Two-qubit formulas with a common optimal ensemble

For every real $F\in[0,1]$, let $\Omega_2=(|0,0\rangle+|1,1\rangle)/\sqrt2$, $\Phi_2=|\Omega_2\rangle\langle\Omega_2|$, and $\rho_F=F\Phi_2+(1-F)(I-\Phi_2)/3$. For a unit vector $\psi\in\mathbb C^2\otimes\mathbb C^2$, define $(A_\psi)_{ij}=\sum_b\psi(i,b)\overline{\psi(j,b)}$, $\tau(\psi)=2(1-\operatorname{Re}\operatorname{tr}(A_\psi^2))$, and $c(\psi)=\sqrt{\tau(\psi)}$; define $T(\sigma)$ and $C(\sigma)$ as the respective infima of the average costs $\tau$ and $c$ over finite ensembles of unit vectors with nonnegative weights summing to $1$ and average matrix $\sigma$, where $|\psi\rangle\langle\psi|$ has entries $\psi(u)\overline{\psi(v)}$. Then

Independently, for every (t\in[1/2,1]$, if$P_2(t)$is the infimum of$2(1-\mu_0^2-\mu_1^2)$over$\mu_0,\mu_1\ge0$with$\mu_0+\mu_1=1$and$\sqrt{\mu_0}+\sqrt{\mu_1}=\sqrt{2t}$, then$P_2(t)=(2t-1)^2$and$\sqrt{P_2(t)}=2t-1$. There exists a single finite ensemble$(w_k,\psi_k)$, with$w_k\ge0$,$\sum_kw_k=1$, and unit vectors$\psi_k$, whose average matrix is$\rho_F$, whose average$\tau$-cost equals$T(\rho_F)$, and whose average$c$-cost equals$C(\rho_F)$; every component of strictly positive weight satisfies both$\tau(\psi_k)=T(\rho_F)$and$c(\psi_k)=C(\rho_F)$. Components of zero weight are not subject to these last equalities.