- Properties of Pauli Matrices

1. Basic Properties of Pauli Matrices #

The Pauli matrices are defined as:

With basis ${|+\rangle, |-\rangle}$, where:

• $\sigma_z |+\rangle = +1 |+\rangle$
• $\sigma_z |-\rangle = -1 |-\rangle$

2. Eigenvalues and Eigenvectors #

For $\sigma_x$: $$\sigma_x(a|+\rangle + b|-\rangle) = \lambda (a|+\rangle + b|-\rangle)$$

This gives us:

• $\lambda a = \langle+|\sigma_x(a|+\rangle + b|-\rangle) = b$
• $\lambda b = \langle-|\sigma_x(a|+\rangle + b|-\rangle) = a$

Since $a = \lambda b$ and $b = \lambda a$, we get $\lambda^2 = 1$, thus $\lambda = \pm 1$.

For $\lambda = 1$: $a = b$, giving normalized eigenvector $|+\rangle_x = \frac{1}{\sqrt{2}}(|+\rangle + |-\rangle)$

For $\lambda = -1$: $a = -b$, giving normalized eigenvector $|-\rangle_x = \frac{1}{\sqrt{2}}(|+\rangle - |-\rangle)$

For $\sigma_y$: Applying the same method:

• $\lambda a = -i b$
• $\lambda b = i a$

Thus $\lambda = \pm 1$

For $\lambda = 1$: $b = i a$, giving normalized eigenvector $|+\rangle_y = \frac{1}{\sqrt{2}}(|+\rangle + i|-\rangle)$

For $\lambda = -1$: $b = -i a$, giving normalized eigenvector $|-\rangle_y = \frac{1}{\sqrt{2}}(|+\rangle - i|-\rangle)$

3. Raising and Lowering Operators #

Defining $\sigma_+ = \sigma_x + i\sigma_y$ and $\sigma_- = \sigma_x - i\sigma_y$:

For $\sigma_+$, since $\langle+|\sigma_+|+\rangle = 0$, $\langle-|\sigma_+|+\rangle = 0\ $, $$\sigma_+ |+\rangle = \langle+|\sigma_+|+\rangle |+\rangle + \langle-|\sigma_+|+\rangle |-\rangle = 0$$

since $\langle+|\sigma_+|-\rangle = 2$, $\langle-|\sigma_+|-\rangle = 0\ $,
$$\sigma_+ |-\rangle = \langle+|\sigma_+|-\rangle |+\rangle + \langle-|\sigma_+|+\rangle |-\rangle = 2|+\rangle$$

Similarly:

• $\sigma_- |+\rangle = 2|-\rangle$
• $\sigma_- |-\rangle = 0$

4. Commutation and Anti-commutation Relations #

The commutator relation: $[\sigma_i, \sigma_j] = 2i \sum_{k} \epsilon_{ijk} \sigma_k$

This shows that Pauli matrices are vector operators.

The anti-commutator relation: $\{\sigma_i, \sigma_j\} = \sigma_i\sigma_j + \sigma_j\sigma_i = 2\delta_{ij}$

Therefore: $$\sigma_i\sigma_j = \frac{1}{2}([\sigma_i, \sigma_j] + \{\sigma_i, \sigma_j\})$$

$$= \delta_{ij} + i\sum_{k}\epsilon_{ijk}\sigma_k$$

This leads to: $(\mathbf{a} \cdot \mathbf{\sigma})(\mathbf{b} \cdot \mathbf{\sigma}) = \mathbf{a} \cdot \mathbf{b} + i(\mathbf{a} \times \mathbf{b}) \cdot \mathbf{\sigma}$

5. Exponential Identity #

The following identity holds: $$e^{-i\theta \mathbf{n} \cdot \mathbf{\sigma}} = \cos\theta - i\sin\theta \mathbf{n} \cdot \mathbf{\sigma}$$

This is a generalization of Euler’s identity $e^{i\theta} = \cos\theta + i\sin\theta$, which represents rotation in the complex plane. When a measurement axis(corresponding to $z$-axis) perpendicular to the rotation axis(corresponding to $y$-axis) $\mathbf{n}$ is rotated by $\theta$, it rotates a vector of the eigenspace of the measurement axis by $\theta$ as well.

6. Choice of Basis Matrices #

The choice of basis matrices affects our understanding of rotation angles:

(1) Using $\{\sigma_x, \sigma_y, \sigma_z\}$ with eigenvalues $\pm 1$: $$e^{-i\theta \mathbf{n} \cdot \mathbf{\sigma}} = \cos\theta - i\sin\theta \mathbf{n} \cdot \mathbf{\sigma}$$

(2) Using $\{\frac{1}{2}\sigma_x, \frac{1}{2}\sigma_y, \frac{1}{2}\sigma_z\}$ with eigenvalues $\pm \frac{1}{2}$: $$e^{-i2\theta \mathbf{n} \cdot \mathbf{\frac{1}{2}\sigma}} = \cos\theta - i\sin\theta \mathbf{n} \cdot \mathbf{\sigma}$$

Or equivalently: $$e^{-i\theta’ \mathbf{n} \cdot \mathbf{\frac{1}{2}\sigma}} = \cos\frac{\theta’}{2} - 2i\sin\frac{\theta’}{2} \mathbf{n} \cdot \mathbf{\frac{1}{2}\sigma}$$

If we define $\mathbf{\tilde{\sigma}} = \mathbf{\frac{1}{2}\sigma}$ and let $\theta’ \to \theta$, then: $$e^{-i\theta \mathbf{n} \cdot \mathbf{\tilde{\sigma}}} = \cos\frac{\theta}{2} - 2i\sin\frac{\theta}{2} \mathbf{n} \cdot \mathbf{\tilde{\sigma}}$$