 2 the source notes that the kernel criterion transfers to every $(X\otimes 1)W(X^{†}\otimes 1)$ with $X$ of full rank; its footnote to Corollary 2 notes that for a kernel vector $\mathrm{vec}(X)$ of full Schmidt rank the trace in eq. (11) would have to be replaced by a state-dependent quantity.