## 7.5 Single qubit channels

The best way to familiarise ourselves with the concept of a quantum channel is to study a few examples, and we will start with the simplest case: **single qubit channels**.
The single qubit case is special since we can visualise the action of the channel by looking at the corresponding deformation of the Bloch sphere.
Recall that an arbitrary density matrix for a single qubit can be written in the form

*A bit-flip with probability*p :\rho \longmapsto (1-p)\rho+pX\rho X. The Kraus operators are\sqrt{1-p}\mathbf{1} and\sqrt{p}X ; the original Bloch sphere shrinks into a prolate spheroid aligned with thex -axis; for the specific case ofp=\frac{1}{2} , the Bloch sphere degenerates to the[-1,1] interval on thex -axis.*A phase-flip with probability*p :\rho \longmapsto (1-p)\rho+pZ\rho Z. The Kraus operators are\sqrt{1-p}\mathbf{1} and\sqrt{p}Z ; the original Bloch sphere shrinks into a prolate spheroid aligned with thez -axis; for the specific case ofp=\frac{1}{2} , the Bloch sphere degenerates to the[-1,1] interval on thez -axis.*The depolarising channel:*\rho\longmapsto (1-p)\rho + \frac{p}{3}\left(X\rho X+Y\rho Y+Z\rho Z\right). Here the qubit remains intact with probability1-p , while a quantum error occurs with probabilityp . The error can be of any one of three types: bit-flipX , phase-flipZ , or and both bit- and phase-flipY ; each type of error is equally likely. Forp<\frac{3}{4} , the original Bloch sphere contracts uniformly under the action of the channel, and the Bloch vector shrinks by the factor1-\frac{4}{3}p ; for the specific case ofp=\frac{3}{4} , the Bloch sphere degenerates to the point at the centre of the sphere; forp>\frac{3}{4} , the Bloch sphere is flipped, and the Bloch vector starts pointing in the opposite direction increasing the magnitude up to\frac{1}{3} (which occurs forp=1 ).

There are two interesting points that must be mentioned here.
The first one is about the interpretation of the action of the channel in terms of Kraus operators: our narrative may change when we switch to a different set of effects.
For example, take the phase-flip channel with *narrative* is different.
The first set of effects tells us that the channel chooses randomly, with the same probability, between the two options: let the qubit pass undisturbed or apply the phase-flip

Describing actions of quantum channels purely in terms of their effects (i.e. Kraus operators) can be ambiguous.

The second interesting point is that not *all* transformations of the Bloch sphere into spheroids are possible.
For example, we cannot deform the Bloch sphere into a pancake-like oblate spheroid.
This is due to *complete* positivity (instead of mere positivity) of quantum channels, which we will explain shortly.