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My question is somehow simple, but not trivial. I'm reading many articles about the ER=EPR correspondence. There are some articles which study this proposal from a field theory point of view (one really obscure by Marolf and Polchinski using eigenvalue thermalization hypothesis, and one by other authors using a random matrix approach), and they conclude that typical entangled states do not present a smooth wormhole connecting the tho theories, while the thermofield double state $|\psi> \sim \sum_i e^{-\beta E_i /2} |i>_L|i>_R$ does. However, this latter state is non typical.

My question is: is there any definition for typical state on a given Hilbert space? In the context I'm talking about, it is somehow clear that the randomness introduced using a random matrix approach or the eigenvalues thermalization hypothesis will consider more states than just the thermofield double, but does the typicality come in play when in general we take an average? Or can a single state be a typical state?

Thanks

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  • $\begingroup$ Hi, interesting question. I am also interested in this topic. I am curious about the entanglement pattern of 'typical states' since this is related with how the spacetime is built behind the blackhole. I think the TFD state is a kind of 'static' state that with the evolution of both sides, the entanglement pattern of the interior of the two sides and the entanglement entropy between them are all 'static', so the wormhole between them is 'smooth'. But for other 'typical states', the entanglement pattern flatuate (though only a little) with time, so the wormhole is not smooth. $\endgroup$ – XXDD Feb 13 '16 at 3:32
  • $\begingroup$ Also L. Susskind call the TFD state as 'maximally entangled', but for me it's strange since the reduced density matrix for each part is $Diag(exp(-\beta H_1))$, but not an identity matrix. Why this is a maximally entangled state? $\endgroup$ – XXDD Feb 13 '16 at 3:35
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Typically, a typical state a state which is chosen according to the so-called Haar measure, or unitarily invariant measure. This is the unique probability measure $p$ on the Hilbert space $\mathcal H$ which satisfies that $p(U\mathcal K)=p(\mathcal K)$ for any (measurable) subset of states $\mathcal K\subset \mathcal H$. One then says that a property holds for a typical state if it holds for most (asymptotically all) state w.r.t. to this measure.

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