bibkey: siudzinska2025twoconstants authors: Katarzyna Siudzińska year: 2025 title: Measures from conical 2-designs depend only on two constants doi: 10.1088/1751-8121/ae0203 url: https://arxiv.org/abs/2506.18211 claim: “The concurrence lower bound of a conical 2-design is eta times (the trace norm of its correlation operator minus xi), with eta = sqrt(2/(d(d-1)))/S and xi = C_max.” strata_touched:

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DOI: 10.1088/1751-8121/ae0203

Source: https://arxiv.org/abs/2506.18211

J. Phys. A: Math. Theor. 58 (2025) 375302. Section 2, Eq. (9), PDF p. 2: the conical-design identity. Section 6, Eq. (59), PDF p. 9: the measurement correlation operator. Section 7, “Concurrence lower bound”, Theorem 6, Eqs. (73)–(74), PDF p. 10: the two-constant formula. The DOI and title match the Crossref record.

The two-constant concurrence formula

Section 2 states:

By definition, $\mathcal{P}$ is a conical 2-design if ${\sum }_{\alpha =1}^N{\sum }_{k=1}^{M_{\alpha }}{P}_{\alpha ,k}\otimes P_{\alpha ,k}={\kappa }_{+}{I}_d\otimes I_d+{\kappa }_{-}{F}_d$, where ${\kappa }_{+}\ge {\kappa }_{-}>0$.

Section 6 defines the correlation operator in Eq. (59) and states:

The elements of $\mathcal{B}(\rho )$ in the basis of ${\omega }_{\alpha ,k}$ form the correlation matrix ${\mathcal{B}}_{\alpha ,k;\beta ,\ell }=\mathrm{Tr}[\rho (P_{\alpha ,k}\otimes P_{\beta ,\ell })]$.

Its matrix in that basis has entries $\mathrm{tr}[\rho (P_i\otimes P_j)]$. Theorem 6 refers to Eq. (60); the defining operator expression is Eq. (59). Section 7, Theorem 6 states:

The concurrence of a mixed bipartite state $\rho$ is lower bounded by \mathcal{N}_{\min}(\rho)=\eta\Big[\|\mathcal{B}(\rho)\|_{\tr}-\xi\Big], where $\mathcal{B}(\rho )$ is the correlation operator from eq. (60) and $\eta =\frac{1}{S}\sqrt{\frac{2}{d(d-1)}},\xi ={\mathcal{C}}_{\max }$.

With $\alpha ={\kappa }_{+}$, $\beta ={\kappa }_{-}$, $S=\beta$ and ${\mathcal{C}}_{\max }=\alpha +\beta$, this is precisely $B_E(\rho )=c_d(\Vert P_E(\rho ){\Vert }_1-\alpha -\beta )/\beta$. The measurement labels are flattened to Fin m; there is no extra normalization of the effects or of the correlation matrix. The comparison uses this bound with the same design on both subsystems.