Entangling and assisted entangling power of bipartite unitary operations

Nonlocal unitary operations can create quantum entanglement between distributed particles, and the quantification of created entanglement is a hard problem. It corresponds to the concepts of entangling and assisted entangling power when the input states are, respectively, product and arbitrary pure states. We analytically derive them for Schmidt-rank-two bipartite unitary and some complex bipartite permutation unitaries. In particular, the entangling power of permutation unitary of Schmidt rank three can take only one of two values: $\log_2 9 - 16/9$ or $\log_2 3$ ebits. The entangling power, assisted entangling power and disentangling power of $2\times d_B$ permutation unitaries of Schmidt rank four are all $2$ ebits. These quantities are also derived for generalized Clifford operators. We further show that any bipartite permutation unitary of Schmidt rank greater than two has entangling power greater than $1.223$ ebits. We construct the generalized controlled-NOT (CNOT) gates whose assisted entangling power reaches the maximum. We quantitatively compare the entangling power and assisted entangling power for general bipartite unitaries, and study their connection to the disentangling power. We also propose a probabilistic protocol for implementing bipartite unitaries.