Complete positivity of nonlinear evolution: A case study

1997/08/31 by Marek Czachor, Maciej Kuna · 32 citations
Computer Science · Mathematics · Physics and Astronomy · #Advanced Thermodynamics and Statistical Mechanics #Applied mathematics #Biology #Computer science #Evolutionary biology #Mathematics #Nonlinear system #Physics #Quantum Information and Cryptography #Quantum Mechanics and Applications #chao-dyn #funct-an #hep-th #math.FA #nlin.CD #quant-ph

paper · pdf · doi:10.1103/physreva.58.128

published in Physical Review A 58(1), 128-134 (American Physical Society) · extended version, 3 appendices added (on mixed states, projection postulate, nonlocality), to be published in Phys. Rev. A

arxiv created 1998/04/07 · openalex publication_date 1998/07/01 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

Simple Hartree-type equations lead to dynamics of a subsystem that is not completely positive in the sense accepted in mathematical literature. In the linear case this would imply that negative probabilities have to appear for some system that contains the subsystem in question. In the nonlinear case this does not happen because the mathematical definition is physically unfitting as shown in a concrete example.

Citations

• Nonlinear Schrödinger equation and two-level atoms
• Density matrix interpretation of solutions of Lie-Nambu equations
• Completely positive dynamical semigroups of N-level systems
• Testing quantum mechanics
• Separability Criterion for Density Matrices
• Nambu-type generalization of the Dirac equation
• Separability criterion and inseparable mixed states with positive partial transposition
• Separability of mixed states: necessary and sufficient conditions