## 9.6 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 ball.

Recall that an arbitrary density matrix for a single qubit can be written in the form
*rotate* the Bloch sphere.
In particular the

**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.**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.**Depolarising channel with probability**p .\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 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.^{182}
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 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.

Recall that Kraus operators are also sometimes called “effects”.↩︎