Approximate Wigner approach to Coulomb entanglement

The electric interaction between two nearby evolving electrons triggers the correlation between their waves and governs the operation of logical devices called Coulomb entanglers. Of technological interest, in the presence of magnetic fields, are multi-spatial evolution scenarios beyond pure state descriptions. The two-electron density matrix becomes eight-dimensional even for two-dimensional spatial cases, and is thus computationally prohibitive. In this work, we present two new approximations of the two-electron Wigner equation that aim at computational feasibility: a BBGKY approach for reducing the number of variables and a field approximation of the Coulomb-Wigner operator. They exhibit different conceptual aspects that illustrate alternative viewpoints on entanglement: only the evolution provided by the latter model satisfies the orthodox definition of entanglement. Our analysis, based on the Fredholm integral representation of the models, allows us to develop an intuitive picture and physical insight into the process. • The computationally prohibitive two-electron problem of Coulomb entanglement is approached with Wigner (phase space) quantum mechanics. • Two approximate methods, which are well-established in the quantum transport theory, are now applied and analyzed with respect to their relevance to describe the process of entanglement. • Projections over subspaces are used to obtain the BBGKY hierarchy of equations, which gives rise to a model violating the orthodox definition of entanglement. However, we show that the evolution is non-linear and non-Markovian, demonstrating correlations between the reduced (single-electron) states. • The field approximation of the Coulomb-Wigner operator gives rise to classical forces that govern the trajectories’ characteristics of the Liouville operator. We show that the latter, being defined in the two-electron phase space, gives rise to entanglement. • The paper focuses on the physical insights provided by the two derived approximate models on different aspects of the process of entanglement, while the mathematical derivations are given in the appendices.