trureturing — Discovering truth

bibkey: braunstein2006laplaciandensity authors: Samuel L. Braunstein; Sibasish Ghosh; Simone Severini year: 2006 title: “The Laplacian of a graph as a density matrix: a basic combinatorial approach to separability of mixed states” doi: 10.1007/s00026-006-0289-3 url: https://arxiv.org/abs/quant-ph/0406165v2 claim: “Conjecture 6.7 asserts that the star maximizes entanglement of formation among connected graph density matrices.” strata_touched:

• D5/S3/Quantum/Entanglement/GraphDensity/FormationBounds
• D5/S3/Quantum/Entanglement/GraphDensity/SelectiveFormationBounds
• D5/S3/Quantum/Entanglement/GraphDensity/BraunsteinGhoshSeveriniStarRefutation license: citation-only triage: anchor

Verified locator

DOI: https://doi.org/10.1007/s00026-006-0289-3

Source: https://arxiv.org/abs/quant-ph/0406165v2

Statement and scope

Definition 2.2, p. 3: “The density matrix of a graph $G$ is the matrix” $$\sigma (G)\stackrel{\mathrm{def}}{=}\frac{1}{d_G}L(G).$$ The degree-sum on p. 2 is $d_G={\sum }_{i=1}^n{d}_G(v_i)$.

Section 2.3, p. 4: “The von Neumann entropy of an $n\times n$ density matrix $\rho$ is $S(\rho )=-{\sum }_{i=1}^n{\lambda }_i(\rho ){\log }_2{\lambda }_i(\rho )$.” The convention is $0{\log }_{2}0=0$.

The paragraph before the conjectures, p. 18: “Let ${\rho }_{AB}$ be a density matrix acting on ${\mathbb{C}}_A^p\otimes {\mathbb{C}}_B^q$, where $pq=n$.” It then defines $$S_{\rho }=\left\{\{p_i,|{\psi }_i⟩:i=1,2,\dots ,N\}:{\rho }_{AB}=\sum _i{p}_i|{\psi }_i{⟩}_{AB}⟨{\psi }_i|,|{\psi }_i{⟩}_{AB}\in {\mathbb{C}}_A^p\otimes {\mathbb{C}}_B^q,0\le p_i\le 1,\sum _i{p}_i=1\right\},$$ The next sentence is: “The entanglement of formation of ${\rho }_{AB}$ is denoted and defined by” $$E_F({\rho }_{AB})=\underset{\{p_i,|{\psi }_i⟩:i=1,2,\dots ,N\}\in S_{\rho }}{\inf }\sum _{i=1}^N{p}_{i}S({\mathrm{tr}}_X(|{\psi }_i{⟩}_{AB}⟨{\psi }_i|)),X=A\text{ or }B.$$ The sentence ends “where $X=A$ or $X=B$.” The state vectors are Euclidean unit vectors. The source’s displayed reconstruction has an $n$ upper index while the ensemble has $N$ terms; the ensemble convention reads the reconstruction as the sum over those $N$ terms.

Conjecture 6.7, p. 18 (the final conjecture):

“Let ${\mathcal{G}}_n^c$ be the set of all connected graphs on $n$ vertices. Let $G\in {\mathcal{G}}_n^c$ ($|V|=pq$). Then ${\max }_{{\mathcal{G}}_n^c}{E}_F(\sigma (G))=E_F(\sigma (K_{1,n-1}))$.”

The product labeling is $i=sq+s^{\prime }$ (Section 3.1); the star root is $v_1$. Lean uses zero-based pairs $(a,b)\in \mathrm{Fin}(p)\times \mathrm{Fin}(q)$, with root $(0,0)$, and connected simple graphs with a nonempty edge set. Loops do not affect the Laplacian. The bit-entropy ensemble infimum uses the literal partial trace over the second factor.