bibkey: guo2023partialnorm authors: Y. Guo year: 2023 title: “Partial-Norm of Entanglement: Entanglement Monotones That are not Monogamous” doi: 10.1088/1367-2630/acf152 url: https://arxiv.org/abs/2212.06521v6 claim: “The partial negativity of a pure state with decreasing Schmidt coefficients λ₁ ≥ λ₂ ≥ ⋯ is λ₁λ₂, with reduced function ĥ(ρ) = √(δ₁δ₂) for the two largest eigenvalues of the reduced state; the paper conjectures that ĥ is concave.” strata_touched:

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

Guo, Partial-Norm of Entanglement: Entanglement Monotones That are not Monogamous

Y. Guo, Partial-Norm of Entanglement: Entanglement Monotones That are not Monogamous, New J. Phys. 25 (2023) 083047; arXiv:2212.06521v6 (31 August 2023; quant-ph).

Verified locator

DOI: 10.1088/1367-2630/acf152. Primary version read: https://arxiv.org/abs/2212.06521v6 (the last arXiv version). The TeX source of v6 supplies the definition of concavity of a reduced function (Eq. h), the ordering of the Schmidt coefficients, Proposition 1 (pro1), the definition of the partial negativity with its reduced function, the conjecture and reference [Suppl].

Source statements

The source macros \ra, \la, \mH, \mS are written out.

Concavity: “$h[\lambda {\rho }_1+(1-\lambda ){\rho }_2]\ge \lambda h({\rho }_1)+(1-\lambda )h({\rho }_2)$ for any states ${\rho }_1$, ${\rho }_2$, and any $0\le \lambda \le 1$.”

Schmidt ordering: “Let $|\psi ⟩={\sum }_{j=1}^r{\lambda }_j|e_j{⟩}^A|e_j{⟩}^B$ be the Schmidt decomposition of $|\psi ⟩\in {\mathcal{H}}^{AB}$, where ${\lambda }_1\ge {\lambda }_2\ge \cdots \ge {\lambda }_r$, and $r$ is the Schmidt rank of $|\psi ⟩$.”

Proposition 1: “Let $E$ be an entanglement measure with the reduced function $h$ defined as Eq. (h). If $E$ is an entanglement monotone, then $h$ is concave.”

Partial negativity: “Then $\hat{N}(|\psi ⟩)={\lambda }_1{\lambda }_2$, and the corresponding reduced function is $\hat{h}({\rho }^A)=\sqrt{{\delta }_1{\delta }_2}$, where ${\delta }_1={\lambda }_1^2$, ${\delta }_2={\lambda }_2^2$.”

Conjecture: “We conjecture that $\hat{h}$ is concave [Suppl]. $\hat{h}$ is strictly concave on $\mathcal{S}(\mathcal{H})$ with $\dim \mathcal{H}=2$ since it reduced to an elementary symmetric function, but it is not true for the higher dimensional case.”

Reference [Suppl]: “We checked by many examples that it is true. Especially, if $[\rho ,\sigma ]=0$, $\hat{h}(\rho /2+\sigma /2)\ge \hat{\rho }/2+\hat{h}(\sigma )/2$ is always true.”

After the conjecture: “We now assume that $\hat{N}$ is an entanglement monotone, then we can conclude the following.”

Scope

The paper studies entanglement monotones whose reduced functions depend on part of the spectrum of the reduced state (partial norms) and shows that several are not monogamous. Y. Guo, Complete Genuine Multipartite Entanglement Monotone, arXiv:2301.00334v3 (Results in Physics 57 (2024) 107430), restates $\hat{h}(\rho )=\sqrt{{\delta }_1{\delta }_2}$ with “${\delta }_1$, ${\delta }_2$ are the two largest eigenvalues of $\rho$”, repeats the conjecture and assumes the concavity of $\hat{h}$ in its later sections.