It’s quite clever to have Dirac notation in QM. For example, the measurement operators are in this ket bra form. It also is connected to the spinor theory where Pauli has these matrices to represent various states on bloch sphere, where gates operations will be invovled.

when it’s not in a pure state, we call it a mixed state. Mixed states are an ensemble of various pure states. All states can be represented with a density matrix.

The purity is measured by the trace of diagonal of the density matrices squared. in pure state, the trace is 1, the maximally mixed state as above example, the purity = 0.5.