A General No-Cloning Theorem

Stefan Weigert

Gerd Niestegge

The quantum mechanical no-cloning theorem for pure states is generalized and transfered to the quantum logics with a conditional probability calculus in a rather abstract, though simple and basic fashion without relying on a tensor product construction or finite dimension as required in other generalizations.

Takayuki Miyadera, Hideki Imai

This paper discusses the no-cloning theorem in a logicoalgebraic approach. In this approach, an orthoalgebra is considered as a general structure for propositions in a physical theory. We proved that an orthoalgebra admits cloning operation if and only if it is a Boolean algebra. That is, only classical theory admits the cloning of states. If unsharp propositions are to be included in the theory, then a notion of effect algebra is considered. We proved that an atomic Archimedean effect algebra admitting cloning operation is a Boolean algebra. This paper also presents a partial result, indicating a relation between the cloning on effect algebras and hidden variables.

Nicholas J. Teh

Gerd Niestegge

The quantum mechanical no-cloning theorem for pure states is generalized and transfered to the quantum logics with a conditional probability calculus. This is, on the one hand, an extension of the classical probability calculus and, on the other hand, a mathematical generalization of the Lueders - von Neumann quantum measurement process. In the non-classical case, a very special type of conditional probability emerges, describing the probability for the transition from a past event to a future event independently of any underlying state. This probability results from the algebraic structure of the quantum logic only and is invariant under algebraic morphisms, which is used to prove the generalized no-cloning theorem in a rather abstract, though simple and basic fashion without relying on a tensor product construction as required in other approaches.

Sibasish Ghosh, Guruprasad Kar, Anirban Roy