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Using the triangle inequality, we have $$||\rho-\Pi_{-z}\rho\Pi_{-z}||_1\leq ||\rho||_1+||\Pi_{-z}\rho\Pi_{-z}||_1=1+\mathrm{Tr}(\Pi_{-z}\rho\Pi_{-z})$$ (the final equality holds because $\rho$ and $\Pi_{-z}\rho\Pi_{-z}$ are positive semidefinite). Then we can use $\Pi_{z}^2=\Pi_{z}$ and cyclicity of the trace to find $$\mathrm{Tr}(\Pi_{-z}\rho\Pi_{-z})=\...


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