DeSitter entropy, quantum entanglement and ADS/CFT

Undergraduate thesis

We review recent progress in understanding the entanglement entropy of gravitational configurations for anti-de Sitter gravity in two and three spacetime dimensions using the AdS/CFT correspondence. We derive simple expressions for the entanglement entropy of two-and three-dimensional black holes. In both cases, the leading term of the entanglement entropy in the large black hole mass expansion reproduces exactly the Bekenstein-Hawking entropy, whereas the subleading term behaves logarithmically. In particular, for the BTZ black hole the leading term of the entanglement entropy can be obtained from the large temperature expansion of the partition function of a broad class of 2D CFTs on the torus.

Physical review letters, 2003

Journal of High Energy Physics, 2011

A. Universality and Odd d 44 1 We are using 'area' to denote the (d-1)-dimensional volume of v. If eq. (1.1) is calculated in a Minkowski signature background, the extremal area is only a saddle point. However, if one first Wick rotates to Euclidean signature, the extremal surface will yield the minimal area. 2 See [10] for related results in d = 4. 3 To be precise, γ d = (−) d−2 2 [6(4π) d−2

2019

We compute and provide formulae as functions of spacetime dimension for the change in holographic entanglement entropy and subregion complexity of spherical boundary subregions in the AdS black hole background up to third order in the black hole mass. We also compute exact numerical expressions for the fourth-order change in holographic entanglement entropy. We verify that the first law of entanglement is satisfied up to second order. We observe that the change in entanglement entropy is positive at odd orders and negative at even orders, whereas the change in subregion complexity is negative at odd orders and positive at even orders (except in three spacetime dimensions, where it vanishes identically). We conjecture a relation analogous to the first law of thermodynamics in which entanglement plays the role of heat and complexity plays the role of work. The relation between work and complexity is non-universal and dimension-dependent indicating that there may exist additional infor...

2016

Abstract: Relative entropy between two states in the same Hilbert space is a fun-damental statistical measure of the distance between these states. Relative entropy is always positive and increasing with the system size. Interestingly, for two states which are infinitesimally different to each other, vanishing of relative entropy gives a powerful equation ∆S = ∆H for the first order variation of the entanglement entropy ∆S and the expectation value of the modular Hamiltonian ∆H. We evaluate relative entropy between the vacuum and other states for spherical regions in the AdS/CFT framework. We check that the relevant equations and inequalities hold for a large class of states, giving a strong support to the holographic entropy formula. We elaborate on potential uses of the equation ∆S = ∆H for vacuum state tomography and obtain modified versions of the Bekenstein bound. ArXiv ePrint: 1305.3182 ar

2016

In 2006, Ryu and Takayanagi (RT) pointed out that (with a suitable cutoff) the entanglement entropy between two complementary regions of an equal-time surface of a d+1-dimensional conformal field theory on the conformal boundary of AdS_{d+2} is, when the AdS radius is appropriately related to the parameters of the CFT, equal to 1/4G times the area of the d-dimensional minimal surface in the AdS bulk which has the junction of those complementary regions as its boundary, where G is the bulk Newton constant. We point out here that the RT-equality implies that, in the quantum theory on the bulk AdS background which is related to the boundary CFT according to Rehren's 1999 algebraic holography theorem, the entanglement entropy between two complementary bulk Rehren wedges is equal to 1/4G times the (suitably cut off) area of their shared ridge. (This follows because of the geometrical fact that, for complementary ball-shaped regions, the RT minimal surface is precisely the shared ridg...

Journal of High Energy Physics, 2011

We provide a derivation of holographic entanglement entropy for spherical entangling surfaces. Our construction relies on conformally mapping the boundary CFT to a hyperbolic geometry and observing that the vacuum state is mapped to a thermal state in the latter geometry. Hence the conformal transformation maps the entanglement entropy to the thermodynamic entropy of this thermal state. The AdS/CFT dictionary allows us to calculate this thermodynamic entropy as the horizon entropy of a certain topological black hole. In even dimensions, we also demonstrate that the universal contribution to the entanglement entropy is given by A-type trace anomaly for any CFT, without reference to holography.

Journal of High Energy Physics, 2008

We consider situations where the renormalized geometric entropy, as defined by the AdS/CFT ansatz of Ryu and Takayanagi, shows extensive behavior in the volume of the entangled region. In general, any holographic geometry that is 'capped' in the infrared region is a candidate for extensivity provided the growth of minimal surfaces saturates at the capping region, and the induced metric at the 'cap' is non-degenerate. Extensivity is well-known to occur for highly thermalized states. In this note, we show that the holographic ansatz predicts the persistence of the extensivity down to vanishing temperature, for the particular case of conformal field theories in 2 + 1 dimensions with a magnetic field and/or electric charge condensates.

2010