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 \begin{aligned} \rho &= \frac{1}{2}\left(\mathbf{1}+\vec{s}\cdot \vec\sigma\right) \\&= \frac{1}{2}\left(\mathbf{1}+s_x X+ s_y Y + s_z Z\right) \end{aligned} where \vec{s} is the Bloch vector of the qubit with components (s_x, s_y, s_z), and X, Y, and Z are the Pauli operators. Recall also that unitary operations rotate the Bloch sphere. In particular the X, Y, and Z operators — viewed as unitary transformations — rotate the Bloch sphere by 180^\circ around the x-, y-, and z-axis, respectively. General quantum channels, however, may deform it further, into spheroids with a displaced centre, as the following examples show.

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 the x-axis; for the specific case of p=\frac{1}{2}, the Bloch sphere degenerates to the [-1,1] interval on the x-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 the z-axis; for the specific case of p=\frac{1}{2}, the Bloch sphere degenerates to the [-1,1] interval on the z-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 probability 1-p, while a quantum error occurs with probability p. The error can be of any one of three types: bit-flip X, phase-flip Z, or both bit- and phase-flip Y; each type of error is equally likely. For p<\frac{3}{4}, the original Bloch sphere contracts uniformly under the action of the channel, and the Bloch vector shrinks by the factor 1-\frac{4}{3}p; for the specific case of p=\frac{3}{4}, the Bloch sphere degenerates to the point at the centre of the sphere; for p>\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 for p=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 p=\frac{1}{2} and switch from the effects E_i to F_j as follows: \left\{ \begin{aligned} E_1 &= \frac{1}{\sqrt{2}}\mathbf{1} \\E_2 &= \frac{1}{\sqrt{2}}Z \end{aligned} \right\} \longmapsto \left\{ \begin{aligned} F_1 &= \frac{1}{\sqrt{2}}(E_1+E_2)=|0\rangle\langle 0| \\F_2 &= \frac{1}{\sqrt{2}}(E_1-E_2)=|1\rangle\langle 1|. \end{aligned} \right\} These two sets of Kraus operators \{E_1,E_2\} and \{F_1,F_2\} describe the same channel, but the 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 Z); the second set tells us that channel essentially performs the measurement in the standard basis, but the outcome of the measurement is not revealed.

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”.↩︎