Schrödinger–HJW theorem

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 by Nicolas Hadjisavvas building upon work by Ed Jaynes,^[3][4] while a significant part of it was likewise independently discovered by N. David Mermin.^[5] Thanks to its complicated history, it is also known by various other names such as the GHJW theorem,^[6] the HJW theorem, and the purification theorem.

Purification of a mixed quantum state[]

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

${\displaystyle |\Psi _{SA}\rangle =\sum _{i}{\sqrt {p_{i}}}|\phi _{i}\rangle |a_{i}\rangle .}$

More precisely, ${\displaystyle \rho =\mathrm {Tr} _{A}|\Psi _{SA}\rangle \langle \Psi _{SA}|}$. The state ${\displaystyle |\Psi _{SA}\rangle }$ is thus called the purification of ${\displaystyle \rho }$. 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.

Schrödinger-HJW theorem[]

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

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

Purification 2: ${\displaystyle |\Psi _{SA}^{2}\rangle =\sum _{j}{\sqrt {q_{j}}}|\varphi _{j}\rangle |b_{j}\rangle }$.

The sets ${\displaystyle \{|a_{i}\rangle \}}$and ${\displaystyle \{|b_{j}\rangle \}}$ 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 ${\displaystyle U_{A}}$such that ${\displaystyle |\Psi _{SA}^{1}\rangle =(I\otimes U_{A})|\Psi _{SA}^{2}\rangle }$.^[7] Therefore, ${\displaystyle \textstyle |\Psi _{SA}^{1}\rangle =\sum _{j}{\sqrt {q_{j}}}|\varphi _{j}\rangle \otimes U_{A}|b_{j}\rangle }$, which means that we can realize the different ensembles of a mixed state just by making different measurements on the purifying system.

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