In quantum mechanics, a pure state and a mixed state are two different descriptions of the state of a quantum system.
A pure state refers to a state in which the quantum system is in a definite and well-defined state. Mathematically, a pure state is represented by a ket vector in a Hilbert space, and it can be described by a superposition of basis states. For example, in the case of a qubit (a two-level quantum system), a pure state can be represented as |ψ⟩ = α|0⟩ + β|1⟩, where α and β are complex probability amplitudes and |0⟩ and |1⟩ are the basis states. The probabilities of measuring the system in the states |0⟩ and |1⟩ are given by |α|^2 and |β|^2, respectively, and they must add up to 1.
On the other hand, a mixed state refers to a statistical ensemble of quantum systems that are not in a pure state. In other words, the system is in a mixed state when we have uncertainty or lack of knowledge about its exact state. Mathematically, a mixed state is represented by a density matrix, which is a positive semi-definite operator. The density matrix contains information about the probabilities of different pure states within the ensemble. A mixed state can be a statistical mixture of different pure states with different probabilities. For example, a mixed state of a qubit can be represented as ρ = p|ψ⟩⟨ψ| + (1 - p)|ϕ⟩⟨ϕ|, where |ψ⟩ and |ϕ⟩ are pure states, and p and (1 - p) are the probabilities associated with each state.
In summary, a pure state represents a quantum system that is in a well-defined state with specific probabilities, while a mixed state represents a statistical ensemble of quantum systems where there is uncertainty or lack of knowledge about the exact state of each individual system.