Hastings has proved that the minimal output entropy is not additive: it may happen that $S_{\mathrm{min}}(\Phi_1 \otimes \Phi_2) < S_{\mathrm{min}}(\Phi_1)+S_{\mathrm{min}}(\Phi_2) $ for quantum channels $\Phi_1, \Phi_2$. His construction is probabilistic, so it does not yield specific channels $\Phi_1, \Phi_2$ for which the inequality holds. According to my understanding it is not easy to verify that for a given realization of his random construction (because the dimensions of the involved Hilbert spaces are huge).

Do we presently know an explicit pair $\Phi_1, \Phi_2$ for which strict inequality has been verified?



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