Entanglement Entropy and Quantum Field Theory: a Non-technical Introduction

There has been considerable interest in formulating measures of entanglement and applying them to extended quantum systems with many degrees of freedom, such as quantum spin chains, especially close to a quantum phase transition (see, e.g., several other reports in these proceedings). One of these measures is entanglement entropy. Suppose the whole system is in a pure quantum state |Ψ〉, with density matrix ρ = |Ψ〉〈Ψ|, and an observer A measures a subset A of a complete set of commuting observables, while another observer B may measure the remainder. A’s reduced density matrix is ρA = TrB ρ. The entanglement entropy is just the von Neumann entropy SA = −TrA ρA log ρA associated with this reduced density matrix. Although there are other measures of entanglement, the entropy is most readily suited to analytic investigation. Lack of space prevents us from discussing other measures of entanglement and for referring properly to all the works about entanglement entropy. For a more complete list of references we refer, e.g., to our papers. In these proceedings we try to give a pedagogical and non-technical introduction to the Quantum Field Theory (QFT) approach to entanglement entropy. Particular attention is devoted to the one space dimensional case, with a linear dispersion relation, that, exactly at the quantum phase transition, can be effectively described by a two-dimensional Conformal Field Theory (CFT). Again, for lack of space, we