Quantum key distribution using Mean King problem with modified measurement schemes

In 2001, Jefferey Bub proposed a quantum key distribution protocol using the Mean King problem. In the protocol, King chooses one of the observables σ_x or σ_z, while Alice is required to measure an observable that solves the Mean King problem for three observables σ_x, σ_y and σ_z. The observable employed by Alice is thus rather complicated. In this paper, we give other simple solutions of the Mean King problem for two observables and examine the quantum key distribution protocols using them. Our result shows that not all the solutions of the Mean King problem are available to construct a secure protocol. In addition, we consider a protocol that is tolerant with three attack scenarios and derive a trade-off inequality between Eve´s information gain and error probability observed by the legitimate users.