Purifications
Let be a mixed quantum state on system
. We say that a pure quantum state
on a composite system
is a purification of
if
The following theorem is a basic result in quantum information theory.
Theorem 1 (unitary equivalence of purifications). If and
are two purifications of
then
for some unitary that acts only on system
.
Schmidt decomposition
The above result is usually proved using the following theorem.
Theorem 2 (Schmidt decomposition). Any bipartite quantum state on
can be written as
for some orthonormal bases and
on systems
and
, respectively, and a probability distribution
.
However, standard proofs of Theorem 1 are often based on the following (wrong) argument: given two spectral decompositions of the same matrix, the corresponding eigenvalues and eigenvectors in both decompositions must be equal. This is true only if the given matrix has a non-degenerate spectrum. Otherwise the eigenvectors corresponding to a repeated eigenvalue are not uniquely determined (any orthonormal basis of the corresponding subspace is a valid set of eigenvectors).
A simple proof of Theorem 1
Here is an elementary proof of Theorem 1 which does not run into problems in case of a degenerate spectrum. By Theorem 2 we can write the first state as
Let us expand the first register of the second state in the basis
:
where are some arbitrary (non-normalized) vectors. Then we have
and
By assumption, both partial traces are equal. Since is an orthonormal basis, the coefficients in both expressions must be the same:
This means that are pairwise orthogonal and
(it could happen that
for some
, but this is not a problem). Equivalently, this equation says that we can find an orthonormal basis
such that
. Then we see that
, where
is the unitary change of basis from
to
.