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Let $V_i=U_i^\dagger(A\otimes I_K)U_i$ and $\sigma=\rho\otimes |0\rangle\langle0|$, then we have that $V_i$ is unitary and $$\|\rho_1-\rho_2\|_1=\|tr_K\left((V_1\sigma V_1^\dagger -V_2\sigma V_2^\dagger\right)\|_1\\ \leq \|V_1\sigma V_1^\dagger -V_2\sigma V_2^\dagger\|_1\\ =\|V_1\sigma V_1^\dagger -V_1\sigma V_2^\dagger+V_1\sigma V_2^\dagger-V_2\sigma V_2^\...


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