= PPT maximally entangled overlap bound
{c}
{title2=$F(|\Phi_d\rangle\langle\Phi_d|,\sigma)\leq d^{-1/2}$}
Every <density operator> $\sigma$ with <positive partial transpose> has squared <quantum fidelity> at most $1/d$ with a fixed <maximally entangled state> on $d\otimes d$. The <partial transpose of a maximally entangled projector> and the <trace-adjoint identity for partial transpose> express that squared overlap as $\operatorname{Tr}(S\sigma^{T_B})/d$. Since $\sigma^{T_B}$ is positive of <trace> one and the <swap operator> has <operator norm> one, <trace duality> gives the bound. A matching <product state> attains equality.
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