Fidelity-only lower bounds for arbitrary mixed states

For every natural number $d\ge2$ and every positive semidefinite complex matrix $\rho$ of trace $1$ on $\mathbb C^d\otimes\mathbb C^d$, let $\Phi_d=|\Omega_d\rangle\langle\Omega_d|$, where $\Omega_d=d^{-1/2}\sum_{i=0}^{d-1}|i,i\rangle$, and set $F=\operatorname{Re}\operatorname{tr}(\Phi_d\rho)$. For a unit vector $\psi$, define $A_\psi$ by $(A_\psi)_{ij}=\sum_b\psi(i,b)\overline{\psi(j,b)}$, and put $\tau(\psi)=2(1-\operatorname{Re}\operatorname{tr}(A_\psi^2))$ and $c(\psi)=\sqrt{\tau(\psi)}$. Here $|\psi\rangle\langle\psi|$ has entries $\psi(u)\overline{\psi(v)}$, and real square roots take value $0$ on negative arguments. For either cost $h=\tau$ or $h=c$, let $V_h(F)$ be the set of numbers $\sum_k w_kh(\psi_k)$ arising from finite ensembles of unit vectors with $w_k\ge0$, $\sum_kw_k=1$, and $\sum_kw_k\operatorname{Re}\operatorname{tr}(\Phi_d|\psi_k\rangle\langle\psi_k|)=F$, and let $R_h(F)=\inf V_h(F)$; these ensembles are constrained only by their mean overlap, with no requirement that their average matrix equal $\rho$. Define $T(\rho)$ and $C(\rho)$ as the respective infima of $\sum_kw_k\tau(\psi_k)$ and $\sum_kw_kc(\psi_k)$ over such finite ensembles whose average matrix is exactly $\rho$. Also define

and, for a real function $f$, let $H_df(t)$ be the infimum of $\sum_kw_kf(t_k)$ over finite scalar mixtures with nonnegative weights summing to $1$, points $t_k\in[1/d,1]$, and mean $\sum_kw_kt_k=t$. The assertion is that $R_\tau(F)$ belongs to $V_\tau(F)$ and is no larger than any member of that set, and likewise $R_c(F)$ belongs to and is least in $V_c(F)$; moreover $R_\tau(F)\le T(\rho)$ and $R_c(F)\le C(\rho)$, and, if $1/d\le F$, then $H_dP_d(F)\le T(\rho)$ and $H_dr_d(F)\le C(\rho)$. All finite mixtures permit zero weights, while their weights summing to $1$ excludes an empty ensemble.