Better Late than Never: Information Retrieval from Black Holes

We show that, in order to preserve the equivalence principle until late times in unitarily evaporating black holes, the thermodynamic entropy of a black hole must be primarily entropy of entanglement across the event horizon. For such black holes, we show that the information entering a black hole becomes encoded in correlations within a tripartite quantum state, the quantum analogue of a one-time pad, and is only decoded into the outgoing radiation very late in the evaporation. This behavior generically describes the unitary evaporation of highly entangled black holes and requires no specially designed evolution. Our work suggests the existence of a matter-field sum rule for any fundamental theory.

The equivalence principle provides an important tenet of black hole physics: that a sufficiently small observer freely falling into a black hole should experience nothing special as she passes the event horizon -the boundary of no return.Similarly, quantum fields impinging on a black hole should exhibit no special behavior at the event horizon.Indeed quantum fields should be entangled across the event horizon just as they would be across the boundary to any volume in flat space.We study this claim using random subsystems as models of black hole evaporation.We find that unless the Bekenstein-Hawking entropy of a black hole is almost entirely entropy of entanglement, then the trans-event horizon entanglement vanishes long before the black hole has evaporated to the Planck scale.This would force quantum fields across the event horizon to be arbitrarily far from the vacuum state; an energetic curtain would have descended around the black hole.
Cutting out a volume of space across which a quantum field propagates yields entanglement between the subsystems so formed.For fields (or condensed matter systems) in or close to the ground state, the entropy of this entanglement scales as the surface area of the boundary [1].One can mentally grow or shrink the volume, and the entanglement responds accordingly, but in a manner which is essentially decoupled from any underlying dynamics.For a black hole, a distinguished boundary, namely the event horizon, is chosen by causality.Classically, the internal degrees of freedom of trans-event horizon entanglement are forbidden from escaping even as the black hole's surface area shrinks due to quantum evaporation [2].The only way for this entanglement to scale down with the black hole's thermodynamic entropy is to leave through some quantum mechanical tunneling process [3].We claim that unlike trans-boundary entanglement in flat space, trans-event horizon entanglement within a black hole must participate within the underlying (evaporative) dynamics.
A powerful tool for studying unitary dynamics in high dimensional systems is random matrix theory.Indeed, numerical simulations show that for the quantities studied here individual randomly selected unitaries yield almost identical results as analytically computed averages.Thus, unless the black hole dynamics is from a vanishingly small set, averages over random unitaries should give an excellent approximation to the actual dynamics.
In particular, Don Page [4] pioneered this approach for modeling black hole evaporation as the 'budding off' of random subsystems from the interior Hilbert space of a black hole to represent outgoing radiation [4,5,6].Page argued that the Hilbert space dimensionality of this interior space should be well approximated by N = e SBH , in terms of the thermodynamic entropy S BH = A 4 of a black hole of area A. He assumed an initially pure state i⟩ int for the black hole interior (int), so evaporation becomes i⟩ int → (U i⟩) RB .Here a random unitary U ∈ U (N ) is followed by the 'emission' of radiation into subsystem R with the remaining interior subsystem B.
When incorporating trans-event horizon entanglement into the evaporative dynamics, Page's model becomes Here ∑ i p i i⟩⟨i is the reduced density matrix of the external (ext) modes neighboring the event horizon.The entropy of entanglement may be quantified with a Rényi entropy H ext (≤ S BH ) for this state.As the logarithm of the dimension of the radiation subsystem grows to ln[dim(R)] = 1 2 S BH + 1 2 H ext + c the initial trans-event horizon entanglement between the external neighborhood and the interior subsystems has virtually vanished, with it appearing instead (with a fidelity of at least 1 − e −c ) as entanglement between external neighborhood modes and the outgoing radiation (see Appendix B).
In an arbitrary system where trans-boundary entanglement has vanished, the quantum field cannot be in or anywhere near its ground state.Applied to black holes, a loss of trans-event horizon entanglement implies fields far from the vacuum state in the vicinity of the event horizon.Were this to happen before the black hole had evaporated to the Planck scale, at ln[dim(R)] ≈ S BH , there would be a manifest failure of the equivalence principle.Therefore either the thermodynamic entropy of black holes is due primarily to entropy of entanglement, i.e., S BH ≈ H ext , or the randomly selected subsystem model of black hole evaporation is missing an important subtlety.
We are not the first to conjecture that a black hole's thermodynamic entropy is entropy of entanglement [7,8,9,10,11,12].Indeed, it unavoidably holds for some models of eternal black holes [10,11] and even resolves some difficulties associated with computing their entropy at the microscopic level [12].The argument above demonstrates that in models of dynamically evolving black holes, unless this conjecture holds at least approximately, it is time to draw a curtain on some of the equivalence principle's cherished predictions.
The author gratefully acknowledges the hospitality of the Frank Graham Research House.

