Fundamental Entangling Operators in Quantum Mechanics and Their Properties

Abstract

For the first time, we introduce so-called fundamental entangling operators \(e^{iQ_{1} P_{2}}\) and \(e^{iP_{1} Q_{2} }\) for composing bipartite entangled states of continuum variables, where Qi and Pi (i = 1, 2) are coordinate and momentum operator, respectively. We then analyze how these entangling operators naturally appear in the quantum image of classical quadratic coordinate transformation (q1, q2) → (Aq1 + Bq2, Cq1 + Dq2), where AD−BC = 1, which means even the basic coordinate transformation (Q1, Q2) → (AQ1 + BQ2, CQ1 + DQ2) involves entangling mechanism. We also analyse their Lie algebraic properties and use the integration technique within an ordered product of operators to show they are also one- and two- mode combinatorial squeezing operators.