Page curves for tripartite systems

Abstract

We investigate information flow and Page curves for tripartite systems. We prepare a tripartite system (say, A, B, and C) of a given number of states and calculate information and entropy contents by assuming random states. Initially, every particle was in A (this means a black hole), and as time goes on, particles move to either B (this means Hawking radiation) or C (this means a broadly defined remnant, including a non-local transport of information, the last burst, an interior large volume, or a bubble universe, etc). If the final number of states of the remnant is smaller than that of Hawking radiation, then information will be stored by both the radiation and the mutual information between the radiation and the remnant, while the remnant itself does not contain information. On the other hand, if the final number of states of the remnant is greater than that of Hawking radiation, then the radiation contains negligible information, while the remnant and the mutual information between the radiation and the remnant contain information. Unless the number of states of the remnant is large enough compared to the entropy of the black hole, Hawking radiation must contain information; and we meet the menace of black hole complementarity again. Therefore, this contrasts the tension between various assumptions and candidates of the resolution of the information loss problem.

Multipartite analysis of average-subsystem entropies

“…The most common applications found in the literature are for bipartite systems [1,10,13,18,19], where subsystem entropies are typically within 1 nat of maximal mixing, but we have argued herein that it is often more useful to look at tri-partite or even multi-partite decompositions of the universe. (Indeed multi-partite decompositions have attracted and continue to attract considerable attention [3,4,6,12,21,83,84]. )…”

A generic unitary black-hole evaporation model based on first principles

“…First, the original Bekenstein-Hawking bound [20,27] S * ∝ A is satisfied only if F ≥ 1 + 4S * /A at a specific entropy density S * /A ≥ 1/4. The factor 1 + 4S * /A coincides with simpler models [21,27,28] and represents both the excess entropy directly due to the Hawking radiation and indirectly due to the associated area reduction. Unfortunately, this constraint corresponds to a separatrix, which does not forbid BHs that are already on the other side from further deviating away, and most extremely, from forming remnants: stable, nearly fully evaporated BHs with a huge amount of residue entropy.…”

“…Therefore the emitted radiation must be maximally entangled with whatever is inside BH. Unless the horizon is somehow leaky, the entanglement entropy piles up and leads to a Planck-size BH that retains a similar amount of entropy as the original one [21]. This conclusion is in direct conflict with Bekenstein's law, and the storage of the excess entanglement entropy in a form other than the horizon quanta, is necessary.…”

Broken bridges: a counter-example of the ER=EPR conjecture

“…However, it was discovered that the original assumptions of black hole complementarity are seriously flawed [8][9][10]. In order to cure this inconsistency, Almheiri, Marolf, Polchinski and Sully (AMPS) [11,12] proposed that around the black hole horizon scale (after the Page time [13][14][15][16]), the equivalence principle must be violated; and they named such a region as the firewall.…”

“…The key argument for the inconsistency of black hole complementarity can be described as follows [17]. First, due to unitarity, a particle from Hawking radiation [18] (say, B) must be entangled (either after the Page time [13][14][15][16] or scrambling time [19], where the details depend on situations) with its counterpart R B , where R B can be either the earlier part of Hawking radiation (for one-sided black holes) or the distilled information that is recovered at the boundary of the opposite side (for two-sided eternal black holes). Second, based on local quantum field theory, any given Hawking radiation B should be entangled with its counterpart A (which is inside the horizon).…”