3.3 The Pauli matrices, algebraically

Matrices form a vector space: you can add them, and you can multiply them by a scalar. One possible choice of a basis in the vector space of (2\times 2) matrices is the set of matrices \{M_{00},M_{01},M_{10},M_{11}\}, where the entries of M_{ij} are all 0 except for the ij-th entry, which is 1 (e.g. M_{01}=\begin{bmatrix}0&1\\0&0\end{bmatrix}). However, it turns out that there is a different basis which offers lots of insights into the structure of the general single-qubit unitary transformations, namely \{\mathbf{1},X,Y,Z\}, i.e. the identity matrix and the three Pauli matrices.50 We have already defined the Pauli operators in Chapter 2, but we recall their definition here along with a different notation that we sometimes use.

Identity \mathbf{1}= \begin{bmatrix}1&0\\0&1\end{bmatrix}
Bit-flip X\equiv\sigma_x = \begin{bmatrix}0&1\\1&0\end{bmatrix}
Bit-phase-flip Y\equiv\sigma_y = \begin{bmatrix}0&-i\\i&0\end{bmatrix}
Phase-flip Z\equiv\sigma_z = \begin{bmatrix}1&0\\0&-1\end{bmatrix}

Recalling Chapter 2, we know that the Pauli operators (as well as the identity operator) are unitary and Hermitian, square to the identity, and anti-commute.51

Any (2\times 2) complex matrix A has a unique expansion in the form \begin{aligned} A &= \begin{bmatrix} a_0 + a_z & a_x - i a_y \\a_x +i a_y & a_0 - a_z \end{bmatrix} \\&= a_0\mathbf{1}+ a_x \sigma_x + a_y \sigma_y + a_z \sigma_z \\&= a_0\mathbf{1}+ \vec{a}\cdot\vec{\sigma}. \end{aligned} for some complex numbers a_0, a_x, a_y, and a_z. Here, \vec{a} is a vector with three complex components (a_x, a_y, a_z), and \vec{\sigma} represents the “vector” of Pauli matrices (\sigma_x,\sigma_y,\sigma_z). The algebraic properties of the Pauli matrices can be neatly compacted (see the exercises) into a single expression:

The multiplication rule: (\vec{a}\cdot\vec{\sigma})\,(\vec{b}\cdot\vec{\sigma}) = (\vec{a}\cdot\vec{b})\,\mathbf{1}+ i(\vec{a}\times \vec{b})\cdot\vec{\sigma}.

We also introduce the inner product of two matrices:

The Hilbert–Schmidt product:52 (A|B) = \frac12 \operatorname{tr}A^\dagger B.

Recall that the trace of a square matrix A, denoted by \operatorname{tr}A, is defined to be the sum of the elements on the main diagonal of A, and defines a linear mapping: for any scalars \alpha and \beta, \operatorname{tr}(\alpha A+\beta B) = \alpha\operatorname{tr}A +\beta\operatorname{tr}B. Moreover, the trace is invariant under cyclic permutations: e.g. \operatorname{tr}(ABC) = \operatorname{tr}(BCA) = \operatorname{tr}(CAB). Note, however, that this does not imply that e.g. \operatorname{tr}(ABC)=\operatorname{tr}(ACB).

  1. In this chapter we are concerned only with the single-qubit Pauli operators. There are analogous multi-qubit Pauli operators, but be careful: these do not satisfy all the same properties! For example, anti-commutativity (explained below) is special to the single-qubit case.↩︎

  2. \begin{aligned}XY+YX&=0,\\XZ+ZX&=0,\\YZ+ZY&=0.\end{aligned}↩︎

  3. The \frac12 coefficient here is simply the normalisation factor, which changes if we consider multi-qubit Pauli operators.↩︎