# Pure state
**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-state|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.

## Density matrix representation
Any pure state $\lvert\psi\rangle$ can be equivalently represented as a [[density-matrix|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).

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