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 (albeit partially) by Nicolas Gisin,[3] and 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 by various other names such as the GHJW theorem,[7] the HJW theorem, and the purification theorem.

Purification of a mixed quantum state

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Let   be a finite-dimensional complex 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  .[8]

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.[9] 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

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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, namely, there exists a unitary matrix   such that  .[10] Therefore,  , which means that we can realize the different ensembles of a mixed state just by making different measurements on the purifying system.

References

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  1. ^ 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.
  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. ^ Gisin, N. (1989). “Stochastic quantum dynamics and relativity”, Helvetica Physica Acta 62, 363–371.
  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. ^ Mermin, N. David (1999). "What Do These Correlations Know about Reality? Nonlocality and the Absurd". Foundations of Physics. 29 (4): 571–587. arXiv:quant-ph/9807055. Bibcode:1998quant.ph..7055M. doi:10.1023/A:1018864225930.
  8. ^ Nielsen, Michael A.; Chuang, Isaac L., "The Schmidt decomposition and purifications", Quantum Computation and Quantum Information, Cambridge: Cambridge University Press, pp. 110–111.
  9. ^ Watrous, John (2018). The Theory of Quantum Information. Cambridge: Cambridge University Press. doi:10.1017/9781316848142. ISBN 978-1-107-18056-7.
  10. ^ 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.