**Entangling power of permutations** - Sibasish Ghosh

The notion of entangling power of unitary matrices was introduced by Zanardi, Zalka and Faoro, where the linearized entropy of subsystem`s density matrix was used as a measure of entanglement of the density matrix of the whole system. Using the same measure of entanglement, we study the entangling power of permutations, given in terms of a combinatorial formula. We show that the permutation matrices with zero entangling power are, up to local unitaries, the indentity and the swap. We construct the permutations with the minimum nonzero entangling power for every dimension. With the use of orthogonal latin squares, we construct the permutations with the maximum entangling power for every dimension. Moreover, we show that the value obtained is maximum over all unitaries of the same dimension, with possible exception for 36. Our result enables us to construct generic examples of 4-qudits maximally entangled states for all dimensions except for 2 and 6. We numerically classify, according to their entangling power, the permutation matrices of dimension 4 and 9, and give some estimates for higher dimensions. Taking the `disentangling` power of any unitary operator, acting on a two-qudit system, as the average (over all maximally entangled states) of the linearized entropy of one subsystem of the two-qudit system which is in a state obtained after the action of the unitary operator on a maximally entangled state of the system, we show that the permutations having maximal entangling powers are also having minimal `disentangling` powers ove rthe set of all unitaries with possible exceptions for dimensions 2 and 6.