In
quantum information theory and
quantum optics
Quantum optics is a branch of atomic, molecular, and optical physics dealing with how individual quanta of light, known as photons, interact with atoms and molecules. It includes the study of the particle-like properties of photons. Photons have ...
, 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
Erwin Rudolf Josef Alexander Schrödinger (, ; ; 12 August 1887 – 4 January 1961), sometimes written as or , was a Nobel Prize-winning Austrian physicist with Irish citizenship who developed a number of fundamental results in quantum theo ...
,
Lane P. Hughston,
Richard Jozsa and
William Wootters
William "Bill" Kent Wootters () is an American theoretical physicist, and one of the founders of the field of quantum information theory. In a 1982 joint paper with Wojciech H. Zurek, Wootters proved the no cloning theorem, at the same time as D ...
. The result was also found independently (albeit partially) by
Nicolas Gisin, and by Nicolas Hadjisavvas building upon work by
Ed Jaynes, while a significant part of it was likewise independently discovered by
N. David Mermin. Thanks to its complicated history, it is also known by various other names such as the GHJW theorem, the HJW theorem, and the purification theorem.
Purification of a mixed quantum state
Let
be a finite-dimensional 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
.
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. 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, viz., there exists a unitary matrix
such that
.
Therefore,
, which means that we can realize the different ensembles of a mixed state just by making different measurements on the purifying system.
References
{{DEFAULTSORT:Schrodinger-HJW theorem
Quantum information theory