# Is purification physicaly meaningful?

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 dimention 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 ?

• It is mostly just a mathematical trick, however it is also often argued that purifications can represent non-local information, i.e. entanglement in your system. In AdS/CFT for example, the thermofield double (TFD) state is dual to an eternal black hole, and the TFD state is merely a purification of a thermal state. One can show that the two CFT's are entangled even though there are no local interactions between them. This observation essentially lead to the famous ER = EPR. – Akoben Aug 21 '18 at 15:47
• @Akoben With a bit more elaboration that could be a very good answer! – ACuriousMind Aug 21 '18 at 15:48