Abstract
In this paper we consider deterministic nonlinear time evolutions satisfying so called convex quasi-linearity condition. Such evolutions preserve the equivalence of ensembles and therefore are free from problems with signaling. We show that if family of linear non-trace-preserving maps satisfies the semigroup property then the generated family of convex quasi-linear operations also possesses the semigroup property. Next we generalize the Gorini-Kossakowski-Sudarshan-Lindblad type equation for the considered evolution. As examples we discuss the general qubit evolution in our model as well as an extension of the Jaynes-Cummings model. We apply our formalism to spin density matrix of a charged particle moving in the electromagnetic field as well as to flavor evolution of solar neutrinos.
Highlights
In last decades many authors tried to generalize the standard quantum mechanical evolution
Recently considered evolutions generated by nonHermitian Hamiltonians [10, 21, 25, 32] belong to this class. It shows that convex quasi-linearity might be used as a principle joining nonlinear quantum mechanics and non-Hermitian quantum mechanics
In this paper we have discussed a generalization of the notion of the quantum operations and the quantum time evolution
Summary
In last decades many authors tried to generalize the standard quantum mechanical evolution. Let us stress that in our approach we do not change anything else in the quantum formalism but only extend admissible set of quantum evolutions by including nonlinear deterministic evolutions that do not admit superluminal signaling. In this paper we demonstrate that there exist a large class of convex quasi-linear evolutions These evolutions are generated by linear non-tracepreserving quantum operations and/or derived from generalized master equation. Recently considered evolutions generated by nonHermitian Hamiltonians [10, 21, 25, 32] belong to this class It shows that convex quasi-linearity might be used as a principle joining nonlinear quantum mechanics and non-Hermitian quantum mechanics.
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