Cryptographic Censorship

TL;DR

The paper argues that when holographic CFT time evolution on a code subspace is sufficiently pseudorandom, an event horizon must form in the bulk within finite time, providing a quantum analogue of weak cosmic censorship (Cryptographic Censorship). It develops a rigorous connection between computational learning limits (YanEng23) and bulk reconstruction, showing that any efficient boundary algorithm capable of reconstructing the causal wedge cannot predict all bulk observables if a horizon is absent, implying horizons exist for typical states. It then introduces a quantitative framework to classify naked singularities by size, showing classical and semi-Planckian singularities can generate gravitational pseudorandomness and hence must be hidden behind horizons, while Planckian cases may escape. The authors propose Quantum Cosmic Censorship, positing that typical quantum-gravity states with emergent semiclassical geometry do not harbor large naked singularities, and outline future paths to a fully quantum formulation of censorship in holography.

Abstract

We formulate and take two large strides towards proving a quantum version of the weak cosmic censorship conjecture. We first prove "Cryptographic Censorship": a theorem showing that when the time evolution operator of a holographic CFT is approximately pseudorandom (or Haar random) on some code subspace, then there must be an event horizon in the corresponding bulk dual. This result provides a general condition that guarantees (in finite time) event horizon formation, with minimal assumptions about the global spacetime structure. Our theorem relies on an extension of a recent quantum learning no-go theorem and is proved using new techniques of pseudorandom measure concentration. To apply this result to cosmic censorship, we separate singularities into classical, semi-Planckian, and Planckian types. We illustrate that classical and semi-Planckian singularities are compatible with approximately pseudorandom CFT time evolution; thus, if such singularities are indeed approximately pseudorandom, by Cryptographic Censorship, they cannot exist in the absence of event horizons. This result provides a sufficient condition guaranteeing that seminal holographic results on quantum chaos and thermalization, whose general applicability relies on typicality of horizons, will not be invalidated by the formation of naked singularities in AdS/CFT.

Figures (16)

• Figure 1: The evaporation point of a black hole is, in some sense that we make precise in Sec. \ref{['sec:size-of-sings']}, "small".
• Figure 2: A spacetime with a Python's Lunch. In the Penrose diagram on the left, the lunch, causal wedge ($\mathcal{W}_{\rm C}$) and the wedge of the outermost QES ($\mathcal{W}_{\rm O}$) are indicated. The left and right green dots are the minimal and outermost QES, respectively. The right panel depicts a timeslice.
• Figure 3: A forbidden situation: a QES in causal contact with the boundary.
• Figure 4: An example where null geodesics fired from $\chi$ are geodesically incomplete, but there is no horizon.
• Figure 5: A QES outside the horizon, leading to a contradiction with the unitary invariance of von Neumann entropy: $S_{\rm vN}[\rho_{\mathscr{I}}] = S_{\rm vN}[U^{\dag}\rho_{\mathscr{I}} U]$.

Theorems & Definitions (29)

• Definition 1: Computational Indistinguishability
• theorem 1
• proof
• Definition 2: Pseudorandom State Ensemble (PRS); Definition 2 in JiLiu18
• Definition 3: Pseudorandom Unitary Ensemble (PRU); Definition 5 in JiLiu18
• theorem 2: Complexity of Learning Pseudorandom Unitaries, Theorem 2 of YanEng23
• Definition 4: Gravitationally Pseudorandom Unitary Ensemble (GPRU)
• theorem 3: Complexity of Learning for Typical Pseudorandom Unitaries
• proof
• Corollary 3.1: Distinguishing Operator at $N\to\infty$