Entangled inputs cannot make imperfect quantum channels perfect

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With OPEN ρ i = ( 𝟙 ⊗ P i ) ​ ρ ​ ( 𝟙 ⊗ P i ) † ) / q i \rho^{i}=(\mathbbm{1}\otimes P_{i})\rho(\mathbbm{1}\otimes P_{i})^{\dagger})/{q_{i}} , one finds C ← ( ρ , B : E ) = ∑ i = 0 k − 1 q i S ( ρ B i | | ρ B ) . \displaystyle C_{\leftarrow}(\rho,B:E)=\sum_{i=0}^{k-1}q_{i}S(\rho^{i}_{B}||\rho_{B}). (23)

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This is numerically evaluated by optimizing over mixed-state ensembles ρ = ∑ j p j ​ ρ j \rho=\sum_{j}p_{j}\rho_{j} using global optimization both based on the routine 𝚏𝚖𝚒𝚗𝚌𝚘𝚗 {\tt fmincon} in Matlab with randomly sampled initial conditions and simulated annealing

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