Pure state is a quantum state that is completely described by a single state vector $\lvert\psi\rangle$. It carries the maximum possible information about a quantum system. When information about the system leaks into the environment the state becomes mixed — this process is called decoherence, and it is one of the primary challenges in quantum computing.

Classical bits can be 0 or 1, encoded as voltages such as $0\,\text{V}$ and $5\,\text{V}$. These voltages are not perfectly precise — they fluctuate due to thermal noise, electromagnetic interference, and current leakage — but digital logic is designed to be resilient to such fluctuations. A pure qubit state $\lvert\psi\rangle = a\lvert 0\rangle + b\lvert 1\rangle$ is far more fragile. Any interaction with the environment perturbs the complex amplitudes $a$ and $b$, entangling the qubit with environmental degrees of freedom and converting the pure state into a mixed one.

Any pure state $\lvert\psi\rangle$ can be equivalently represented as a density matrix $\rho = \lvert\psi\rangle\langle\psi\rvert$. Pure states are exactly the states whose density matrix satisfies $\text{tr}(\rho^2) = 1$. On the Bloch sphere, pure states correspond to points on the surface of the unit sphere (the pure states are the ones “on” the sphere, while mixed states are “inside” the sphere).

$$\rho = \lvert\psi\rangle\langle\psi\rvert, \qquad \text{tr}(\rho^2) = 1$$