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arXiv:2110.13898 · Journal of High Energy Physics

Entanglement entropy and vacuum states in Schwarzschild geometry

Yoshinori Matsuo

Published Jun 20, 2022• 12 citations
arXivHTMLPDF

Abstract

Abstract Recently, it was proposed that there must be either large violation of the additivity conjecture or a set of disentangled states of the black hole in the AdS/CFT correspondence. In this paper, we study the additivity conjecture for quantum states of fields around the Schwarzschild black hole. In the eternal Schwarzschild spacetime, the entanglement entropy of the Hawking radiation is calculated assuming that the vacuum state is the Hartle-Hawking vacuum. In the additivity conjecture, we need to consider the state which gives minimal output entropy of a quantum channel. The Hartle-Hawking vacuum state does not give the minimal output entropy which is consistent with the additivity conjecture. We study the entanglement entropy in other static vacua and show that it is consistent with the additivity conjecture.

DOI
10.1007/jhep06(2022)109

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Abstract Recently, it was proposed that there must be either large violation of the additivity conjecture or a set of disentangled states of the black hole in the AdS/CFT correspondence. In this paper, we study the additivity conjecture for quantum states of fields around the Schwarzschild black hole. In the eternal Schwarzschild spacetime, the entanglement entropy of the Hawking radiation is calculated assuming that the vacuum state is the Hartle-Hawking vacuum. In the additivity conjecture, we need to consider the state which gives minimal output entropy of a quantum channel. The Hartle-Hawking vacuum state does not give the minimal output entropy which is consistent with the additivity conjecture. We study the entanglement entropy in other static vacua and show that it is consistent with the additivity conjecture.
We argue that one of the following statements must be true: (a) extensive violations of quantum information theory's additivity conjectures exist or (b) there exists a set of `disentangled' black hole microstates that can account for the entire Bekenstein-Hawking entropy, up to at most a subleading O(1)O(1)O(1) correction. Possibility (a) would be a significant result in quantum communication theory, demonstrating that entanglement can enhance the ability to transmit information much more than has currently been established. Option (b) would provide new insight into the microphysics of black holes. In particular, the disentangled microstates would have to have nontrivial structure at or outside the black hole horizon, assuming the validity of the quantum extremal surface prescription for calculating entanglement entropy in AdS/CFT.
The relationship between Bekenstein-Hawking and entanglement entropy is one \nof the most intriguing problems of black hole and theoretical physics. Although both entropies have the same geometric characterization (they scale as an area), they have a different origin. The Bekenstein-Hawking entropy has it’s roots in the thermal correlations whereas the entanglement entropy describes the quantum correlations. In this thesis we address this problem from a holographic \nperspective, making large use of the AdS/CFT correspondence. \nWe first look at the BTZ black hole in which case the black hole horizon and \nthe conical singularity are related to each other. Using the modular transformations of the dual 2D CFT, we first obtain an expression of the holographicentanglement entropy for the Euclidean BTZ black hole, AdS3 vacua and conical singularity and then analyse the behaviour of the leading terms in the expansions of the holographic entanglement entropy for the BTZ black hole and the \nconical singularity. From these calculations, we have extracted the “signatures” through which entanglement entropy differentiates between the horizon and the conical singularity. When we deal with CFT at a finite temperature then it is well known that, entanglement entropy fails as a measure and hence it is replaced by entanglement negativity which is able to separate the thermal/classical \ncorrelations from the quantum correlations thereby capturing “distillable” entanglement. We address the problem of finding a suitable holographic prescription \nto calculate the entanglement negativity for two adjacent intervals when the CFT is at a finite temperature and in turn dual to a black hole in the bulk. We first propose a conjecture for the holographic entanglement negativity for two adjacent intervals in the \nAdS3/CFT2 setup and perform calculations in the \nvarious cases to support this conjecture. We further push forward the validity of this conjecture by calculating the holographic entanglement negativity for two \nadjacent subsystems when we have RN-AdS black holes in the bulk which in \nturn are dual to CFT with a conserved charge.
Using the AdS/CFT correspondence we derive a formula for the entanglement entropy of the anti-de Sitter black hole in two spacetime dimensions. The leading term in the large black hole mass expansion of our formula reproduces exactly the Bekenstein–Hawking entropy SBH, whereas the subleading term behaves as lnSBH. This subleading term has the universal form typical for the entanglement entropy of physical systems described by effective conformal fields theories (e.g. one-dimensional statistical models at the critical point). The well-known form of the entanglement entropy for a two-dimensional conformal field theory is obtained as analytic continuation of our result and is related with the entanglement entropy of a black hole with negative mass.
Abstract We studied the upper bounds of the holographic entanglement entropy growth rate for thermofield double (TFD) states. By comparing the cases of vacuum AdS and charged AdS black holes, we conjecture: for all static planar or spherically symmetric asymptotically Schwarzschild-AdS black holes of same mass density or entropy density, the vacuum AdS black hole gives the maximum entanglement entropy growth rate. We gave proofs by assuming dominant energy condition. We also considered the AdS black hole spacetime with real scalar fields case, where the scalar fields violate the dominant energy condition and the bulk geometry is not asymptotically Schwarzschild-AdS. Numerical results show that this case vacuum black hole still has maximal growth rate if we fixed entropy. However, in the case of fixed energy, vacuum case has maximal growth rate of entanglement entropy only under standard quantization scheme.
The definition of entanglement entropy in quantum systems relies on the factorization of the total Hilbert space into subsystems, which allows the construction of reduced density matrices through the partial trace operation. However, there exist important physical systems for which such a factorization is not available. In these cases, reduced density matrices and entanglement entropy are not well defined in the conventional sense. In this thesis, we develop an operator-algebraic framework for describing entanglement in such systems using the theory of von Neumann algebras. As a primary example, we consider eternal Schwarzschild black holes in Anti-de Sitter (AdS) spacetime. Starting from the dual boundary theory, we show how a suitable notion of trace can be defined, leading to a well-defined von Neumann entropy. We then reproduce the same entropy from the bulk gravitational description and demonstrate the agreement between the two calculations. Finally, we show that this quantity can be naturally interpreted as the generalized entropy.
We study thermodynamic properties and the entanglement island of a black hole in asymptotically safe quantum gravity, analyzing key thermodynamic quantities such as the Hawking temperature, heat capacity, and entropy, as well as the mass-horizon radius relation. Unlike Schwarzschild black holes, the temperature decreases with mass near the evaporation endpoint, signaling a phase transition and possible stable remnant. The entanglement entropy of Hawking radiation is obtained both with and without island contributions. Without islands, the radiation entropy grows linearly indefinitely, leading to the information paradox. By including island contributions and extremizing the generalized entropy functional, we resolve this paradox. At late times, the radiation entropy saturates at the Bekenstein-Hawking entropy, confirming unitary evolution. From this, we derive the Page time and scrambling time by equating early- and late-time entanglement entropies. The result of this study establishes the finiteness of the radiation entropy and consistency with quantum mechanics.