# Gisin-Hughston-Jozsa-Wootters theorem

Jump to navigation Jump to search

In quantum information theory and quantum optics, Gisin-Hughston-Jozsa-Wootters (GHJW) theorem is a result about the realization of a mixed state of the quantum system as an ensemble of pure quantum states and the relation between the corresponding purification of the density operators. The theorem is named after physicists and mathematicians Nicolas Gisin, Lane P. Hughston, Richard Jozsa and William Wootters.

## Purification of a mixed quantum state

Consider a mixed state $\rho =\sum _{i}p_{i}|\phi _{i}\rangle \langle \phi _{i}|$ of the system $S$ , where the states $|\phi _{i}\rangle$ are not assumed to be mutually orthogonal. We can add an auxiliary space ${\mathcal {H}}_{A}$ with an orthonormal basis $\{|a_{i}\rangle \}$ , then the mixed state can be obtained as reduced density operator from the pure bipartite state

$|\Psi _{SA}\rangle =\sum _{i}{\sqrt {p_{i}}}|\phi _{i}\rangle |a_{i}\rangle .$ More precisely, $\rho =\mathrm {Tr} _{A}|\Psi _{SA}\rangle \langle \Psi _{SA}|$ . The state $|\Psi _{SA}\rangle$ is thus called the purification of $\rho$ . 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.

## Details of the Gisin-Hughston-Jozsa-Wootters (GHJW) theorem

Consider a mixed quantum state $\rho$ with two different realizations as ensemble of pure states as $\rho =\sum _{i}p_{i}|\phi _{i}\rangle \langle \phi _{i}|$ and $\rho =\sum _{j}q_{j}|\varphi _{j}\rangle \langle \varphi _{j}|$ , note that here both $|\phi _{i}\rangle$ and $|\varphi _{j}\rangle$ are not assumed to be mutually orthogonal. There will be two corresponding purifications of the mixed state $\rho$ reading

Purification 1: $|\Psi _{SA}^{1}\rangle =\sum _{i}{\sqrt {p_{i}}}|\phi _{i}\rangle |a_{i}\rangle$ ;

Purification 2: $|\Psi _{SA}^{2}\rangle =\sum _{j}{\sqrt {q_{j}}}|\varphi _{j}\rangle |b_{j}\rangle$ ;where $\{|a_{i}\rangle \}$ and $\{|b_{j}\rangle \}$ are two collections of orthonormal bases of the respective auxiliary spaces. It is natural to ask that what is the relation between these two purifications. The answer is that they only differ by a unitary transformation acting on the auxiliary space, viz., there exists a unitary matrix $U_{A}$ such that $|\Psi _{SA}^{1}\rangle =I\otimes U_{A}|\Psi _{SA}^{2}\rangle$ . Therefore, $|\Psi _{SA}^{1}\rangle =\sum _{j}{\sqrt {q_{j}}}|\varphi _{i}\rangle \otimes U_{A}|b_{j}\rangle$ , where means that we can realize the different ensembles of a mixed state just by choosing to measure different observables of one given purification, this result is known as GHJW theorem.