Certifying optimality for convex quantum channel optimization problems

Bryan Coutts1,2, Mark Girard1, and John Watrous1,2,3

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

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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.

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

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

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

[4] Álvaro M. Alhambra, Georgios Styliaris, Nayeli A. Rodriguez-Briones, Jamie Sikora, and Eduardo Martin-Martinez, "Fundamental limitations to local energy extraction in quantum systems", arXiv:1902.02357.

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

[6] Samuel S. Cree and Jonathan Sorce, "Geometric conditions for saturating the data processing inequality", arXiv:2011.03473.

The above citations are from SAO/NASA ADS (last updated successfully 2021-05-06 19:41:26). The list may be incomplete as not all publishers provide suitable and complete citation data.

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