A defining property of Hawking radiation is that states with very low entanglement masquerade as highly mixed states; this property is captured by a quantum computational phenomenon known as spoofing entanglement. Motivated by the potential implications for black hole information and the emergence of spacetime connectivity, as well as possible applications of spoofing entanglement, we investigate the geometrization of two types of entanglement spoofers in AdS/CFT: so-called EFI pairs and pseudoentangled state ensembles. We show that (a strengthened version of) EFI pairs with a semiclassical bulk dual have a Python’s Lunch; the maximally mixed state over the pseudoentangled state ensemble likewise features a Python’s Lunch. Since a Python’s Lunch must lie behind an event horizon, we find that black holes are the exclusive gravitational source of entanglement spoofing in the semiclassical limit. Finally, we use an extant construction of holographic pseudorandom states to yield a candidate example of a pseudoentangled state ensemble with a semiclassical bulk dual.

United States

Netta Engelhardt

Åsmund Folkestad

Adam Levine

Evita Verheijden

Lisa Yang

Spoofing entanglement in holography

It has been recently proposed that the naive semiclassical prediction of non-unitary black hole evaporation can be understood in the fundamental description of the black hole as a consequence of ignorance of high-complexity information. Validity of this conjecture implies that any algorithm which is polynomially bounded in computational complexity cannot accurately reconstruct the black hole dynamics. In this work, we prove that such bounded quantum algorithms cannot accurately predict (pseudo)random unitary dynamics, even if they are given access to an arbitrary set of polynomially complex observables under this time evolution; this shows that “learning” a (pseudo)random unitary is computationally hard. We use the common simplification of modeling black holes and more generally chaotic systems via (pseudo)random dynamics. The quantum algorithms that we consider are completely general, and their attempted guess for the time evolution of black holes is likewise unconstrained: it need not be a linear operator, and may be as general as an arbitrary (e.g. decohering) quantum channel.

The complexity of learning (pseudo)random dynamics of black holes and other chaotic systems

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.

Cryptographic Censorship

We propose an algebraic definition of ER=EPR in the G  N  → 0 limit, which associates bulk spacetime connectivity/disconnectivity to the operator algebraic structure of a quantum gravity system. The new formulation not only includes information on the amount of entanglement, but also more importantly the structure of entanglement. We give an independent definition of a quantum wormhole as part of the proposal. This algebraic version of ER=EPR sheds light on a recent puzzle regarding spacetime disconnectivity in holographic systems with   <math> <mi>O</mi> </math>  $$ \mathcal{O} $$ (1/G  N ) entanglement. We discuss the emergence of quantum connectivity in the context of black hole evaporation and further argue that at the Page time, the black hole-radiation system undergoes a transition involving the transfer of an emergent type III1 subalgebra of high complexity operators from the black hole to radiation. We argue this is a general phenomenon that occurs whenever there is an exchange of dominance between two competing quantum extremal surfaces.

Hong Liu

Algebraic ER=EPR and complexity transfer

Quantum error correction has given us a natural language for the emergence of spacetime, but the black hole interior poses a challenge for this framework: at late times the apparent number of interior degrees of freedom in effective field theory can vastly exceed the true number of fundamental degrees of freedom, so there can be no isometric (i.e. inner-product preserving) encoding of the former into the latter. In this paper we explain how quantum error correction nonetheless can be used to explain the emergence of the black hole interior, via the idea of “non-isometric codes protected by computational complexity”. We show that many previous ideas, such as the existence of a large number of “null states”, a breakdown of effective field theory for operations of exponential complexity, the quantum extremal surface calculation of the Page curve, post-selection, “state-dependent/state-specific” operator reconstruction, and the “simple entropy” approach to complexity coarse-graining, all fit naturally into this framework, and we illustrate all of these phenomena simultaneously in a soluble model.

