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arXiv:hep-th/0507228 · Canadian Journal of Physics

Is entanglement entropy proportional to area?

Morteza Ahmadi, Saurya Das, S Shankaranarayanan

Published Jan 15, 2006• 9 citations
arXivHTMLPDF

Abstract

It is known that the entanglement entropy of a scalar field, found by tracing over its degrees of freedom inside a sphere of radius [Formula: see text], is proportional to the area of the sphere (and not its volume). This suggests that the origin of black-hole entropy, also proportional to its horizon area, may lie in the entanglement between the degrees of freedom inside and outside the horizon. We examine this proposal carefully by including excited states, to check probable deviations from the area law.PACS Nos.: 04.60.–m, 04.62, 04.70.–s, 03.65.Ud

DOI
10.1139/p06-002

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It is known that the entanglement entropy of a scalar field, found by tracing over its degrees of freedom inside a sphere of radius [Formula: see text], is proportional to the area of the sphere (and not its volume). This suggests that the origin of black-hole entropy, also proportional to its horizon area, may lie in the entanglement between the degrees of freedom inside and outside the horizon. We examine this proposal carefully by including excited states, to check probable deviations from the area law.PACS Nos.: 04.60.–m, 04.62, 04.70.–s, 03.65.Ud
We revisit the problem of finding the entanglement entropy of a scalar field on a lattice by tracing over its degrees of freedom inside a sphere. It is known that this entropy satisfies the area law---entropy proportional to the area of the sphere---when the field is assumed to be in its ground state. We show that the area law continues to hold when the scalar field degrees of freedom are in generic coherent states and a class of squeezed states. However, when excited states are considered, the entropy scales as a lower power of the area. This suggests that, for large horizons, the ground state entropy dominates, whereas entropy due to excited states gives power-law corrections. We discuss possible implications of this result to black hole entropy.
We reexamine the idea that the origin of black-hole entropy may lie in the entanglement of quantum fields between the inside and outside of the horizon. Motivated by the observation that certain modes of gravitational fluctuations in a black-hole background behave as scalar fields, we compute the entanglement entropy of such a field, by tracing over its degrees of freedom inside a sphere. We show that while this entropy is proportional to the area of the sphere when the field is in its ground state, a correction term proportional to a fractional power of area results when the field is in a superposition of ground and excited states. The area law is thus recovered for large areas. Further, we identify the location of the degrees of freedom that give rise to the above entropy.
We are using entanglement entropy approach to study how the entropy depends upon the location of degrees of freedom near and far from the horizon. We consider quantum fields in the vicinity of black holes space-time, the entropy receives log- arithmic and power law corrections in area law. The power law corrections arises when the fields are in excited or mixed state (superposition of ground and excited state)a but logarithmic corrections are present in ground state. In this paper, we try to understand physically the deviation from the area law for the excited and mixed state. We show that the degrees of freedom near to the horizon contribute the total entropy, but when degrees of freedom far away from the horizon are more significant for excited/mixed state instead of ground state.
It has been known for decades that black holes behave as thermodynamic objects and, as such, have an entropy, which was acquired in exact form as the Bekenstein-Hawking entropy. However, the origin of this entropy, and in particular its proportionality to the area of the horizon, remained unexplained. Motivated by this mystery, several authors examined the possibility of entanglement entropy as a source of black hole entropy, by considering a scalar ?eld in the ground state and\ntracing out the degrees of freedom inside an imaginary sphere, analogous to the interior of the black hole. The resulting entanglement entropy between the two regions turned out to scale with the area of their mutual boundary. In the simplest model the scalar ?eld is represented by a system of coupled harmonic oscillators. After an introduction to the phenomenon of quantum entanglement and the associated entanglement entropy, we will review the problem of two coupled oscillators in the ground state and derive the entanglement entropy using an approach in Fock space, as opposed to the standard method involving integrals in positon space. This will also\nallow us to look at some excited states of the system. Subsequently we will try to extend this approach to a system of N coupled harmonic oscillators and derive the entanglement entropy between an inner and outer region of oscillators.