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.

  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.↩︎