Choi Matrices and Dual Functionals

Kyung Hoon Han, Seung-Hyeok Kye, Erling Størmer

Nathaniel Johnston, David W. Kribs

We provide new results for computing and comparing the quantum gate fidelity of quantum channels via their Choi matrices. We extend recent work that showed that there exist non-dual pairs of quantum channels with equal gate fidelity by providing an explicit characterization of all such channels. We use our characterization to show that when the dimension is 2 (or 3, under slightly stronger hypotheses), the gate fidelity of two channels is equal if and only if their difference equals the difference of some unital map and its dual—a fact that has been shown to be false when the dimension is 4 or larger. We also present a formula for the minimum gate fidelity of a channel in terms of a well-studied norm on a compression of its Choi matrix. As a consequence, several new ways of bounding and approximating the minimum gate fidelity follow, including a simple semidefinite program to compute it for qubit channels.

Kyung Hoon Han, Seung-Hyeok Kye

In this paper, we consider all possible variants of Choi matrices of linear maps, and show that they are determined by non-degenerate bilinear forms on the domain space. We will do this in the setting of finite dimensional vector spaces. In case of matrix algebras, we characterize all variants of Choi matrices which retain the usual correspondences between k-superpositivity and Schmidt number ≤k as well as k-positivity and k-block-positivity. We also compare de Pillis’ definition [Pac. J. Math. 23, 129–137 (1967)] and Choi’s definition [Linear Algebra Appl. 10, 285–290 (1975)], which arise from different bilinear forms.

Alexander Ya. Helemskii

BoGyu Jang

Muneerah Al Nuwairan

In this paper we introduce EPOSIC channels, a class of SU(2)-covariant quantum channels. We give their definition, a Kraus representation of them, and compute their Choi matrices. We show that these channels form the set of extreme points of all SU(2)-irreducibly covariant channels. We also compute their complementary channels, and their dual maps. As application of these channel, we get a new example of positive map that is not completely positive.