A class of optimal positive maps in M

Optimality of generalized Choi maps in M 3

“…Note that the spanning property is only a sufficient condition for optimality, but not a necessary one. There exist optimal maps without spanning property [30][31][32][33][34][35]. For example, the Choi map is optimal without having the spanning property [30,31].…”

“…where ∆ 2 denotes the 2-simplex with vertices x i . Below we provide the conditions under which the map Φ W is positive (for details see [1]).…”

“…Again, if w ij = C ij is cirulant, then condition ( 12) is trivially satisfied, and theorem 2 reduces to theorem 1. Now, followed by [1], we consider the new parameterization of the matrix W. As W/w is a doubly stochastic matrix the Birkhoff theorem implies (13) with {a, b, c, d, e, f} satisfying a + b + c + d + e + f = w. Note that the parameters {d, e, f} perturb the circulant part {C ij }. However, this representation is not unique.…”

“…Moreover, it serves as a powerful tool for characterizing quantum entanglement and hence plays a key role in various aspects of quantum information theory. Although there exist several results regarding the classification of positive maps [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18], the complete analysis of positive maps still remains an open question. In this direction, examples of positive linear maps given by Choi [19][20][21] play a prominent role in investigating the structure of the positive cones of positive linear maps.…”

“…In this paper, we consider a generalization of the family of linear positive maps in M 3 proposed by Cho et al [22] which are generalizations of the Choi maps. The positivity of such maps is rigorously discussed in [1]. Our main focus is to find the condition for optimality.…”