Gisin–Hughston–Jozsa–Wootters theorem

In quantum information theory and quantum optics, the Gisin–Hughston–Jozsa–Wootters (GHJW) 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 Nicolas Gisin, Lane P. Hughston, Richard Jozsa and William Wootters, though much of it was established decades earlier by Erwin Schrödinger. The result was also found independently 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 as the HJW theorem and the Schrödinger–HJW theorem.

Purification of a mixed quantum state

Consider a mixed state [math]\displaystyle{ \rho=\sum_ip_i|\phi_i\rangle\langle\phi_i| }[/math] of the system [math]\displaystyle{ S }[/math], where the states [math]\displaystyle{ |\phi_i\rangle }[/math] are not assumed to be mutually orthogonal. We can add an auxiliary space [math]\displaystyle{ \mathcal{H}_A }[/math] with an orthonormal basis [math]\displaystyle{ \{|a_i\rangle\} }[/math], then the mixed state can be obtained as reduced density operator from the pure bipartite state

[math]\displaystyle{ |\Psi_{SA}\rangle=\sum_i\sqrt{p_i}|\phi_i\rangle|a_i\rangle. }[/math]

More precisely, [math]\displaystyle{ \rho=\mathrm{Tr}_A|\Psi_{SA}\rangle\langle\Psi_{SA}| }[/math]. The state [math]\displaystyle{ |\Psi_{SA}\rangle }[/math] is thus called the purification of [math]\displaystyle{ \rho }[/math]. 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.

GHJW theorem

Consider a mixed quantum state [math]\displaystyle{ \rho }[/math] with two different realizations as ensemble of pure states as [math]\displaystyle{ \rho=\sum_ip_i|\phi_i\rangle\langle\phi_i| }[/math] and [math]\displaystyle{ \rho=\sum_jq_j|\varphi_j\rangle\langle\varphi_j| }[/math]. Here both [math]\displaystyle{ |\phi_i\rangle }[/math]and [math]\displaystyle{ |\varphi_j\rangle }[/math] are not assumed to be mutually orthogonal. There will be two corresponding purifications of the mixed state [math]\displaystyle{ \rho }[/math] reading as follows:

Purification 1: [math]\displaystyle{ |\Psi_{SA}^1\rangle=\sum_i\sqrt{p_i}|\phi_i\rangle|a_i\rangle }[/math];

Purification 2: [math]\displaystyle{ |\Psi_{SA}^2\rangle=\sum_j\sqrt{q_j}|\varphi_j\rangle|b_j\rangle }[/math].

The sets [math]\displaystyle{ \{|a_i\rangle\} }[/math]and [math]\displaystyle{ \{|b_j\rangle\} }[/math] 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 [math]\displaystyle{ U_A }[/math]such that [math]\displaystyle{ |\Psi^1_{SA}\rangle=I\otimes U_A|\Psi^2_{SA}\rangle }[/math]. Therefore, [math]\displaystyle{ |\Psi_{SA}^1\rangle=\sum_j\sqrt{q_j}|\varphi_i\rangle\otimes U_A|b_j\rangle }[/math], which means that we can realize the different ensembles of a mixed state just by choosing to measure different observables of one given purification.