APPENDIX A
Before considering the general entangled state described by the manuscript let us understand it for a simpler model, one for which the behavior is available in the literature.In particular, consider the model for black hole evaporation with trans-event horizon entanglement as Here ln E is the entropy of entanglement between the external (ext) modes neighboring the event horizon and the interior of the black hole.Except for the interpretation of the source of entanglement, this model has been recently analyzed by Hayden and Preskill [5].We may therefore quote their key result in our terms: As the logarithm of the dimension of the radiation subsystem grows to 1 2 S BH + 1 2 ln E + c the initial trans-event horizon entanglement between the external neighborhood and the interior subsystems has virtually vanished, with it appearing instead (with a fidelity of at least 1 − e −c ) as entanglement between external neighborhood modes and the outgoing radiation.Here (as in the manuscript) c is a free parameter, but will be dwarfed by any of the entropies involved.
Below we shall see that when the uniform entanglement of the above analysis is replaced with general trans-event horizon entanglement, the measure of entanglement ln E is replaced by the Rényi entropy H ext .

APPENDIX B
The Rényi entropy of a state is defined Note that all Rényi entropies are bounded above by the logarithm of the Hilbert space dimension, so 0 ≤ H (q) (ρ ext ) ≤ S BH for the state we study.Of particular interest to us here will be the fractional Rényi entropy for q = 1 2 , so Our key result is based on a generalization of the decoupling theorem of Ref. 13.Consider now the tripartite state where the joint subsystems Y = Y 1 Y 2 will be decomposed as either the radiation modes and interior black holes modes RB or vice-versa BR.This allows us to define The mother-in-law of all decoupling theorems: where Here, to go from Eq. ( 6) to Eq. ( 7), we assume ρ XZ = ρ X ⊗ ρ Z ; and to go from Eq. ( 7) to Eq. ( 8), we assume ρ XY Z is pure and we take 2ν = 2µ = 1 2 .
Proof: Using the Cauchy-Schwarz inequality we may write where without loss of generality we may assume that ρ ν X and ρ µ Z are invertible; then using the methods already outlined in Ref. 13 the results are easily obtained.∎ We note that the statement of the result reduces to the conventional decoupling theorem for the choice ν = 0 and subsystem Z is one-dimensional.
Of particular interest here is the case where 2ν = 1 2 and ρ ext,Y is pure, which gives where the trace norm is defined by X 1 ≡ tr X and the fidelity by As a consequence, the fidelity with which the initial trans-event horizon entanglement is encoded within the combined ext, Y 1 subsystem is bounded below by [6] 1 − e Hext Y 2 Y 1 .Now allowing this in turn to be bounded from below by 1 − e −c and choosing Y 1 = R and Y 2 = B gives the result quoted in the manuscript.
Interestingly, the opposite choice Y 1 = B and Y 2 = R tells us that for the logarithm of the dimension of the radiation subsystem less than 1  2 (S BH − H ext ) − c the initial trans-event horizon entanglement remains encoded between the external neighborhood and the interior subsystems with fidelity of at least 1 − e −c .This effectively gives the number of qunats (ln 2 times the number of qubits) that must be radiated before trans-event horizon entanglement begins to be depleted; for H ext ≈ S BH this occurs almost immediately.
Including matter: A recent paper [6] considers matter (entangled with some distant reference, ref, subsystem) which collapses to form a black hole, which itself exhibits trans-event horizon entanglement, via where here k = ln K is the number of qunats of quantum information in the in-fallen matter and where again we assume the initial dimensionality of the black hole interior is well approximated by its thermodynamic entropy, so N = dim(int) = RB = e SBH .It will also be convenient to define which we shall here interpret as approximating the number of unentangled qunats initially within the interior of the black hole.Note, that 0 ≤ H ext ≤ S BH − k for this model, so 0 ≤ χ ≤ S BH − k.
It is now straightforward to apply the generalized decoupling theorem above to show that the when the logarithm of the size of the radiation subsystem reaches S BH − 1 2 χ + c, the trans-event horizon entanglement has effectively vanished and instead has been transferred to entanglement between the external neighborhood modes and the outgoing radiation, with a fidelity of at least 1 − e −c .Thus, unless the black hole entropy is primarily entropy of entanglement, i.e., χ ⋘ S BH , there would be a manifest failure of the equivalence principle long before the black hole evaporated to the Planck scale.
We may also apply our generalized decoupling theorem to learn more details about encoding and decoding into the quantum one-time pad described in Ref. 6.In particular, prior to the first 1 2 χ − c qunats radiated, the information about the in-fallen matter is still encoded solely within the black hole interior, with a fidelity of at least 1 − e −c .Similarly, within the final 1 2 χ − c qunats radiated, the information about the in-fallen matter is encoded within the out-going radiation, with a fidelity of at least 1 − e −c .Now, with these results and those of Ref. 6, we can determine the encoding (and decoding) time for the black hole to transform all the information about the in-fallen matter into its quantum one-time pad.Both the encoding and decoding occur during the radiation of k + 1 2 (H ext − Hext ) + 2c qunats, where Hext ≡ H (2) (ρ ext ) is another Rényi entropy.Since typically H ext − Hext ≲ O(1), and this quantity cannot become negative, this implies that the encoding (and decoding) of the black hole's quantum one-time pad occurs at roughly the radiation emission rate.
Black hole entropy sum rule?We recall (see, e.g., T. Nishioka et al. arXiv:0905.0932) that the entropy of entanglement of quantum fields piercing a black hole's event horizon is expected to be proportional to the number of matter fields, however, the black hole's thermodynamic entropy is purely geometric.To preserve the equivalence principle, therefore, our result would appear to imply that black hole entropy provides a "sum rule" quantifying the number of matter fields.The tools necessary to calculate such a sum rule in a cut-off independent manner do not yet appear to be available.