We show that if an electromagnetic energy pulse in a multimode coherent state with average photon number $\overline{n}$ is used to carry out the same quantum logical operation on a set of $N$ atoms, either simultaneously or sequentially, the overall error probability in the worst-case scenario (i.e., maximized over all the possible initial atomic states) scales as $N^2∕\overline{n}$. This means that in order to keep the error probability bounded by $Nϵ$, with $ϵ\sim 1∕\overline{n}$, one needs to use $N\overline{n}$ photons or, equivalently, $N$ separate “minimum-energy” pulses: in this sense the pulses cannot, in general, be shared. The origin of this phenomenon is found in atom-field entanglement. These results may have important consequences for quantum logic and, in particular, for large-scale quantum computation.

• Received 11 September 2006

DOI:https://doi.org/10.1103/PhysRevA.74.060301