  It is known that any bipartite unitary operator of Schmidt rank three is
equivalent to a controlled unitary under local unitaries. We propose a standard
form of such operators. Using the form we improve the upper bound for the
entanglement cost to implement such operators under local operations and
classical communications (LOCC), and provide a corresponding protocol. A part
of our protocol is based on a recursive-control protocol which is helpful for
implementing other unitary operators. We show that any bipartite permutation
unitary of Schmidt rank three can be implemented using LOCC and two ebits. We
give two protocols for implementing bipartite permutation unitaries of any
Schmidt rank $r$, and showed that one of the protocol uses $O(r)$ ebits of
entanglement and $O(r)$ bits of classical communication, while these two types
of costs for the other protocol scale as $O(r\log r)$ but the actual values are
smaller for all $r<1100$. Based on this we obtain upper bounds of the number of
nonlocal CNOT gates needed to implement bipartite classical reversible maps
using classical circuits under two different conditions. We also quantify the
entangling power of bipartite permutation unitaries of Schmidt rank two and
three. We show that they are respectively $1$ ebit and some value between
$\log_2 9 - 16/9$ and $\log_2 3$ ebits.
