The ghost in the radiation: Robust encodings of the black hole interior

TL;DR

The black hole firewall puzzle is reconsidered, emphasizing that quantum error- correction, computational complexity, and pseudorandomness are crucial concepts for understanding the black hole interior.

Abstract

We reconsider the black hole firewall puzzle, emphasizing that quantum error-correction, computational complexity, and pseudorandomness are crucial concepts for understanding the black hole interior. We assume that the Hawking radiation emitted by an old black hole is pseudorandom, meaning that it cannot be distinguished from a perfectly thermal state by any efficient quantum computation acting on the radiation alone. We then infer the existence of a subspace of the radiation system which we interpret as an encoding of the black hole interior. This encoded interior is entangled with the late outgoing Hawking quanta emitted by the old black hole, and is inaccessible to computationally bounded observers who are outside the black hole. Specifically, efficient operations acting on the radiation, those with quantum computational complexity polynomial in the entropy of the remaining black hole, commute with a complete set of logical operators acting on the encoded interior, up to corrections which are exponentially small in the entropy. Thus, under our pseudorandomness assumption, the black hole interior is well protected from exterior observers as long as the remaining black hole is macroscopic. On the other hand, if the radiation is not pseudorandom, an exterior observer may be able to create a firewall by applying a polynomial-time quantum computation to the radiation.

Figures (12)

• Figure 1: A black hole forms due to the gravitational collapse of an infalling state of matter. The black hole then evaporates for a while, emitting the "early" radiation $E$ and the "late" radiation $B$; the formation and partial evaporation of the black hole are described by the unitary transformation $U_{\text{bh}}$. An observer $O$ interacts with the early radiation and a probe system $P$, where the unitary transformation $U_{\mathcal{E}}$ (enclosed by the dotted line) has quantum complexity which scales polynomially with the size $|H|$ of the remaining black hole (essentially its entropy $S_{\text{bh}}$). If the radiation is pseudorandom, then $O$ is unable to distinguish $E$ from a perfectly thermal state.
• Figure 2: An observer samples from a distribution and attempts to decide whether the distribution is uniformly random or not.
• Figure 3: Our toy model of a partially evaporated black hole, where $EB$ is the Hawking radiation emitted so far, and $H$ is the remaining black hole. The initial state $|\phi_{\text{matter}}\rangle$ of the gravitationally collapsing matter is modeled as a product state. We conjecture that the unitary black hole dynamics prepares a pseudorandom state of $EB$.
• Figure 4: Graphical depiction of the decoupling bound, which follows from the pseudorandomness of the Hawking radiation emitted by an old black hole. On the left, a black hole forms from collapse and partially evaporates; the emitted radiation is $EB$ and the remaining black hole is $H$. Then an observer $O$ and probe $P$ interact with the radiation subsystem $E$ for a time that scales polynomially with the initial black hole entropy $S_{\text{bh}}$. On the right, the unitary transformation describing the interaction of $OPE$ is the same as on the left, but the state of the Hawking radiation is replaced by a maximally mixed state of $EB$. The decoupling bound asserts that the final state of $OB$ is the same in both cases, up to an error that is exponentially small in $|H|$, the size of the remaining black hole, provided that $|O| \ll |H|$.
• Figure 5: The definition of the encoding of $\tilde{B}$ into $EH$, with $\Psi$ defined as in Figure \ref{['fig:psi']}.

Theorems & Definitions (31)

• Lemma 4.1: Ji2018, Lemma 1
• Definition 6.1
• Theorem 7.1: BenyOreshkov10, Theorem 1
• Lemma 7.2
• Definition 8.1
• Definition 8.2
• Lemma 8.3
• proof
• Theorem 8.4
• proof