On contraction coefficients, partial orders and approximation of capacities for quantum channels

Christoph Hirche1, Cambyse Rouzé2,3, and Daniel Stilck França2,3

1QMATH, Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen, Denmark
2Department of Mathematics, Technische Universität München, 85748 Garching, Germany
3Munich Center for Quantum Science and Technology (MCQST), München, Germany

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Abstract

The data processing inequality is the most basic requirement for any meaningful measure of information. It essentially states that distinguishability measures between states decrease if we apply a quantum channel and is the centerpiece of many results in information theory. Moreover, it justifies the operational interpretation of most entropic quantities. In this work, we revisit the notion of contraction coefficients of quantum channels, which provide sharper and specialized versions of the data processing inequality. A concept closely related to data processing is partial orders on quantum channels. First, we discuss several quantum extensions of the well-known less noisy ordering and relate them to contraction coefficients. We further define approximate versions of the partial orders and show how they can give strengthened and conceptually simple proofs of several results on approximating capacities. Moreover, we investigate the relation to other partial orders in the literature and their properties, particularly with regard to tensorization. We then examine the relation between contraction coefficients with other properties of quantum channels such as hypercontractivity. Next, we extend the framework of contraction coefficients to general f-divergences and prove several structural results. Finally, we consider two important classes of quantum channels, namely Weyl-covariant and bosonic Gaussian channels. For those, we determine new contraction coefficients and relations for various partial orders.

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[2] Samson Wang, Enrico Fontana, M. Cerezo, Kunal Sharma, Akira Sone, Lukasz Cincio, and Patrick J. Coles, "Noise-induced barren plateaus in variational quantum algorithms", Nature Communications 12, 6961 (2021).

[3] Ryuji Takagi, Hiroyasu Tajima, and Mile Gu, "Universal sampling lower bounds for quantum error mitigation", arXiv:2208.09178, (2022).

[4] Giacomo De Palma, Milad Marvian, Cambyse Rouzé, and Daniel Stilck França, "Limitations of Variational Quantum Algorithms: A Quantum Optimal Transport Approach", PRX Quantum 4 1, 010309 (2023).

[5] Abhinav Deshpande, Pradeep Niroula, Oles Shtanko, Alexey V. Gorshkov, Bill Fefferman, and Michael J. Gullans, "Tight Bounds on the Convergence of Noisy Random Circuits to the Uniform Distribution", PRX Quantum 3 4, 040329 (2022).

[6] Supanut Thanasilp, Samson Wang, M. Cerezo, and Zoë Holmes, "Exponential concentration and untrainability in quantum kernel methods", arXiv:2208.11060, (2022).

[7] Christoph Hirche, Cambyse Rouzé, and Daniel Stilck França, "Quantum Differential Privacy: An Information Theory Perspective", arXiv:2202.10717, (2022).

[8] Chi-Fang Chen, Kohtaro Kato, and Fernando G. S. L. Brandão, "Matrix Product Density Operators: when do they have a local parent Hamiltonian?", arXiv:2010.14682, (2020).

[9] Daniel Stilck França and Raul Garcia-Patron, "A game of quantum advantage: linking verification and simulation", arXiv:2011.12173, (2020).

[10] Christoph Hirche and Felix Leditzky, "Bounding quantum capacities via partial orders and complementarity", arXiv:2202.11688, (2022).

[11] Li Gao, Marius Junge, Nicholas LaRacuente, and Haojian Li, "Complete order and relative entropy decay rates", arXiv:2209.11684, (2022).

[12] Daniel Stilck França and Raul Garcia-Patron, "A game of quantum advantage: linking verification and simulation", Quantum 6, 753 (2022).

The above citations are from Crossref's cited-by service (last updated successfully 2023-02-07 03:15:56) and SAO/NASA ADS (last updated successfully 2023-02-07 03:15:57). The list may be incomplete as not all publishers provide suitable and complete citation data.