arXiv:1110.1239v1 [hep-th] 6 Oct 2011 Entanglement entropy in all dimensions Samuel L. Braunstein 1 , ∗ Saurya Das 2 , † and S. Shankaranarayanan 3‡ 1 Computer Science, University of York, York Y010 5DD, UK 2 Theoretical Physics Group, University of Lethbridge, 4401 University Drive, Lethbridge, Alberta, Canada T1K 3M4 and 3 School of Physics, Indian Institute of Science Education and Research, CET Campus, Thiruvananthapuram 695016, India It has long been conjectured that the entropy of quantum fields across boundaries scales as the boundary area. This conjecture has not been easy to test in spacetime dimensions greater than four because of divergences in the von Neumann entropy. Here we show that the R´ enyi entropy provides a convergent alternative, yielding a quantitative measure of entanglement between quantum field theoretic degrees of freedom inside and outside hypersurfaces. For the first time, we show that the entanglement entropy in higher dimensions is proportional to the higher dimensional area. We also show that the R´ enyi entropy diverges at specific values of the R´ enyi parameter q in each dimension, but this divergence can be tamed by introducing a mass of the quantum field. PACS numbers: 03.67.Mn, 05.50.+q, 11.10.-z, 04.50.Gh Entanglement, a term first coined by Schr¨ odinger, is an intriguing and quintessentially quantum mechanical property, which correlates microscopic systems in a pre- cise way, even if they are separated by large distances. On the one hand it gives rise to apparent contradictions (such as the EPR paradox) and on the other, hides enormous untapped resources for computation and communication (e.g., teleportation). A mathematically precise way of measuring entanglement has remained elusive however, except in the simplest cases where the combined system is in a pure state, i.e., for which all quantum numbers are known. Usually, the entanglement entropy is com- puted as the von Neumann entropy associated with ρ: S vN = −tr (ρ ln ρ). For recent reviews, see [1, 2]. Entanglement as a source of entropy has also been ex- amined in an entirely different context, that of the mi- croscopic origin of black hole entropy. The basic idea is that the long-range entanglement of the quantum fields across a black-hole horizon can leave its mark on the re- duced density matrix of the external degrees of freedom (DOF) which, in turn, accounts for the black-hole en- tropy. Starting with [3, 4], it was demonstrated that to leading order, and for a scalar field in its global ground state, the entanglement entropy between its DOF inside and outside a black hole horizon is proportional to the area of the horizon. Recently, it was demonstrated that the DOF at or near the horizon contribute to most of this entropy [5] and that excited states of the field lead to sub-leading order power law corrections [6, 7]. However, entanglement as a source of black-hole en- tropy has a couple of drawbacks: (i) The proportionality constant depends on the ultra-violet cut-off and the num- ber of fields present (also these are in general independent of each other, although it was recently heuristically ar- gued that the requirement of the stability of the cosmos related the two and naturally related the cut-off to the Planck mass [8]), and (ii) it is not evident that the area proportionality holds for higher spacetime dimensions, say D +2 > 4 (throughout D + 1 denotes the number of spatial dimensions)? Although there have been attempts to obtain uni- versal expressions for higher dimensions from the two- dimensional entropy c-functions [1], such is yet to be shown from first principles. For higher dimensional black-holes, the brick-wall entropy contains extra diver- gent terms other than from the ultra-violet modes [9]. Following Srednicki [4], if one regularizes the entropy function by introducing a radial lattice, the sum of par- tial waves does not converge and the entropy turns out to be infinite in higher dimensions. Here, we show that the entropy-area relation can be obtained for all dimensions by using a different measure of entropy, and furthermore, that the divergences are similar in nature to the infrared divergences in QED, which can be tamed by introducing a mass to the field. While the von Neumann entropy is the most common measure of entanglement, it is neither the most general, nor unique. There are other measures, such as the R´ enyi and Tsallis entropies, which under certain limits reduce to the von Neumann entropy. In this work we study entanglement via the R´ enyi entropy defined as S (q) ≡ 1 1 − q ln  n  i=1 p q i  . (1) In the limit that the parameter q → 1, the R´ enyi en- tropy reduces to its von Neumann counterpart. Also, like von Neumann entropy (and unlike Tsallis entropy) R´ enyi entropy is additive and has maximum value ln(n) for p i =1/n. Consider a free, massless real scalar field propagating in a (D + 2)-dimensional flat spacetime with action 1 2  dt dr r D dΩ D (η ab ∂ a Φ ∂ b Φ+ g θnθn ∂ θn Φ ∂ θn Φ), (2)