Aja viide. Aja lugu.

In quantum information theory, quantum state purification refers to the process of representing a mixed state as a pure quantum state of higher-dimensional Hilbert space. The purification allows the original mixed state to be recovered by taking the partial trace over the additional degrees of freedom. The purification is not unique, the different purifications that can lead to the same mixed states are limited by the Schrödinger–HJW theorem.

Purification is used in algorithms such as entanglement distillation, magic state distillation and algorithmic cooling.

Description

Let ${\mathcal {H}}_{S}$ be a finite-dimensional complex Hilbert space, and consider a generic (possibly mixed) quantum state $\rho$ defined on ${\mathcal {H}}_{S}$ and admitting a decomposition of the form $\rho =\sum _{i}p_{i}|\phi _{i}\rangle \langle \phi _{i}|$ for a collection of (not necessarily mutually orthogonal) states $|\phi _{i}\rangle \in {\mathcal {H}}_{S}$ and coefficients $p_{i}\geq 0$ such that ${\textstyle \sum _{i}p_{i}=1}$ . Note that any quantum state can be written in such a way for some $\{|\phi _{i}\rangle \}_{i}$ and $\{p_{i}\}_{i}$ .^[1]

Any such $\rho$ 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 ${\mathcal {H}}_{A}$ and a pure state $|\Psi _{SA}\rangle \in {\mathcal {H}}_{S}\otimes {\mathcal {H}}_{A}$ such that $\rho =\operatorname {Tr} _{A}{\big (}|\Psi _{SA}\rangle \langle \Psi _{SA}|{\big )}$ . Furthermore, the states $|\Psi _{SA}\rangle$ satisfying this are all and only those of the form $|\Psi _{SA}\rangle =\sum _{i}{\sqrt {p_{i}}}|\phi _{i}\rangle \otimes |a_{i}\rangle$ for some orthonormal basis $\{|a_{i}\rangle \}_{i}\subset {\mathcal {H}}_{A}$ . The state $|\Psi _{SA}\rangle$ is then referred to as 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.^[2] Because all of them admit a decomposition in the form given above, given any pair of purifications $|\Psi \rangle ,|\Psi '\rangle \in {\mathcal {H}}_{S}\otimes {\mathcal {H}}_{A}$ , there is always some unitary operation $U:{\mathcal {H}}_{A}\to {\mathcal {H}}_{A}$ such that $|\Psi '\rangle =(I\otimes U)|\Psi \rangle .$

Schrödinger–HJW theorem

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 Erwin Schrödinger who proved it in 1936,^[3] and after Lane P. Hughston, Richard Jozsa and William Wootters who rediscovered in 1993.^[4] The result was also found independently (albeit partially) by Nicolas Gisin in 1989,^[5] and by Nicolas Hadjisavvas building upon work by E. T. Jaynes of 1957,^[6]^[7] while a significant part of it was likewise independently discovered by N. David Mermin in 1999 who discovered the link with Schrödinger's work.^[8] Thanks to its complicated history, it is also known by various other names such as the GHJW theorem,^[9] the HJW theorem, and the purification theorem.

Consider a mixed quantum state $\rho$ with two different realizations as ensemble of pure states as ${\textstyle \rho =\sum _{i}p_{i}|\phi _{i}\rangle \langle \phi _{i}|}$ and ${\textstyle \rho =\sum _{j}q_{j}|\varphi _{j}\rangle \langle \varphi _{j}|}$ . 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 as follows:

Purification 1:

|\Psi _{SA}^{1}\rangle =\sum _{i}{\sqrt {p_{i}}}|\phi _{i}\rangle \otimes |a_{i}\rangle

;

Purification 2:

|\Psi _{SA}^{2}\rangle =\sum _{j}{\sqrt {q_{j}}}|\varphi _{j}\rangle \otimes |b_{j}\rangle

.

The sets $\{|a_{i}\rangle \}$ and $\{|b_{j}\rangle \}$ 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, namely, there exists a unitary matrix $U_{A}$ such that $|\Psi _{SA}^{1}\rangle =(I\otimes U_{A})|\Psi _{SA}^{2}\rangle$ .^[10] Therefore, ${\textstyle |\Psi _{SA}^{1}\rangle =\sum _{j}{\sqrt {q_{j}}}|\varphi _{j}\rangle \otimes U_{A}|b_{j}\rangle }$ , which means that we can realize the different ensembles of a mixed state just by making different measurements on the purifying system.

References

^ Nielsen, Michael A.; Chuang, Isaac L., "The Schmidt decomposition and purifications", Quantum Computation and Quantum Information, Cambridge: Cambridge University Press, pp. 110–111.
^ Watrous, John (2018). The Theory of Quantum Information. Cambridge: Cambridge University Press. doi:10.1017/9781316848142. ISBN 978-1-107-18056-7.
^ Schrödinger, Erwin (1936). "Probability relations between separated systems". Mathematical Proceedings of the Cambridge Philosophical Society. 32 (3): 446–452. Bibcode:1936PCPS...32..446S. doi:10.1017/S0305004100019137.
^ 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.
^ Gisin, N. (1989). “Stochastic quantum dynamics and relativity”, Helvetica Physica Acta 62, 363–371.
^ 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.
^ 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.
^ 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.
^ 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.
^ 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.

[1] Nielsen, Michael A.; Chuang, Isaac L., "The Schmidt decomposition and purifications", Quantum Computation and Quantum Information, Cambridge: Cambridge University Press, pp. 110–111.

[2] Watrous, John (2018). The Theory of Quantum Information. Cambridge: Cambridge University Press. doi:10.1017/9781316848142. ISBN 978-1-107-18056-7.

[3] Schrödinger, Erwin (1936). "Probability relations between separated systems". Mathematical Proceedings of the Cambridge Philosophical Society. 32 (3): 446–452. Bibcode:1936PCPS...32..446S. doi:10.1017/S0305004100019137.

[4] 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.

[5] Gisin, N. (1989). “Stochastic quantum dynamics and relativity”, Helvetica Physica Acta 62, 363–371.

[6] 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.

[7] 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.

[8] 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.

[9] 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.

[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.

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[10]

Aja viide. Aja lugu.

Värsked postitused

Most Used Categories

Helsingi kõrval on piirkond, kus krunte ei taheta isegi tasuta ja politsei alustas tõhustatud valvet

Soome saabub külm ilm – nii kaua see kestab

Tartu Tamme gümnaasium hakkab pakkuma veterinaaria õppesuunda

Trumpi lähiringkond ei soovita tal enne nõusolekut rahule Putiniga suhelda

Neid asju nõuab Venemaa Soomelt suhete taastamiseks

VIDEO: poeg tervitab rindelt saabunud isa

PILTUUDIS: Venemaal on hakatud kortermaju droonirünnakute eest võrkudesse mässima

Soome president: keegi Euroopa juhtidest peab rääkima Putiniga

Viipekeelsed uudised

Kaja Kallas

Description

Schrödinger–HJW theorem

References