The duality of a single particle with an n-dimensional internal degree of freedom

The wave–particle duality of a single particle with an n-dimensional internal degree of freedom is re-examined theoretically in a Mach–Zehnder interferometer. The famous duality relation D²+V² ≤ 1 is always valid in this situation, where D is the distinguishability and V is the visibility. However, the sum of the particle information and the wave information, D²+V², can be smaller than one for the input of a pure state if this initial pure state includes the internal degree of freedom of the particle, while the quantity D²+V² is always equal to one when the internal degree of freedom of the particle is excluded.