Let be a
finite-dimensionalcomplexHilbert space, and consider a generic (possibly
mixed)
quantum state defined on and admitting a decomposition of the form
for a collection of (not necessarily mutually orthogonal) states and coefficients such that . Note that any quantum state can be written in such a way for some and .[8]
Any such can be purified, that is, represented as the
partial trace of a
pure state defined in a larger Hilbert space. More precisely, it is always possible to find a (finite-dimensional) Hilbert space and a pure state such that . Furthermore, the states satisfying this are all and only those of the form
for some orthonormal basis . The state is then referred to as the "purification of ". Since the auxiliary space and the basis can be chosen arbitrarily, the purification of a mixed state is not unique; in fact, there are infinitely many purifications of a given mixed state.[9] Because all of them admit a decomposition in the form given above, given any pair of purifications , there is always some unitary operation such that
Theorem
Consider a mixed quantum state with two different realizations as ensemble of pure states as and . Here both and are not assumed to be mutually orthogonal. There will be two corresponding purifications of the mixed state reading as follows:
Purification 1: ;
Purification 2: .
The sets and are two collections of orthonormal bases of the respective auxiliary spaces. These two purifications only differ by a unitary transformation acting on the auxiliary space, namely, there exists a unitary matrix such that .[10] Therefore, , which means that we can realize the different ensembles of a mixed state just by making different measurements on the purifying system.