We develop the probabilistic implementation of a nonlocal gate $\mathrm{exp}[i\ensuremath{\xi}{\ensuremath{\sigma}}_{{n}_{A}}{\ensuremath{\sigma}}_{{n}_{B}}]$ and $\ensuremath{\xi}∊[0,(\ensuremath{\pi}∕4)]$, by using a single nonmaximally entangled state. We prove that nonlocal gates can be implemented with a fidelity of
>79.3 and a consumption of
<0.969ebits and two classical bits, when $\ensuremath{\xi}\ensuremath{\leqslant}0.353$. This provides a higher bound for the feasible operation compared to the former techniques. Besides, gates with $\ensuremath{\xi}\ensuremath{\geqslant}0.353$ can be implemented with the probability 79.3% and a consumption of 0.969 ebits, which is the same efficiency as the distillation-based protocol, while our method saves extra classical resources. Gates with $\ensuremath{\xi}\ensuremath{\rightarrow}0$ can be implemented with nearly unit probability and a small entanglement. We also generalize some applications to the multiple system, where we find it is possible to implement certain nonlocal gates between many nonentangled partners using a nonmaximally multiple entangled state.
We propose and study a method for using nonmaximally entangled states to probabilistically implement desired nonlocal gates. Unlike distillation-based protocols, this method does not generate a maximally entangled state at intermediate stages of the process. As a consequence, the method becomes more efficient at a certain range of parameters. Gates of the form $\mathrm{exp}[i\ensuremath{\xi}{\ensuremath{\sigma}}_{{n}_{A}}{\ensuremath{\sigma}}_{{n}_{B}}]$ with $\ensuremath{\xi}⪡1$ can be implemented with nearly unit probability and with vanishingly small entanglement, while for the distillation-based method, the gate is produced with a vanishing success probability. We also derive an upper bound to the optimal success probability and show that, in the small-entanglement limit, the bound is tight.