Concept in quantum information theory
In quantum information theory and quantum optics, the Schrödinger–HJW theorem is a result about the realization of a mixed state of a quantum system as an ensemble of pure quantum states and the relation between the corresponding purifications of the density operators. The theorem is named after physicists and mathematicians Erwin Schrödinger,[1] Lane P. Hughston, Richard Jozsa and William Wootters.[2] The result was also found independently (albeit partially) by Nicolas Gisin,[3] and by Nicolas Hadjisavvas building upon work by Ed Jaynes,[4][5] while a significant part of it was likewise independently discovered by N. David Mermin.[6] Thanks to its complicated history, it is also known by various other names such as the GHJW theorem,[7] the HJW theorem, and the purification theorem.
Purification of a mixed quantum state
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
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.
References