Simulation of non-Pauli channels

We consider the simulation of a quantum channel by two parties who share a resource state and may apply local operations assisted by classical communication (LOCC). One specific type of such LOCC is standard teleportation, which is however limited to the simulation of Pauli channels. Here we show how we can easily enlarge this class by means of a minimal perturbation of the teleportation protocol, where we introduce noise in the classical communication channel between the remote parties. By adopting this noisy protocol, we provide a necessary condition for simulating a non-Pauli channel. In particular, we characterize the set of channels that are generated assuming the Choi matrix of an amplitude damping channel as a resource state. Within this set, we identify a class of Pauli-damping channels for which we bound the two-way quantum and private capacities.

Figure 1

Possible Pauli channels when $S_{30}=0.5$ and $\gamma =0.6$, including the shrinking effect of Eq. (64). The hollow tetrahedron is $\mathcal{T}$ characterizing all Pauli channels, whilst the shaded region is the allowable values of $\mathbf{q}$ bounded by $(\frac{2}{\sqrt{7}},\pm \frac{1}{\sqrt{7}},\mp \frac{2}{7}),(\pm \frac{1}{\sqrt{7}},\frac{2}{\sqrt{7}},\mp \frac{2}{7}),(-\frac{2}{\sqrt{7}},\pm \frac{1}{\sqrt{7}},\pm \frac{2}{7}),\text{and}(\pm \frac{1}{\sqrt{7}},-\frac{2}{\sqrt{7}},\pm \frac{2}{7})$ for these particular values.

Figure 2

Upper and lower bounds for the two-way private capacity $P_2$ and the two-way quantum capacity $Q_2$ of the squared channel ${\mathcal{E}}_{\text{sq}}$, in terms of its parameter $\eta$ which is the square of the amplitude damping parameter $\gamma$ associated with its resource state.