# Mixed state
**Mixed state** is a quantum state that cannot be described by a single state vector $\lvert\psi\rangle$. It arises when a quantum system is entangled with its environment or when there is classical uncertainty about how the system was prepared. Mixed states are represented by density matrices $\rho$.

A pure state has density matrix $\rho = \lvert\psi\rangle\langle\psi\rvert$ with $\text{tr}(\rho^2) = 1$. A mixed state instead represents a classical probability distribution over pure states. For example, if a qubit was prepared in state $\lvert 0\rangle$ with probability $p$ and in state $\lvert 1\rangle$ with probability $1-p$, its density matrix is:

$$\rho = p\lvert 0\rangle\langle 0\rvert + (1-p)\lvert 1\rangle\langle 1\rvert = \begin{pmatrix} p & 0 \\ 0 & 1-p \end{pmatrix}$$

## Purity
The **purity** of a state $\rho$ is defined as $\gamma = \text{tr}(\rho^2)$ and lies in the range $(0, 1]$. A pure state satisfies $\text{tr}(\rho^2) = 1$, while a maximally mixed state (a uniform distribution over all basis states) satisfies $\text{tr}(\rho^2) = 1/d$ where $d$ is the Hilbert space dimension. For a single qubit, the maximally mixed state has purity $1/2$ and density matrix $I/2$.

$$\rho_{\text{max mixed}} = \frac{1}{2}\begin{pmatrix}1 & 0 \\ 0 & 1\end{pmatrix} = \frac{I}{2}$$

## Bloch sphere interpretation
On the [[bloch-sphere]], pure states correspond to points on the surface where $x^2 + y^2 + z^2 = 1$. Mixed states correspond to points strictly inside the sphere where $x^2 + y^2 + z^2 < 1$. The maximally mixed state corresponds to the origin $(0, 0, 0)$.

## Decoherence
In quantum computing, pure states are required to perform coherent computation. Interaction with the environment causes pure states to become mixed through a process called decoherence. This loss of purity is one of the primary error mechanisms in quantum hardware, which is why quantum computers must be carefully isolated from thermal noise and electromagnetic interference.