the norm of a state on the full matrix

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Let $M$ be a 2 by 2 full matrix. $\phi$ is a state on $M$ with density $(\beta, 1-\beta)$, then we can find only one projection $e\in M$ with $\|\phi_{eM(1-e)}\|=|1-2\beta|$.

How to choose the projection $e$?