Robustness of quantum gates in the presence of noise

We define several quantitative measures of the robustness of a quantum gate against noise. Exact analytic expressions for the robustness against depolarizing noise are obtained for all bipartite unitary quantum gates, and it is found that the controlled-NOT gate is the most robust two-qubit quantum gate, in the sense that it is the quantum gate which can tolerate the most depolarizing noise and still generate entanglement. Our results enable us to place several analytic upper bounds on the value of the threshold for quantum computation, with the best bound in the most pessimistic error model being $p_{\mathrm{th}}<~\mathrm{0.5.}$