Pure state is a state that completely encodes the information about a quantum system. If a state is pure it can be represented by a state vector $\lvert\psi\rangle$ in Hilbert space $\mathcal{H}$. If this information leaks into the environment the state is called “mixed”. The state vector alone is no longer sufficient to describe the quantum system and is upgraded $\rho$. This is why the environment is a major enemy of quantum computing. If we consider the Hilbert space that combines both the environment quantum system we're looking at $\mathcal{H}\otimex\mathcal{H}_\text{Env}$

Consider a classical bit: