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Entanglement and Area
The connection between entanglement and area has been found to be more general than first thought.
The concept of entropy in classical physics describes the accuracy to which we have specified the state of a physical system. If we put, say, many atoms in a container, we typically do not have a complete description at hand of the movement of every atom.
This is, however, not required to meaningfully talk about macroscopic properties of a collection of systems. The entropy of such a classical system is related to the precision with which one has information about the state. It is essentially the number of microscopic states that is consistent with the macroscopic state that we observe.
In quantum mechanics, there is yet another origin of entropy, very different from this mere lack of resolution. If we have a composite system in a pure state, a state with vanishing entropy, then the quantum state of a part will typically have a positive entropy. This is not due to a lack of resolution, but entirely due to the fact that we only have access to a smaller number of observables associated with the subsystem. This entropy quantifies correlations between the parts, and if the whole was in a pure state of which we have perfect knowledge, the degree of entanglement between the parts.
In quantum field theory, the concept of geometric entropy is just due to the accessability to a limited part of a composite system [1,2]. Imagine that we have a free scalar field, in a pure state, and distinguish a certain region.
What is the entropy associated with that region? And how does it scale with the size of the region? With the volume (as entropy is an extensive quantity)? In the quantum field context this question is particularly exciting, as this relationship between geometrical properties and entropy are thought to be related to this relationship in black holes. While numerical studies suggested the former [1,2], a strict proof of this property for general spatial dimensions and cubic areas was outstanding
Recently however, it was shown analytically that in lattice versions of this question, the entropy indeed scales as the boundary area of the region [3,4,5].
It can be proven that this holds true for arbitrary geometrical regions, and for entanglement measures that have been developed in quantum information science over the last few years, even when the system is not in a ground state, but a thermal state [3,4]. Not the degrees of freedom in the interiour count, but only a thin layer of the size of the classical correlation length contributes to the entropy. This result generalizes earlier findings, and shows that this connection between entanglement and area is more general than previously thought.
[1] M. Srednicki, Phys. Rev. Lett. 71, 66 (1993).
[2] C. Holzhey, F. Larsen and F. Wilczek, Nucl. Phys. B 424, 443 (1995).
[3] M. Plenio, J. Eisert, J. Dreissig, and M. Cramer, Phys. Rev. Lett. 94, 060503 (2005).
[4] M. Cramer, J. Eisert, M. Plenio, and J. Dreissig, in preparation.
[5] P. Calabrese and J. Cardy, J. Stat. 0406, P002 (2004).
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