Entanglement of harmonic systems in squeezed states

Abstract

The entanglement entropy of a free scalar field in its ground state is dominated by an area law term. It is noteworthy, however, that the study of entanglement in scalar field theory has not advanced far beyond the ground state. In this paper, we extend the study of entanglement of harmonic systems, which include free scalar field theory as a continuum limit, to the case of the most general Gaussian states, namely the squeezed states. We find the eigenstates and the spectrum of the reduced density matrix and we calculate the entanglement entropy. We show that our method is equivalent to the correlation matrix method. Finally, we apply our method to free scalar field theory in 1+1 dimensions and show that, for very squeezed states, the entanglement entropy is dominated by a volume term, unlike the ground-state case. Even though the state of the system is time-dependent in a non-trivial manner, this volume term is time-independent. We expect this behaviour to hold in higher dimensions as well, as it emerges in a large-squeezing expansion of the entanglement entropy for a general harmonic system.

Entanglement in cosmology

“…The original calculation focused on the leading term in the entropy, while the logarithmic correction was computed in [24]. The basic formalism for the generalization of the calculation to the case of a Friedmann-Robertson-Walker (FRW) background was developed in [35,36] and was applied to a toy (1+1)-dimensional model. In this work we carry out the analysis for a (3+1)-dimensional cosmological background that undergoes a transition from an inflationary era to a period of radiation domination.…”

“…We also provide a concise summary of the method developed in refs. [35,36] for the calculation of the reduced density matrix and its eigenvalues, from which the entanglement entropy is deduced. In section 3 we present the results of a numerical calculation of the entanglement entropy in a time-dependent background that evolves from an inflationary era to radiation domination.…”

“…In this section we summarize the formalism that we employ for the calculation of the entanglement entropy. We follow the approach of Srednicki [16], which was generalized to the case of an expanding background in [35,36]. We refer the reader to these publications for the details.…”

“…The quantization procedure treats the field as a collection of quantum harmonic oscillators. In order to make the analogy with the language of quantum mechanics clearer, we follow the notation of previous work [16,35,36] and use the variable x instead of f . It must be kept in mind that it is the field value that is treated as a harmonic oscillator.…”

“…where the vector x denotes collectively the field values f lm,j , following the notation of [16,35,36], as we explained in the previous subsection. The matrix W is given by…”