Let be a finite-dimensional complex Hilbert 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