Quantum States and the Bloch Sphere
Quantum States and the Bloch Sphere
A qubit — the quantum analogue of a classical bit — can be in a superposition of $|0\rangle$ and $|1\rangle$. The most general single-qubit pure state is:
$$|\psi\rangle = \cos\!\frac{\theta}{2}|0\rangle + e^{i\phi}\sin\!\frac{\theta}{2}|1\rangle$$
where $\theta \in [0, \pi]$ is the polar angle and $\phi \in [0, 2\pi)$ is the azimuthal phase. This state corresponds to a unique point on the Bloch sphere — a unit sphere whose north pole is $|0\rangle$ and south pole is $|1\rangle$.
For the state above, the spin expectation values are:
$$\langle\hat{S}_x\rangle = \frac{\hbar}{2}\sin\theta\cos\phi, \qquad \langle\hat{S}_y\rangle = \frac{\hbar}{2}\sin\theta\sin\phi, \qquad \langle\hat{S}_z\rangle = \frac{\hbar}{2}\cos\theta$$
The Bloch vector $(\sin\theta\cos\phi,\, \sin\theta\sin\phi,\, \cos\theta)$ points from the center of the sphere to the state.
Quantum Gates
Quantum gates are unitary operations that rotate the Bloch vector. The six Pauli-based gates are:
| Gate | Matrix | Effect |
|---|---|---|
| H (Hadamard) | $\frac{1}{\sqrt{2}}\begin{pmatrix}1&1\\1&-1\end{pmatrix}$ | $|0\rangle\leftrightarrow|+\rangle$ |
| X (NOT) | $\begin{pmatrix}0&1\\1&0\end{pmatrix}$ | Bit flip: $|0\rangle\leftrightarrow|1\rangle$ |
| Z | $\begin{pmatrix}1&0\\0&-1\end{pmatrix}$ | Phase flip |
| S | $\begin{pmatrix}1&0\\0&i\end{pmatrix}$ | $\pi/2$ phase rotation |
| T | $\begin{pmatrix}1&0\\0&e^{i\pi/4}\end{pmatrix}$ | $\pi/4$ phase rotation |
| R_z(\theta) | $\begin{pmatrix}e^{-i\theta/2}&0\\0&e^{i\theta/2}\end{pmatrix}$ | Z-axis rotation |
Density Matrix and Mixed States
When a qubit interacts with its environment (decoherence), it transitions from a pure state to a mixed state, described by a $2\times2$ density matrix $\rho$. The key quantities are:
- Purity: $\text{Tr}(\rho^2) = \frac{1+r^2}{2}$ where $r = |\mathbf{r}|$ is the Bloch vector length (1 = pure, 0.5 = maximally mixed)
- Von Neumann entropy: $S = -\text{Tr}(\rho \log_2 \rho)$ (0 for pure, 1 bit for maximally mixed)
- Coherence: $|\rho_{01}|$ — off-diagonal magnitude