Duality Games and Operational Duality Relations

We give operational meaning to wave-particle duality in terms of discrimination games. Duality arises as a constraint on the probability of winning these games. The games are played with the aid of an $n$-port interferometer, and involve 3 parties, Alice and Bob, who cooperate, and the House, who supervises the game. In one game called ways they attempt to determine the path of a particle in the interferometer. In another, called phases, they attempt to determine which set of known phases have been applied to the different paths. The House determines which game is to be played by flipping a coin. We find a tight wave-particle duality relation that allows us to relate the probabilities of winning these games, and use it to find an upper bound on the probability of winning the combined game. This procedure allows us to express wave-particle duality in terms of discrimination probabilities.

Figure 1

Schematic representation of the games. Operations that are basis dependent, as they meet requirements specific to the path basis, are colored blue.

Figure 2

Various regions involved in Theorem 2, for $n=2$ (union of pale blue, yellow, and orange regions), $n=9$ (union of yellow and orange regions), and $n\to \infty$ (orange region). The region defined by the bound in Ref. [23] coincides with the $n=2$ region.