Open Quantum Systems and the Parametric Representation: From Entanglement to Berry’s Phase

Abstract

Open quantum systems (OQS) are usually treated, in particular in the realm of quantum information theory and quantum computation, in terms of reduced density matrices, which provide a definition of the state of the principal system via the partial trace operation over the environment. This approach is a powerful tool to investigate relevant features of the open system evolution, especially when the physical situation allows for Markovian-like approximation schemes. On the other hand, the density matrix formulation and the subsequent approximation schemes induce an uncontrollable loss of information about the environmental structure, preventing some phenomena to be properly described. In this work we propose an alternative description of OQS, based on a parametric representation of the environment, as obtained in terms of generalized coherent states. The representation is used to describe a prototypical composite system, made of a spin- $\frac{1}{2}$ (the principal system) and a spin-S (the environment), interacting via a Heisenberg Hamiltonian. The resulting description shows that the emergence of a geometric (Berry) phase for a spin in an external magnetic field does follow from the fact that the true physical set up, of which the “spin in a field” is just a semiclassical-like parametric representation, is that of a quantum composite system in an entangled state. In fact, the Von Neumann entropy of the spin- $\frac{1}{2}$ , which is finite due to the existence of the environment (the spin-S), turns out to be the binary entropy of the normalized Berry’s phase.