# Fidelity of quantum strategies with applications to cryptography

Gus Gutoski1, Ansis Rosmanis2,3, and Jamie Sikora2,4

1Perimeter Institute for Theoretical Physics, ON, Canada
2Centre for Quantum Technologies, National University of Singapore, Singapore
3School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore
4MajuLab, CNRS-UNS-NUS-NTU International Joint Research Unit, UMI 3654, Singapore

### Abstract

We introduce a definition of the fidelity function for multi-round quantum strategies, which we call the $\textit{strategy fidelity}$, that is a generalization of the fidelity function for quantum states. We provide many properties of the strategy fidelity including a Fuchs-van de Graaf relationship with the strategy norm. We also provide a general monotinicity result for both the strategy fidelity and strategy norm under the actions of strategy-to-strategy linear maps. We illustrate an operational interpretation of the strategy fidelity in the spirit of Uhlmann's Theorem and discuss its application to the security analysis of quantum protocols for interactive cryptographic tasks such as bit-commitment and oblivious string transfer. Our analysis is general in the sense that the actions of the protocol need not be fully specified, which is in stark contrast to most other security proofs. Lastly, we provide a semidefinite programming formulation of the strategy fidelity.

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[3] Xin Wang and Mark M. Wilde, "Resource theory of asymmetric distinguishability for quantum channels", Physical Review Research 1 3, 033169 (2019).

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[10] Gilad Gour and Carlo Maria Scandolo, "Dynamical Entanglement", Physical Review Letters 125 18, 180505 (2020).

[11] Mark Girard, Martin Plávala, and Jamie Sikora, "Jordan products of quantum channels and their compatibility", Nature Communications 12 1, 2129 (2021).

[12] Manuel B. Santos, Paulo Mateus, and Armando N. Pinto, "Quantum Oblivious Transfer: A Short Review", Entropy 24 7, 945 (2022).

[13] Graeme D. Berk, Andrew J. P. Garner, Benjamin Yadin, Kavan Modi, and Felix A. Pollock, "Resource theories of multi-time processes: A window into quantum non-Markovianity", arXiv:1907.07003, (2019).

[14] Guang Ping He, "Unconditionally secure quantum bit commitment based on the uncertainty principle", Proceedings of the Royal Society of London Series A 475 2222, 20180543 (2019).

[15] Guang Ping He, "An optical implementation of quantum bit commitment using infinite-dimensional systems", arXiv:1909.09865, (2019).

The above citations are from Crossref's cited-by service (last updated successfully 2023-02-03 19:54:48) and SAO/NASA ADS (last updated successfully 2023-02-03 19:54:49). The list may be incomplete as not all publishers provide suitable and complete citation data.