Consider a quantum system with Hilbert space $\mathscr{H}$ and suppose the quantum state is specified by a density operator $\rho$. Since it is Hermitian, it has a spectral decomposition: $$\rho = \sum p_i |\phi_i\rangle \langle \phi_i |.$$

Now take another quantum system with Hilbert space $\mathscr{H}'$ with dimension at least equal to the first. Take any basis $|\psi_i\rangle$ and consider the state $$|\Psi\rangle = \sum \sqrt{p_i} |\phi_i\rangle \otimes \lvert\psi_i\rangle.$$

A partial trace over the second system yields the first state. This is the purification. A mixed state is always a partial trace of some pure state in a composite system.

There are issues, however: (1) the purification is highly non-unique, any Hilbert space of dimension equal or higher to the first will work, and we can pick any basis we want yielding distinct pure states. (2) this is a mathematical construction. The purifying system seems to have no true meaning physically, this seems to be further implied by the non-uniqueness described in (1).

So is purification a purely mathematical construction with no physical meaning, or it indeed has some physical meaning ? If so, what is the physical meaning of the purification?