A mathematical framework for quantum information systems

It is well known that quantum mechanics is explained in quantum logic and orthomodular lattices. However, these logic and algebraic structure have not always succeeded in explaining behavior of quantum information systems. A qubit |Ψ>=α|0>+β|1> in quantum information systems is extension of classical concept "bit", where |0> and |1> are basis of a two dimensional quantum system, α and β are probabilistic amplitudes in C (complex numbers). Then, one qubit can have infinite number of values in contrast with classical one bit. In this paper, to analyze various infinite number of quantum states, we establish a discrete algebraic structure as a model of qubit space, which is isomorphic to Kleene algebra 3=< {0, 1/2, 1}, ∼, ∧, ∨ >. Furthermore, we propose weak Kleenean non-additive measures and weak Kleene-Choquet integrals. Then, we show that we can analyze quantum communication channels effectively by the proposed framework.