Quantum States & the Bloch Sphere

A classical bit is the simplest unit of information: it is either $0$ or $1$. A qubit (quantum bit) is a two-level quantum system, and its state is described by a vector in a two-dimensional complex Hilbert space.

The general state of a qubit can be written as a superposition of the computational basis states $|0\rangle$ and $|1\rangle$:

• • The state vector $|\psi\rangle = \alpha|0\rangle + \beta|1\rangle$ with $\alpha, \beta \in \mathbb{C}$.
• • Normalization requires $|\alpha|^2 + |\beta|^2 = 1$.
• • Global phase is physically irrelevant: $|\psi\rangle$ and $e^{i\gamma}|\psi\rangle$ represent the same physical state.
• • The computational basis $\{|0\rangle, |1\rangle\}$ is the standard orthonormal basis for a single qubit.

Because the global phase is irrelevant and the norm is fixed, a single qubit has only two real degrees of freedom. These can be parameterized by two angles, $\theta$ and $\phi$, giving the Bloch sphere representation.

Every pure state of a single qubit corresponds to a unique point on the surface of the unit sphere. The north pole represents $|0\rangle$, the south pole represents $|1\rangle$, and the equator contains equal superpositions.

• • Parameterization: $\alpha = \cos(\theta/2)$ and $\beta = e^{i\phi}\sin(\theta/2)$.
• • $\theta \in [0, \pi]$ is the polar angle from the $z$-axis.
• • $\phi \in [0, 2\pi)$ is the azimuthal angle in the $xy$-plane.
• • The state on the Bloch sphere is $|\psi\rangle = \cos\frac{\theta}{2}|0\rangle + e^{i\phi}\sin\frac{\theta}{2}|1\rangle$.

When a quantum system is in a definite state $|\psi\rangle$, we call it a pure state. The density matrix (or density operator) provides a unified way to describe both pure and mixed states.

For a pure state, the density matrix is $\rho = |\psi\rangle\langle\psi|$. For a mixed state, the system is in a probabilistic mixture of pure states, and the density matrix becomes a convex combination.

• • Pure state: $\rho = |\psi\rangle\langle\psi|$.
• • Hermitian: $\rho = \rho^\dagger$.
• • Unit trace: $\text{tr}(\rho) = 1$.
• • Pure state idempotent: $\rho^2 = \rho$.
• • Mixed state: $\rho = \sum_i p_i |\psi_i\rangle\langle\psi_i|$ with $\sum_i p_i = 1$.

Any single-qubit density matrix can be written in terms of the Pauli matrices and the Bloch vector $\vec{r} = (r_x, r_y, r_z)$. This provides a direct geometric interpretation: pure states live on the surface of the Bloch sphere, while mixed states live in the interior.

The length of the Bloch vector quantifies the purity of the state. A maximally mixed state sits at the origin and carries no directional information.

• • General form: $\rho = \frac{1}{2}(I + \vec{r} \cdot \vec{\sigma})$.
• • For pure states: $|\vec{r}| = 1$.
• • For mixed states: $|\vec{r}| < 1$.
• • Maximally mixed state: $\rho = \frac{I}{2}$, with $\vec{r} = \vec{0}$.

When dealing with composite systems, we often need to describe the state of a subsystem alone. The partial trace is the operation that removes degrees of freedom of one subsystem to yield the reduced density matrix of the other.

Consider a bipartite system $AB$ in a state $\rho_{AB}$. The reduced density matrix for subsystem $A$ is obtained by tracing out subsystem $B$: $\rho_A = \text{tr}_B(\rho_{AB})$.

• • Composite pure state: $|\psi\rangle_{AB}$.
• • Reduced density matrix: $\rho_A = \text{tr}_B(\rho_{AB})$.
• • Tracing out one qubit of a Bell state yields the maximally mixed state $\frac{I}{2}$.
• • The partial trace is the unique operation consistent with measuring observables on a subsystem.

Qubits are not just mathematical abstractions; they are realized experimentally using a variety of two-level quantum systems. Each platform has its own trade-offs in coherence time, gate fidelity, and scalability.

While a deep discussion of hardware is beyond this module, it is useful to know the leading candidates and what physical degrees of freedom encode the $|0\rangle$ and $|1\rangle$ states.

• • Nuclear spins: the spin-up and spin-down states of a nucleus in a magnetic field.
• • Photon polarization: horizontal and vertical polarizations of a single photon.
• • Superconducting qubits: quantized energy levels in an anharmonic LC circuit (e.g., transmon).
• • Trapped ions: hyperfine or Zeeman sublevels of an ion confined in an electromagnetic trap.