= Known-bit quantum money forgery
{title2=$p_{\rm pass}^{\rm opt}=(1+1/\sqrt2)/2$}
If a known classical bit is independently encoded in a secret uniformly chosen X or Z eigenbasis, the averaged verification operator is $A_b=I/2+(-1)^b(X+Z)/4$. Its largest <eigenvalue> is the displayed optimal acceptance <probability> for a <known-bit quantum money forgery>[counterfeit] prepared without the original <qubit>. The maximizing state bisects the two candidate Bloch directions. For $n$ independent secret bases, the maximal all-pass <probability> is the $n$th <power>, because the total acceptance operator is a <tensor product>. Randomly choosing one of the legitimate encodings instead passes with <probability> $3/4$ per bit.
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