Quantum Operation Basics

Quantum operation is the language for changing and reading quantum states. In the simplest single-qubit setting, a state is represented by a normalized vector

\[ |\psi\rangle = \alpha |0\rangle + \beta |1\rangle, \qquad |\alpha|^2 + |\beta|^2 = 1. \]

The coefficients are probability amplitudes. Their squared magnitudes give measurement probabilities in the computational basis:

\[ P(0) = |\alpha|^2, \qquad P(1) = |\beta|^2. \]

Unitary Operations

An ideal closed-system gate is a unitary matrix (U). It maps one normalized state to another normalized state:

\[ |\psi'\rangle = U|\psi\rangle, \qquad U^\dagger U = I. \]

The unitary condition preserves inner products, lengths, and total probability. This is the linear-algebra reason that reversible quantum gates can be drawn as rotations of the state representation.

Measurement Operations

Measurement is not only a rotation. A projective measurement uses operators such as

\[ \Pi_0 = |0\rangle\langle 0|, \qquad \Pi_1 = |1\rangle\langle 1|. \]

The probability of an outcome is computed by applying the corresponding projector:

\[ p_k = \langle \psi|\Pi_k|\psi\rangle. \]

After a measurement, the state is updated according to the observed outcome. That state update is why measurement is treated separately from ordinary gate rotation.

Coordinates and Pictures

The same operation can be described in several coordinate systems. A column vector is convenient for matrix multiplication, spherical coordinates are useful for geometric intuition, and the Bloch sphere gives a compact picture of single-qubit pure states.