The wave-particle duality of the qudit-based quantum space demonstrated by the wave-like quantum functionals

Abstract

The wave-particle duality of the qudit-based quantum space is demonstrated by the wave-like quantum functionals and the particle-like quantum states. The quantum functionals are quantum objects generated by the basis qudit functionals, which are the duals of the basis qudit states. The relation between the quantum states and quantum functionals is analogous to the relation between the position and momentum in fundamental quantum physics. In particular, the quantum states and quantum functionals are related by a Fourier transform, and the quantum functionals have wave-like interpretations. The quantum functionals are not just mathematical constructs but have clear physical meanings and quantum circuit realizations. Any arbitrary qudit-based quantum state has dual characters of wave and particle, and its wave character can be evaluated by an observable and realized by a quantum circuit