in *any* EIT-based nonlinear optics systems. In fact, both the conditional and the unconditional gate fidelity approach $1$ in the limit of very small $g_j$, but this limit yields a CPS which becomes appreciable only for unrealistically long gas cells. For the parameters of Fig \[fig:CPSlong\], $v_g = 5.8\times 10^6$, which requires a cell of length $v_g \times t_{int} = 5.8\times 10^6 \textrm{ m/s} \times (500/\gamma) \simeq 81$ m. Therefore a larger CPS requires a larger ratio $g_j\sqrt{N_a}/\Omega_j$. This condition however increases the population of the collective atomic states $|e_1^{(n_p,n_t)}\rangle $ and $| e_5^{(n_p,n_t)}\rangle $ at the expense of the initial atomic state ${\left|3\right\rangle}$, thus unavoidably decreasing the gate fidelity. Similar conclusions hold for other options, such as increased detunings $\delta_j$, or adjusting two-photon detunings $\epsilon_{ij}$. Therefore, just the pure coherent unitary evolution of the system, governed by the Hamiltonian of Eq. (\[eq:coll\_ham\]) causes this inherent trade-off.

Short Interaction Time {#sec:SSShort}
----------------------

To further illustrate our findings, we calculate the CPS and the gate fidelities in the range of parameters where the total interaction time is an order of magnitude smaller than in Sec. \[sec:SSLong\]. The CPS and the gate fidelities are calculated as described in Sec. \[sec:SSLong\], and the 