Product Łukasiewicz Quantum Logic

M. Dalla Chiara, Roberto Giuntini, Richard J. Greechie

Jarosław Pykacz

Abstract It is shown that any quantum logic in the Birkhoff - von Neumann sense that possesses an ordering set of probability measures can be isomorphically represented as a particular kind of infinite-valued Łukasiewicz logic with partially defined conjunction and disjunction. It is argued that such non-classical features of BvN quantum logic as lack of distributivity or validity of the orthomodular law follow from properties of Łukasiewicz conjunction and disjunction.

Kenji Tokuo

Jarosław Pykacz

Morrison Turnansky

We present the Lukasiewicz logic as a viable system for an implication algebra on a system of qubits. Our results show that the three valued Lukasiewicz logic can be embedded in the stabilized space of an arbitrary quantum error correcting stabilizer code. We then fully characterize the non trivial errors that may occur up to group isomorphism. Lastly, we demonstrate by explicit algorithmic example, how any algorithm consistent with the Lukasiewicz logic can immediately run on a quantum system and utilize the indeterminate state.

Leopoldo Rom�n, B. Rumbos