3.3 The Pauli matrices, algebraically

Matrices (of a fixed size, with entries in a fixed field) 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 (over any field) 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}=\left[\begin{smallmatrix}0&1\\0&0\end{smallmatrix}\right]). 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.61 We have already defined the Pauli operators (Section 2.7), but we recall their definition here.

Identity \mathbf{1}= \begin{bmatrix}1&0\\0&1\end{bmatrix} \begin{array}{lcr}|0\rangle&\longmapsto&|0\rangle\\|1\rangle&\longmapsto&|1\rangle\end{array}
Bit-flip X = \begin{bmatrix}0&1\\1&0\end{bmatrix} \begin{array}{lcr}|0\rangle&\longmapsto&|1\rangle\\|1\rangle&\longmapsto&|0\rangle\end{array}
Bit-phase-flip Y = \begin{bmatrix}0&-i\\i&0\end{bmatrix} \begin{array}{lcr}|0\rangle&\longmapsto&i|1\rangle\\|1\rangle&\longmapsto&-i|0\rangle\end{array}
Phase-flip Z = \begin{bmatrix}1&0\\0&-1\end{bmatrix} \begin{array}{lcr}|0\rangle&\longmapsto&|0\rangle\\|1\rangle&\longmapsto&-|1\rangle\end{array}

Recall that the Pauli operators (as well as the identity operator) are unitary and Hermitian, square to the identity, and anti-commute.62

The fact that \{\mathbf{1},X,Y,Z\} forms a basis for (2\times2) complex matrices is equivalent to the statement that 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. \end{aligned} for some complex numbers a_0, a_x, a_y, and a_z.

If we define vectors \vec{a}=(a_x, a_y, a_z) and \vec{\sigma}=(\sigma_x,\sigma_y,\sigma_z), then we can write the above expansion very concisely: A = a_0\mathbf{1}+ \vec{a}\cdot\vec{\sigma}. The algebraic properties of the Pauli matrices can then be neatly compacted (see Exercise 3.7.4) 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}.

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).

We can also define an inner product on the vector space of matrices:63

The Hilbert–Schmidt product of A and B is given by64 (A|B) = \frac{1}{2} \operatorname{tr}A^\dagger B.

We will return to the algebraic structure of these Pauli matrices in Chapter 8, before explaining how they turn out to be useful for things such as quantum error correction.

  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 \frac{1}{2} coefficient here is simply the normalisation factor, which changes if we consider multi-qubit Pauli operators.↩︎

  4. The factor of \frac{1}{2} isn’t necessary, but simplifies some calculations.↩︎