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

Qudits are the fundamental building blocks of quantum information and quantum computation – usually the 2-level qubits are used, but recently there are growing interests in generalizing to higher d-level qudit systems. In this work, we propose the wave-particle duality of the qudit-based quantum space and the wave-like quantum functionals are new fundamental physical realities that have not been previously considered. The theory was developed by recognizing the deep connection between the usual momentum-position duality lying at the foundation of quantum physics, and the new duality between the qudit functionals and qudit states – both are examples of the mathematical concept of Pontryagin duality. The quantum functionals are quantum objects generated by the basis qudit functionals, which are the duals of the basis qudit states. The quantum states and quantum functionals are related by a Fourier transform and an entropic uncertainty principle can be defined between the dual representations. The quantum functionals are not just mathematical constructs but have clear physical meanings and quantum circuit realizations. Connecting the partition interpretation of the quantum functionals to the effects of quantum gates may allow systematic understanding of quantum circuits.