Quantum Gravity

I will introduce a quantity from noncommutative probability theory called the free mutual information, and discuss how it can be used as a new diagnostic of quantum many-body chaos. This quantity captures a notion of the spreading of the operator in the abstract space of all possible time-evolved operators, rather than the more familiar notion of spreading in the physical space of degrees of freedom. I will discuss a precise relation between the free mutual information and the higher-point out-of-time-ordered correlators which applies in any system. I will further discuss the behaviour of the free mutual information in a few representative models of chaotic systems, and how it is sensitive to features of the dynamics beyond those captured by the four-point out-of-time-ordered correlator.

A scrambling unitary never destroys information according to quantum information/Shannon theory. However, this framework alone doesn’t capture the fact that scrambled information can be effectively inaccessible. This limitation points to the need for a new kind of information theory—one that quantifies how much information is scrambled, rather than how much is lost to noise. To address this, we propose introducing a new family of entropies into physics: free entropy. Unlike conventional quantum entropies, which are extensive under tensor independence, free entropy has the defining feature of extensivity under freeness—the appropriate notion of independence pertaining to quantum scrambling. I will present a preliminary result showing how free entropy naturally arises in a variant of Schumacher compression, providing it with an operational interpretation as the quantum minimum description length of quantum states. I will sketch how this interpretation extends to observables and unitaries, allowing free entropy to capture an operational aspect of quantum scrambling. Finally, I will highlight striking parallels between free entropy and von Neumann entropy, suggesting that free entropy may form the foundation of a new, complementary information theory.

I will discuss a concrete realization in N=4 SYM of the mechanism of cryptographic censorship: that sufficiently random time evolution in a holographic CFT incurs an event horizon in the bulk dual. I will show that perturbing half-BPS states by randomly distributing a large number of defects on them corresponds in the gravitational dual to exciting an extremal (horizonless) black hole to near-extremality. This random distribution of defects corresponds to acting with a typical random isometry on the half-BPS subspace. This allows us to interpret this process in the field theory as an extension of cryptographic censorship.

Our search for a quantum theory of gravity is aided by a unique and perplexing feature of the classical theory: General Relativity “knows” about its own quantum states (the entropy of a black hole), and about those of all matter (via the Quantum Focusing Conjecture). The results we are able to extract from classical gravity are inherently non-perturbative and increasingly sophisticated. Recent breakthroughs include a derivation of the entropy of Hawking radiation, a computation of the exact integer number of states of some black holes, a proof of the QFC, and the construction of gravitational holograms in general spacetimes. The nature of the oracle, and its full power, remain unknown.

The Ryu-Takayanagi formula implies that the entropy, a non-linear function of the state, is an observable in the semiclassical limit. This is a general phenomenon in a class of quantum error-correcting codes (QECC), but few specific area operators in the boundary CFT. In this work, I define a specific code in a holographic 2d CFT that includes all thermofield double states with arbitrary amounts of time evolution, but no bulk local degrees of freedom. While the naive area operator vanishes, it is possible to write down an area operator for appropriately classical states; the answer matches with previous results obtained from bulk considerations. Interestingly, this area operator involves an explicit coarse-graining. This suggests a construction for a non-isometric holographic map for these states.

The entanglement entropy for regions in a BMS field theory living at null infinity has been proposed to be holographically dual to certain ‘swing surfaces’ in flat space. We lift this construction to AdS/CFT and revisit both bulk and boundary aspects of this proposal.

In quantum gravity, the gravitational path integral involves a sum over topologies, representing the joining and splitting of multiple universes. To account for topology change, one is led to allow the creation and annihilation of both closed and open universes in a framework often called third quantization or universe field theory. We argue that since topology change in gravity is a rare event, its contribution to late-time physics should be universally governed by a Poisson distribution. In the Fock space of closed baby universes, this Poisson distribution corresponds to the statistics of the number operator in a coherent state, whereas allowing for the creation of asymptotic open universes calls for a non-commutative generalization of a Poisson process. We propose such an operator algebraic framework, called Poissonization, which takes as input the observable algebra and a (unnormalized) state of a quantum system and outputs a von Neumann algebra represented on its symmetric Fock space. Physically, our construction is a generalization of the coherent state vacua of bipartite quantum systems.

We construct algebras of diff-invariant observables in a global de Sitter universe with two observers and a free scalar QFT in two dimensions. In the limit when the observers have infinite mass and are localized along geodesics at the North and South poles, it was shown in previous work (CLPW) that their algebras are mutually commuting type II_1 factors. Away from this limit, we show that the algebras fail to commute and that they are type I non-factors. Physically, this is because the observers' trajectories are uncertain and state-dependent, and they may come into causal contact. We compute out-of-time-ordered correlators along an observer's worldline, and observe a Lyapunov exponent given by 4πβ_dS, as a result of observer recoil and de Sitter expansion. This should be contrasted with results from AdS gravity, and exceeds the chaos bound associated with the de Sitter temperature by a factor of two. We also discuss how the cosmological horizon emerges in the large mass limit and comment on implications for de Sitter holography.