Is purification physically meaningful?

Question

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?

Answer 1

This really depends whether you believe in the "church of the larger Hilbert space". If you feel that pure states are more fundamental than mixed states, then you might argue that any mixed state is just a lack of knowledge, and somewhere out there is the missing piece of the system which will give you full information (i.e., a pure state). Even though you don't know what it is, you know it is out there. (As you can see, this is really more a matter of interpretation of quantum mechanics, since mathematically, the two perspectives are equivalent.)

There are many cases where this is a very reasonable perspective on mixed states regardless of what you believe, e.g. when you have a pure state which becomes mixed by coupling to the environment: While the state of the system looks mixed to you, it has just unitarily interacted with the environment, so the overall state system + environment will be pure, and the environment will hold the purification of your system.

Clearly, this is true more generally as long as your initial global state is pure and you consider part of an isolated sytem (i.e. unitary dynamics).

Beyond that, purifications are of course also a powerful mathematical tool.