Old Black Holes and Decoding

An old black hole—one already highly entangled with early radiation—can act as an information mirror in a random-unitary model. After a diary is mixed into the remaining black hole, slightly more new radiation than the diary’s size can make the diary recoverable from the union of early and new radiation. The onset is delayed by scrambling, while the existence of a recovery map must be kept separate from the cost of implementing it.

Required background. Quantum Information supplies purification, decoupling, trace distance, Haar randomization, and recovery channels used in the derivation.

Helpful background. Semiclassical Gravity supplies Page and scrambling scales; Quantum Fields in Curved Spacetime supplies outgoing radiation modes; Causality and Global Structure supplies the exterior collection and horizon-crossing regions.

Systems in the mirror model

Let $B$ be the old black hole and $E$ its early radiation. Idealize $BE$ as nearly maximally entangled on the support of $B$. A diary $D$, of dimension $d_D$, is maximally entangled with an inaccessible reference $Q$. After $D$ falls in, a scrambling unitary acts as

$$U: D\otimes B\longrightarrow B'\otimes L,$$

where $L$ is newly emitted late radiation and $B'$ is the remaining black hole. The exterior decoder has $EL$; the inaccessible complement is $B'$. All spaces are finite dimensional, and the model ignores energy, charge, locality, greybody factors, and backreaction except through the chosen dimensions.

The goal is not merely to make $L$ depend on the diary. It is to recover entanglement with $Q$: there should be a channel $\mathcal R_{EL\to\widehat D}$ such that

$$(\mathcal R\otimes\mathrm{id}_Q)(\rho_{ELQ}) \simeq\lvert\Phi\rangle\langle\Phi\rvert_{\widehat DQ}.$$

This reference-sensitive criterion prevents a decoder from succeeding only on a preferred classical basis.

Decoupling derivation

For the diary to be recoverable from $EL$, its reference $Q$ must become approximately decoupled from the discarded system $B'$:

$$\rho_{QB'}\simeq\rho_Q\otimes\rho_{B'}.$$

Purity then implies, through the decoupling–recovery theorem, that a suitable isometry on $EL$ can reconstruct the diary. The ideal exactly maximally old case gives a particularly clean bound: take $BE$ and $DQ$ to be independent maximally entangled pairs and draw $U$ from normalized Haar measure. Then $DB$ is maximally mixed, so $\rho_{B'}=I_{B'}/d_{B'}$ for every $U$. Define the full trace-norm error (twice the conventional trace distance) by

$$\epsilon_U=\left\lVert \rho_{QB'}-\rho_Q\otimes\rho_{B'} \right\rVert_1.$$

The Haar second-moment estimate of Hayden and Preskill gives $\mathbb E_U\epsilon_U^2\leq(d_D/d_L)^2$, hence by Cauchy–Schwarz

$$\mathbb E_U\epsilon_U\leq\frac{d_D}{d_L}.$$

This bound concerns an ensemble average, not every unitary. Thus choose

$$d_L\gtrsim d_D\,2^c.$$

For $d_L=d_D2^c$, the average decoupling error is at most $2^{-c}$. A suitable decoder has entanglement fidelity $F_U=\langle\Phi|\rho_{\widehat DQ}|\Phi\rangle\geq1-\epsilon_U$, so $\mathbb E_U(1-F_U)\leq2^{-c}$. If the diary contains $k=\log_2d_D$ qubits, $k+c$ late qubits therefore suffice for this average infidelity target. A target on the recovered state’s trace distance has a different conversion, $T\leq\sqrt{1-F_U}$. Imperfect initial old-hole entanglement adds an error that must be bounded separately. The source distinguishes precisely these norms and fidelities Hayden and Preskill 2007, § 3, Eqs. (4)–(7).

The early radiation is essential. It supplies the other half of the entanglement resource: late radiation need only reveal how the diary was mixed into a system whose prior state is already known through $E$. For a young black hole with little early entanglement, the same small late subsystem generally cannot decode the diary.

Scrambling before emission

The Haar unitary is a statistical model, not a time evolution. A physical black hole must first spread diary operators across its active degrees of freedom. A common estimate is

$$t_{\mathrm{scr}}\sim\frac{\beta}{2\pi}\ln S_{\mathrm{BH}},$$

after which the required number of late qubits can begin to emerge. Emission and collection add further time. Hayden and Preskill distinguish scrambling dynamics from the efficiency of recovery Hayden and Preskill 2007, §§ 4–5.

Fast scrambling does not imply a fast decoder. The inverse of a known circuit is efficient only if the observer has an adequate microscopic description and access to the correct outputs. A generic recovery map guaranteed by decoupling can have prohibitive circuit complexity. Errors in the black-hole unitary, incomplete early radiation, and gravitational backreaction can also degrade fidelity; the distinction between information-theoretic recovery and its physical implementation is emphasized in Harlow 2016, §§ 5.6 and 8.1.

Independent checks and failure cases

If $d_D=1$, there is no diary information and no additional radiation is required. Increasing $d_L$ by one qubit should reduce a dimension-ratio error by approximately a factor of two, fixing the direction of the bound. If $E$ is discarded, the decoder loses the preexisting purifier and the old-hole advantage disappears.

As an adverse ensemble, let $U$ act trivially on $D$ while scrambling only $B$. No amount of early radiation reveals the diary from a small $L$ that never couples to it. Equal dimensions and black-hole age are therefore insufficient; the message must actually participate in the scrambling dynamics.

The decoupling statement is average over the selected ensemble. Concentration can make most Haar unitaries good at large dimension, but a particular black-hole Hamiltonian requires an independent dynamical argument. Exact symmetries or conserved charges split the Hilbert space and modify the dimension count sector by sector.

Interpretation boundary

The mirror result answers when formal recovery becomes possible in an idealized old-hole channel. It does not prove that the recovered radiation state can be compared with an independent interior copy, nor that the decoder finishes without high energy or backreaction. Generic decoupling theory belongs to Quantum Information, emission dynamics to Quantum Fields in Curved Spacetime, and scrambling-time geometry to Semiclassical Gravity. Explicit holographic decoders are outside this page.

Exercise

A $k=12$ qubit diary falls into the ideal exactly maximally old black hole. How many late qubits suffice to bound the Haar-average entanglement infidelity by $2^{-20}$? Does the same count guarantee recovered-state trace distance $2^{-20}$?

Solution

Choose $c=20$, so $d_L/d_D=2^{20}$ and $32$ late qubits suffice for the average infidelity bound. Jensen’s inequality and $T\leq\sqrt{1-F_U}$ then bound the average recovered-state trace distance only by $2^{-10}$. The same sufficient estimate reaches $2^{-20}$ trace distance with $c=40$, or $52$ late qubits, if the model has that many available. These are ensemble-average guarantees, not runtimes or worst-case claims for a particular black-hole Hamiltonian; unavailable early radiation, imperfect entanglement, or ineffective scrambling can invalidate their assumptions.

References

• Harlow, Daniel. 2016. “Jerusalem Lectures on Black Holes and Quantum Information.” Reviews of Modern Physics 88: 015002. Open text.
• Hayden, Patrick, and John Preskill. 2007. “Black Holes as Mirrors: Quantum Information in Random Subsystems.” Journal of High Energy Physics 2007 (09): 120. Open text.