Qudit Swap Gate

We have seen that the contolled-X (C-NOT) operation in d-state systems acts as





Then the Controlled - operation acts as





Using these operators we can make the Qudit Swap Gate.

Now consider the following gates
Call this as Gate M1.



Call this as Gate M2



Now we can see that operation M1 is






And operation M2 is





Also





So

Now by adding an operation at the end we can form the Qudit swap gates as shown





Since the inverse of Swap operation is also Swap operation we have the following implementations also