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,[1] Lane P. Hughston, Richard Jozsa and William Wootters,[2] though much of it was established decades earlier by Erwin Schrödinger.[3] The result was also found independently 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 as the HJW theorem and the Schrödinger–HJW theorem.

Purification of a mixed quantum state

Consider a mixed state ${\displaystyle \rho =\sum _{i}p_{i}|\phi _{i}\rangle \langle \phi _{i}|}$ 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.

GHJW theorem

Consider a mixed quantum state ${\displaystyle \rho }$ with two different realizations as ensemble of pure states as ${\displaystyle \rho =\sum _{i}p_{i}|\phi _{i}\rangle \langle \phi _{i}|}$ and ${\displaystyle \rho =\sum _{j}q_{j}|\varphi _{j}\rangle \langle \varphi _{j}|}$. 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, ${\displaystyle |\Psi _{SA}^{1}\rangle =\sum _{j}{\sqrt {q_{j}}}|\varphi _{j}\rangle \otimes U_{A}|b_{j}\rangle }$, which means that we can realize the different ensembles of a mixed state just by choosing to measure different observables of one given purification.

References

1. Gisin, N. (1984-05-07). "Quantum Measurements and Stochastic Processes". Physical Review Letters. 52 (19): 1657–1660. Bibcode:1984PhRvL..52.1657G. doi:10.1103/physrevlett.52.1657. ISSN 0031-9007.
2. Hughston, Lane P.; Jozsa, Richard; Wootters, William K. (November 1993). "A complete classification of quantum ensembles having a given density matrix". Physics Letters A. 183 (1): 14–18. Bibcode:1993PhLA..183...14H. doi:10.1016/0375-9601(93)90880-9. ISSN 0375-9601.
3. Schrödinger, Erwin (1936). "Probability relations between separated systems". Proceedings of the Cambridge Philosophical Society. 32 (3): 446–452. Bibcode:1936PCPS...32..446S. doi:10.1017/S0305004100019137.
4. Hadjisavvas, Nicolas (1981). "Properties of mixtures on non-orthogonal states". Letters in Mathematical Physics. 5 (4): 327–332. Bibcode:1981LMaPh...5..327H. doi:10.1007/BF00401481.
5. Jaynes, E. T. (1957). "Information theory and statistical mechanics. II". Physical Review. 108 (2): 171–190. Bibcode:1957PhRv..108..171J. doi:10.1103/PhysRev.108.171.
6. Fuchs, Christopher A. (2011). Coming of Age with Quantum Information: Notes on a Paulian Idea. Cambridge: Cambridge University Press. ISBN 978-0-521-19926-1. OCLC 535491156.
7. Kirkpatrick, K. A. (February 2006). "The Schrödinger-HJW Theorem". Foundations of Physics Letters. 19 (1): 95–102. arXiv:quant-ph/0305068. Bibcode:2006FoPhL..19...95K. doi:10.1007/s10702-006-1852-1. ISSN 0894-9875.