Can quantum entanglement violate the no hair theorem?

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Matterwave

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Yea, so I was thinking...the no hair theorem says that you can only know 3 things about the matter inside black holes - mass, angular momentum, and charge right? But what if I create a pair of entangled particles, and throw one of them into a black hole...will I then know information about that particle once I observe the entangled pair?

Does this violate the no hair theorem? o_O I'm guessing not, but I'd like an explanation why not (in theory please, not practicality arguments). Thank you. :)
 
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This is essentially just the Black Hole Information Paradox. The Black hole is in a unique state, parametrized by its mass, angular momentum and charge. Therefore, anything thrown into the black hole will always end up in one specific state.

This is in principle a violation of unitarity. Quantum mechanics (or QFT, whatever you prefer) is a unitary theory, meaning that states evolve in a unique way. But if you throw something into the black hole the evolution of the state does not depend on the initial state of that something. The black hole destroys the information associated to the initial state - and if this state is an entangled one, well, that will pose some problems.

The paradox has not been resolved though (there are some possible resolvements by the more exotic theories such as ADS/CFT or string theory). Maybe the no-hair theorem breaks down at the quantum level, or, as Hawking seems to put it, the information 'leaks' to parallel universes thereby storing it somewhere else. In our universe unitarity would be violated, but together with the parallel universes it is still valid.
 

Demystifier

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Matterwave, in your case I would not really say that you know information about the inside particle my measuring the outside particle, unless you know the total initial wave function before dropping one entangled particle to the interior.
On the other hand, if you know initial information, than you can can also avoid the no-hair theorem by a purely classical mechanism. For example, you can drop two cars into the black hole, and you will know that the black hole contains two cars, even though you cannot see them from the outside. The case with entangled particles is similar; if you do not take into account your prior knowledge, then measurement of the outside particle by itself says nothing about the inside particle.
 
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Antony Valentini has an interesting 'experimentally testable' proposal on this in terms of hidden variables theories (e.g. de Broglie-Bohm pilot-wave theory).

See "Black holes, information loss, and hidden variables" http://uk.arxiv.org/abs/hep-th/0407032"
and "Extreme test of quantum theory with black holes" http://uk.arxiv.org/abs/astro-ph/0412503"

"We consider black-hole evaporation from a hidden-variables perspective. It is suggested that Hawking information loss, associated with the transition from a pure to a mixed quantum state, is compensated for by the creation of deviations from Born-rule probabilities outside the event horizon. The resulting states have non-standard or 'nonequilibrium' distributions of hidden variables, with a specific observable signature - a breakdown of the sinusoidal modulation of quantum probabilities for two-state systems. Outgoing Hawking radiation is predicted to contain statistical anomalies outside the domain of the quantum formalism. Further, it is argued that even for a macroscopic black hole, if one half of an entangled EPR-pair should fall behind the event horizon, the other half will develop similar statistical anomalies. We propose a simple rule, whereby the relative entropy of the nonequilibrum (hidden-variable) distribution generated outside the horizon balances the increase in von Neumann entropy associated with the pure-to-mixed transition. It is argued that there are relationships between hidden-variable and von Neumann entropies even in non-gravitational physics. We consider the possibility of observing anomalous polarisation probabilities, in the radiation from primordial black holes, and in the atomic cascade emission of entangled photon pairs from black-hole accretion discs."
 
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Demystifier

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Zenith8, another proposal (for solving the black-hole information paradox) that emerged from a research of the Bohmian interpretation (although does not really rest on this interpretation) is
http://xxx.lanl.gov/abs/0905.0538
 

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