Certifying optimality for convex quantum channel optimization problems

Bryan Coutts; Mark Girard; John Watrous

doi:10.22331/q-2021-05-01-448

Certifying optimality for convex quantum channel optimization problems

Bryan Coutts^1,2, Mark Girard¹, and John Watrous^1,2,3

¹Institute for Quantum Computing, University of Waterloo, Canada
²School of Computer Science, University of Waterloo, Canada
³Canadian Institute for Advanced Research, Toronto, Canada

Published:	2021-05-01, volume 5, page 448
Eprint:	arXiv:1810.13295v4
Doi:	https://doi.org/10.22331/q-2021-05-01-448
Citation:	Quantum 5, 448 (2021).

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Abstract

We identify necessary and sufficient conditions for a quantum channel to be optimal for any convex optimization problem in which the optimization is taken over the set of all quantum channels of a fixed size. Optimality conditions for convex optimization problems over the set of all quantum measurements of a given system having a fixed number of measurement outcomes are obtained as a special case. In the case of linear objective functions for measurement optimization problems, our conditions reduce to the well-known Holevo-Yuen-Kennedy-Lax measurement optimality conditions. We illustrate how our conditions can be applied to various state transformation problems having non-linear objective functions based on the fidelity, trace distance, and quantum relative entropy.

► BibTeX data

@article{Coutts2021certifying,
  doi = {10.22331/q-2021-05-01-448},
  url = {https://doi.org/10.22331/q-2021-05-01-448},
  title = {Certifying optimality for convex quantum channel optimization problems},
  author = {Coutts, Bryan and Girard, Mark and Watrous, John},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {5},
  pages = {448},
  month = may,
  year = {2021}
}

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[1] Sam Cree and Jonathan Sorce, "Geometric conditions for saturating the data processing inequality", Journal of Physics A: Mathematical and Theoretical 55 13, 135202 (2022).

[2] Leonardo Banchi, Jason Pereira, Seth Lloyd, and Stefano Pirandola, "Convex optimization of programmable quantum computers", npj Quantum Information 6, 42 (2020).

[3] Álvaro M. Alhambra, Georgios Styliaris, Nayeli A. Rodríguez-Briones, Jamie Sikora, and Eduardo Martín-Martínez, "Fundamental Limitations to Local Energy Extraction in Quantum Systems", Physical Review Letters 123 19, 190601 (2019).

[4] Sam Cree and Jamie Sikora, "A fidelity measure for quantum states based on the matrix geometric mean", arXiv:2006.06918, (2020).

[5] Leonardo Banchi, Jason Pereira, Seth Lloyd, and Stefano Pirandola, "Optimization and learning of quantum programs", arXiv:1905.01318, (2019).

[6] Sam Cree and Jonathan Sorce, "Geometric conditions for saturating the data processing inequality", arXiv:2011.03473, (2020).

[7] Leonid Faybusovich and Cunlu Zhou, "Long-Step Path-Following Algorithm for Quantum Information Theory: Some Numerical Aspects and Applications", arXiv:1906.00037, (2019).

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