Spin and Pauli Matrix Conventions

The default spin-$1/2$ convention uses the $S_z$ eigenbasis

$$\lvert+\rangle_z = \begin{pmatrix} 1\\ 0 \end{pmatrix}, \qquad \lvert-\rangle_z = \begin{pmatrix} 0\\ 1 \end{pmatrix}.$$

In this basis, the Pauli matrices are

$$\sigma_x= \begin{pmatrix} 0&1\\ 1&0 \end{pmatrix}, \qquad \sigma_y= \begin{pmatrix} 0&-i\\ i&0 \end{pmatrix}, \qquad \sigma_z= \begin{pmatrix} 1&0\\ 0&-1 \end{pmatrix}.$$

The physical spin operators are

$$S_i=\frac{\hbar}{2}\sigma_i.$$

Algebra

The Pauli matrices satisfy

$$\sigma_i\sigma_j =\delta_{ij}I+i\sum_k\epsilon_{ijk}\sigma_k,$$

so

$$[\sigma_i,\sigma_j] =2i\sum_k\epsilon_{ijk}\sigma_k, \qquad \{\sigma_i,\sigma_j\}=2\delta_{ij}I.$$

The spin operators satisfy

$$[S_i,S_j] =i\hbar\sum_k\epsilon_{ijk}S_k.$$

The Levi-Civita convention is $\epsilon_{xyz}=+1$.

Raising and Lowering Operators

The default spin raising and lowering operators are

$$S_\pm=S_x\pm iS_y.$$

For Pauli matrices,

$$\sigma_\pm=\frac12(\sigma_x\pm i\sigma_y),$$

so that

$$S_\pm=\hbar\sigma_\pm.$$

With the basis above, $S_+\lvert-\rangle_z=\hbar\lvert+\rangle_z$ and $S_-\lvert+\rangle_z=\hbar\lvert-\rangle_z$.

Bloch-Vector Convention

For a qubit density matrix, the default Bloch form is

$$\rho=\frac12(I+\mathbf r\cdot\boldsymbol\sigma),$$

where $\mathbf r$ is a real vector with $\lVert\mathbf r\rVert\le 1$. Pure states have $\lVert\mathbf r\rVert=1$.

Common Mistakes

• Reversing the sign in $\sigma_y$.
• Confusing dimensionless $\sigma_i$ with physical spin $S_i$.
• Forgetting the factor $\hbar/2$ in spin measurement eigenvalues.
• Using $\sigma_\pm$ and $S_\pm$ interchangeably without the factor of $\hbar$.
• Treating the $S_z$ basis as automatically the energy basis.
• Forgetting that some quantum-information sources use different labels or ordering conventions for qubit basis states.

References

• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
• M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, 2010.

Exercises

1. Verify that $\sigma_y$ is Hermitian under the convention above.

Solution

Taking the conjugate transpose swaps the off-diagonal entries and complex conjugates them. The conjugate of $-i$ is $i$, and the conjugate of $i$ is $-i$, so the matrix returns to itself. Therefore $\sigma_y^\dagger=\sigma_y$.

2. Compute $S_z\lvert+\rangle_z$.

Solution

Since

$$S_z=\frac{\hbar}{2}\sigma_z,$$

and $\sigma_z\lvert+\rangle_z=\lvert+\rangle_z$,

$$S_z\lvert+\rangle_z =\frac{\hbar}{2}\lvert+\rangle_z.$$