# Fidelity of quantum strategies with applications to cryptography

## TL;DR

This paper introduces a new fidelity measure for multi-round quantum strategies, generalizing quantum state fidelity, with applications in cryptography and a semidefinite programming approach.

## Contribution

It defines the strategy fidelity, explores its properties, and applies it to security analysis in quantum cryptographic protocols, offering a general and operational framework.

## Key findings

- Strategy fidelity generalizes quantum state fidelity.
- Provides a semidefinite programming formulation.
- Applications to cryptographic protocol security.

## Abstract

We introduce a definition of the fidelity function for multi-round quantum strategies, which we call the 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.