Holistic and Compositional Logics Based on the Bertini Gate

The theory of logical gates in quantum computation has inspired the development of new forms of quantum logic where the meaning of a formula is identified with a density operator and the logical connectives are interpreted as operations defined in terms of quantum gates. We show some relations between the Bertini gate and many valued connectives by probability values. On this basis, one can deal with quantum circuits as expressions in an algebraic environment such as product many valued algebra for combinational circuits. As can be expected, we show that the compositional logic characterized by the qubit semantics is stronger than the compositional Lukasiewicz quantum computational logic by a counterexample. But, in the holistic case, we conjecture that they can characterize the same logic.