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

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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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