Approximate simulation of quantum channels

Earlier, we proved a duality between two optimizations problems [Phys. Rev. Lett. 104, 120501 (2010)]. The primary one is, given two quantum channels $\mathcal{M}$ and $\mathcal{N}$, to find a quantum channel $\mathcal{R}$ such that $\mathcal{R}\circ \mathcal{N}$ is optimally close to $\mathcal{M}$ as measured by the worst-case entanglement fidelity. The dual problem involves the information obtained by the environment through the so-called complementary channels $\widehat{\mathcal{M}}$ and $\widehat{\mathcal{N}}$, and consists in finding a quantum channel ${\mathcal{R}}^{\prime }$ such that ${\mathcal{R}}^{\prime }\circ \widehat{\mathcal{M}}$ is optimally close to $\widehat{\mathcal{N}}$. It turns out to be easier to find an approximate solution to the dual problem in certain important situations, notably when $\mathcal{M}$ is the identity channel—the problem of quantum error correction—yielding a good near-optimal worst-case entanglement fidelity as well as the corresponding near-optimal correcting channel. Here we provide more detailed proofs of these results. In addition, we generalize the main theorem to the case where there are certain constraints on the implementation of $\mathcal{R}$, namely, on the number of Kraus operators. We also offer a simple algebraic form for the near-optimal correction channel in the case $\mathcal{M}=\mathrm{id}$. For approximate error correction, we show that any $\varepsilon$-correctable channel is, up to appending an ancilla, $\varepsilon$-close to an exactly correctable one. We also demonstrate an application of our theorem to the problem of minimax state discrimination.