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 |/spl Psi/>=/spl alpha/|0>+/spl beta/|1> in quantum information systems is extension of classical concept "bit", where |0> and |1> are basis of a two dimensional quantum system, /spl alpha/ and /spl beta/ 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}, /spl sim/, /spl and/, /spl or/ >. 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.
[1]
Janusz Kacprzyk,et al.
Beyond two: theory and applications of multiple-valued logic
,
2003
.
[2]
Jozef Gruska,et al.
Quantum Computing
,
2008,
Wiley Encyclopedia of Computer Science and Engineering.
[3]
M. Sugeno,et al.
Fuzzy Measures and Integrals: Theory and Applications
,
2000
.
[4]
M. Mukaidono.
A set of independent and complete axioms for a fuzzy algebra (Kleene algebra)
,
1981
.
[5]
Masao Mukaidono.
The B-ternary logic and its applications to the detection of hazards in combinational switching circuits
,
1978,
MVL '78.
[6]
Masao Mukaidono,et al.
On a Kleenean extension of fuzzy measure
,
2001,
Proceedings 31st IEEE International Symposium on Multiple-Valued Logic.
[7]
Masao Mukaidono,et al.
Ternary Kleenean non-additive measures
,
2003
.