Analysis and optimization of quantum adaptive measurement protocols with the framework of preparation games

A preparation game is a task whereby a player sequentially sends a number of quantum states to a referee, who probes each of them and announces the measurement result. Many experimental tasks in quantum information, such as entanglement quantification or magic state detection, can be cast as preparation games. In this paper, we introduce general methods to design n-round preparation games, with tight bounds on the performance achievable by players with arbitrarily constrained preparation devices. We illustrate our results by devising new adaptive measurement protocols for entanglement detection and quantification. Surprisingly, we find that the standard procedure in entanglement detection, namely, estimating n times the average value of a given entanglement witness, is in general suboptimal for detecting the entanglement of a specific quantum state. On the contrary, there exist n-round experimental scenarios where detecting the entanglement of a known state optimally requires adaptive measurement schemes.