Information flow versus divisibility for qubit evolution

We study the relation between lack of information backflow and completely positive divisibility for noninvertible qubit dynamical maps. Recently, these two concepts were shown to be fully equivalent for the so-called image nonincreasing dynamical maps. Here we show that this equivalence is universal for any qubit dynamical map. A key ingredient in our proof is the observation that there does not exist a completely positive and trace-preserving projector onto a three-dimensional subspace spanned by qubit density operators. Our analysis is illustrated by several examples of qubit evolution, including dynamical maps which are not image nonincreasing.

Figure 1

(a) Bloch ball representation of the action of the PTP map $\mathrm{\Psi }$ and the CPTP map ${\mathcal{S}}_3$. The equatorial brown disk and the thick blue $z$ axis represent the set of density matrices lying in the image of maps $\mathrm{\Psi }$ and ${\mathcal{S}}_3$, respectively. (b, c, d) The allowed structures of density matrices lying in the image of qubit dynamical maps. (b) When the map is invertible the image is an ellipsoid. (c) When the map is noninvertible and its image is two-dimensional it forms a line within the Bloch ball. (d) When the map is noninvertible and its image has dimension 1 it forms a point.