Chris Akers

Daniel Harlow

Geoff Penington

Shreya Vardhan

The black hole interior from non-isometric codes and complexity

The Python’s Lunch conjecture for the complexity of bulk reconstruction involves two types of nonminimal quantum extremal surfaces (QESs): bulges and throats, which differ by their local properties. The conjecture relies on the connection between bulk spatial geometry and quantum codes: a constricting geometry from bulge to throat encodes the bulk state nonisometrically, and so requires an exponentially complex Grover search to decode. However, thus far, the Python’s Lunch conjecture is only defined for spacetimes where all QESs are spacelike-separated from one another. Here we explicitly construct (time-reflection symmetric) spacetimes featuring both timelike-separated bulges and timelike-separated throats. Interestingly, all our examples also feature a third type of QES, locally resembling a de Sitter bifurcation surface, which we name a bounce. By analyzing the Hessian of generalized entropy at a QES, we argue that this classification into throats, bulges and bounces is exhaustive. We then propose an updated Python’s Lunch conjecture that can accommodate general timelike-separated QESs and bounces. Notably, our proposal suggests that the gravitational analogue of a tensor network is not necessarily the time-reflection symmetric slice, even when one exists.

Arvin Shahbazi-Moghaddam

Twice upon a time: timelike-separated quantum extremal surfaces

We find a new on-shell replica wormhole in a computation of the generating functional of JT gravity coupled to matter. We show that this saddle has lower action than the disconnected one, and that it is stable under restriction to real Lorentzian sections, but can be unstable otherwise. The behavior of the classical generating functional thus may be strongly dependent on the signature of allowed perturbations. As part of our analysis, we give an LM-style construction for computing the on-shell action of replicated manifolds even as the number of boundaries approaches zero, including a type of one-step replica symmetry breaking that is necessary to capture the contribution of the new saddle. Our results are robust against quantum corrections; in fact, we find evidence that such corrections may sometimes stabilize this new saddle.

Venkatesa Chandrasekaran

Canada

Sebastian Fischetti

Sergio Hernández-Cuenca

A tale of two saddles

We show that the vacuum-subtracted maximal volume, the proposed holographic dual to complexity of formation, can be negative when contributions from compact directions are included. We construct explicit solutions with arbitrarily negative complexity of formation in asymptotically AdS4 × S 7 SUGRA. These examples rely critically on the compact directions, specifically the fact that the full eleven-dimensional spacetime is not asymptotically AdS11. While there is some ambiguity in the extension of the holographic complexity proposal to the compact directions, we show that the two natural candidates can both have arbitrarily negative complexity of formation in SUGRA solutions. We further find examples in which complexity can even decrease at late times, including cases of both single-sided geometries and two-sided wormholes. In particular, we construct a cosmological wormhole with simultaneously negative and decreasing complexity of formation (as computed by volume) at late times. We find a distinguished role for relevant primaries in these constructions and comment on possible interpretations.

Negative complexity of formation: the compact dimensions strike back

We show that the canonical purification of an evaporating black hole after the Page time consists of a short, connected, Lorentzian wormhole between two asymptotic boundaries, one of which is unitarily related to the radiation. This provides a quantitative and general realization of the predictions of <math><mrow><mi>ER</mi><mo>=</mo><mi>EPR</mi></mrow></math> in an evaporating black hole after the Page time; this further gives a standard <math><mrow><mi>AdS</mi><mo>/</mo><mi>CFT</mi></mrow></math> calculation of the entropy of the radiation (without modifications of the homology constraint). Before the Page time, the canonical purification consists of two disconnected, semiclassical black holes. From this, we construct two bipartite entangled holographic CFT states, with equal (and large) amount of entanglement, where the semiclassical dual of one has a connected ERB and the other does not. From this example, we speculate that measures of multipartite entanglement may offer a more complete picture into the emergence of spacetime.

Canonical purification of evaporating black holes

We prove a positive volume theorem for asymptotically AdS spacetimes: the maximal volume slice has nonnegative vacuum-subtracted volume, and the vacuum-subtracted volume vanishes if and only if the spacetime is identically pure AdS. Under the Complexity=Volume proposal, this constitutes a positive holographic complexity theorem. The result features a number of parallels with the positive energy theorem, including the assumption of an energy condition that excludes false vacuum decay (the AdS weak energy condition). Our proof is rigorously established in broad generality in four bulk dimensions, and we provide strong evidence in favor of a generalization to arbitrary dimensions. Our techniques also yield a holographic proof of Lloyd’s bound for a class of bulk spacetimes. We further establish a partial rigidity result for wormholes: wormholes with a given throat size are more complex than AdS-Schwarzschild with the same throat size.