Approximating quantum channels by completely positive maps with small Kraus rank

Cécilia Lancien; Andreas Winter

doi:10.22331/q-2024-04-30-1320

Approximating quantum channels by completely positive maps with small Kraus rank

Cécilia Lancien¹ and Andreas Winter^2,3

¹Institut Fourier & CNRS, Université Grenoble Alpes, 38610 Gières, France
²Departament de Física: Grup d’Informació Quàntica, Universitat Autònoma de Barcelona, 08193 Bellaterra (Barcelona), Spain
³Institució Catalana de Recerca i Estudis Avançats, 08010 Barcelona, Spain

Published:	2024-04-30, volume 8, page 1320
Eprint:	arXiv:1711.00697v3
Doi:	https://doi.org/10.22331/q-2024-04-30-1320
Citation:	Quantum 8, 1320 (2024).

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.

Abstract

We study the problem of approximating a quantum channel by one with as few Kraus operators as possible (in the sense that, for any input state, the output states of the two channels should be close to one another). Our main result is that any quantum channel mapping states on some input Hilbert space $\mathrm{A}$ to states on some output Hilbert space $\mathrm{B}$ can be compressed into one with order $d\log d$ Kraus operators, where $d=\max(|\mathrm{A}|,|\mathrm{B}|)$, hence much less than $|\mathrm{A}||\mathrm{B}|$. In the case where the channel's outputs are all very mixed, this can be improved to order $d$. We discuss the optimality of this result as well as some consequences.

► BibTeX data

@article{Lancien2024approximating,
  doi = {10.22331/q-2024-04-30-1320},
  url = {https://doi.org/10.22331/q-2024-04-30-1320},
  title = {Approximating quantum channels by completely positive maps with small {K}raus rank},
  author = {Lancien, C{\'{e}}cilia and Winter, Andreas},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {8},
  pages = {1320},
  month = apr,
  year = {2024}
}

► References

[1] Dorit Aharonov, Alexei Kitaev, Noam Nisan. Quantum circuits with mixed states. In Proc. of the 13th Annual ACM Symposium on Theory of Computing (STOC'98), 20–30, 1998.
https://doi.org/10.1145/276698.276708

[2] Guillaume Aubrun. On almost randomizing channels with a short Kraus decomposition. Commun. Math. Phys. 288(3):1103–1116, 2009.
https://doi.org/10.1007/s00220-008-0695-y

[3] Koenraad M.R. Audenaert. A sharp Fannes-type inequality for the von Neumann entropy. J. Phys. A 40(28):8127–8136, 2007.
https://doi.org/10.1088/1751-8113/40/28/S18

[4] Fernando G.S.L. Brandão, Aram W. Harrow, Michał Horodecki. Local random quantum circuits are approximate polynomial-designs. Commun. Math. Phys. 346(2):397–434, 2016.
https://doi.org/10.1007/s00220-016-2706-8

[5] Djalil Chafaï, Olivier Guédon, Guillaume Lecué, Alain Pajor. Interactions between compressed sensing, random matrices and high dimensional geometry. Panoramas et synthèse 37, Société Mathématique de France, 2012. Pdf available here.
https://lecueguillaume.github.io/assets/CSbook.pdf

[6] Man-Duen Choi. Completely positive linear maps on complex matrices. Lin. Alg. Appl. 10(3):285–290, 1975.
https://doi.org/10.1016/0024-3795(75)90075-0

[7] Vittorio Giovannetti, Alexander S. Holevo. Quantum channels and their entropic characteristics. Rep. Prog. Phys. 75(046001), 2012.
https://doi.org/10.1088/0034-4885/75/4/046001

[8] Patrick Hayden, Debbie W. Leung, Peter W. Shor, Andreas Winter. Randomizing quantum states: Constructions and applications. Commun. Math. Phys. 250(2):371–391, 2004.
https://doi.org/10.1007/s00220-004-1087-6

[9] Cécilia Lancien, Andreas Winter. Distinguishing multi-partite states by local measurements. Commun. Math. Phys. 323(2):555–573, 2013.
https://doi.org/10.1007/s00220-013-1779-x

[10] Gilles Pisier. The Volume of Convex Bodies and Banach Spaces Geometry. Cambridge Tracts in Mathematics 94, Cambridge University Press, Cambridge, 1989.
https://doi.org/10.1017/CBO9780511662454

[11] William F. Stinespring. Positive functions on C*-algebras. In Proc. of the American Mathematical Society 6(2):211–216, 1955.
https://doi.org/10.2307/2032342

[12] Andreas Winter. Tight uniform continuity bounds for quantum entropies: conditional entropy, relative entropy distance and energy constraints. Commun. Math. Phys. 347(1):291–313, 2016.
https://doi.org/10.1007/s00220-016-2609-8

Cited by

[1] Bin Yan and Nikolai A. Sinitsyn, "Randomized channel-state duality", arXiv:2210.03723, (2022).

[2] Cécilia Lancien and Christian Majenz, "Weak approximate unitary designs and applications to quantum encryption", Quantum 4, 313 (2020).

[3] Hachem Kadri, Stéphane Ayache, Riikka Huusari, Alain Rakotomamonjy, and Liva Ralaivola, "Partial Trace Regression and Low-Rank Kraus Decomposition", arXiv:2007.00935, (2020).

[4] Anne Broadbent, Carlos E. González-Guillén, and Christine Schuknecht, "Quantum private broadcasting", Physical Review A 105 2, 022606 (2022).

The above citations are from SAO/NASA ADS (last updated successfully 2024-05-17 01:18:37). The list may be incomplete as not all publishers provide suitable and complete citation data.

On Crossref's cited-by service no data on citing works was found (last attempt 2024-05-17 01:18:36).

This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.