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1 Is entanglement entropy proportional to area?

Morteza Ahmadi, Saurya Das, S. Shankaranarayanan

Published Jan 1, 2005• 7 citations
OpenAlex

Abstract

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 R, 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

Related papers

Is entanglement entropy proportional to area?

Morteza Ahmadi, Saurya Das, S Shankaranarayanan

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

arXiv:hep-th/0507228•Canadian Journal of Physics•Jan 15, 2006•9 citations

Power-law corrections to entanglement entropy of horizons

Saurya Das, S. Shankaranarayanan, Sourav Sur

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.

arXiv:0705.2070•Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields•Mar 10, 2008•138 citations

How robust is the entanglement entropy-area relation?

Saurya Das, S. Shankaranarayanan

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.

arXiv:gr-qc/0511066•Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields•Jun 28, 2006•70 citations

Entropy and area

Mark Srednicki

The ground-state density matrix for a massless free field is traced over the degrees of freedom residing inside an imaginary sphere; the resulting entropy is shown to be proportional to the area (and not the volume) of the sphere. Possible connections with the physics of black holes are discussed.

arXiv:hep-th/9303048•Physical Review Letters•Aug 2, 1993•1,857 citations

Entanglement entropy of coupled harmonic oscillators: an approach in Fock space

M. Jonker

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.

•Jan 1, 2016•0 citations

Thermodynamics of entanglement in Schwarzschild spacetime

Shinji Mukohyama A, Masafumi Seriu, Hideo Kodama

Extending the analysis in our previous paper, we construct the entanglement thermodynamics for a massless scalar field on Schwarzschild spacetime. Contrary to the flat case, the entanglement energy ${E}_{\mathrm{ent}}$ turns out to be proportional to the area radius of the boundary if it is near the horizon. This peculiar behavior of ${E}_{\mathrm{ent}}$ can be understood by the redshift effect caused by the curved background. Combined with the behavior of the entanglement entropy, this result yields, quite surprisingly, the entanglement thermodynamics of the same structure as black hole thermodynamics. On the basis of these results, we discuss the relevance of the concept of entanglement as the microscopic origin of black hole thermodynamics.

arXiv:gr-qc/9712018•Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields•Jul 31, 1998•26 citations
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 R, 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
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 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 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.
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.
Extending the analysis in our previous paper, we construct the entanglement thermodynamics for a massless scalar field on Schwarzschild spacetime. Contrary to the flat case, the entanglement energy Eent{E}_{\mathrm{ent}}Eent​ turns out to be proportional to the area radius of the boundary if it is near the horizon. This peculiar behavior of Eent{E}_{\mathrm{ent}}Eent​ can be understood by the redshift effect caused by the curved background. Combined with the behavior of the entanglement entropy, this result yields, quite surprisingly, the entanglement thermodynamics of the same structure as black hole thermodynamics. On the basis of these results, we discuss the relevance of the concept of entanglement as the microscopic origin of black hole thermodynamics.