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"Dong, Xi"
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The gravity dual of Rényi entropy
A remarkable yet mysterious property of black holes is that their entropy is proportional to the horizon area. This area law inspired the holographic principle, which was later realized concretely in gauge-gravity duality. In this context, entanglement entropy is given by the area of a minimal surface in a dual spacetime. However, discussions of area laws have been constrained to entanglement entropy, whereas a full understanding of a quantum state requires Rényi entropies. Here we show that all Rényi entropies satisfy a similar area law in holographic theories and are given by the areas of dual cosmic branes. This geometric prescription is a one-parameter generalization of the minimal surface prescription for entanglement entropy. Applying this we provide the first holographic calculation of mutual Rényi information between two disks of arbitrary dimension. Our results provide a framework for efficiently studying Rényi entropies and understanding entanglement structures in strongly coupled systems and quantum gravity.
The discovery that the entropy of black holes is given by their horizon area inspired the holographic principle and led to gauge-gravity duality. Here, the author shows that all Rényi entropies satisfy a similar area law in holographic theories and are given by the areas of dual cosmic branes.
Journal Article
One-loop universality of holographic codes
A
bstract
Recent work showed holographic error correcting codes to have simple universal features at
O
(1
/G
). In particular, states of fixed Ryu-Takayanagi (RT) area in such codes are associated with flat entanglement spectra indicating maximal entanglement between appropriate subspaces. We extend such results to one-loop order (
O
(1) corrections) by controlling both higher-derivative corrections to the bulk effective action and dynamical quantum fluctuations below the cutoff. This result clarifies the relation between the bulk path integral and the quantum code, and implies that i) simple tensor network models of holography continue to match the behavior of holographic CFTs beyond leading order in
G
, ii) the relation between bulk and boundary modular Hamiltonians derived by Jafferis, Lewkowycz, Maldacena, and Suh holds as an operator equation on the code subspace and not just in code-subspace expectation values, and iii) the code subspace is invariant under an appropriate notion of modular flow. A final corollary requires interesting cancelations to occur in the bulk renormalization-group flow of holographic quantum codes. Intermediate technical results include showing the Lewkowycz-Maldacena computation of RT entropy to take the form of a Hamilton-Jacobi variation of the action with respect to boundary conditions, corresponding results for higher-derivative actions, and generalizations to allow RT surfaces with finite conical angles.
Journal Article
Entropy, extremality, euclidean variations, and the equations of motion
A
bstract
We study the Euclidean gravitational path integral computing the Rényi entropy and analyze its behavior under small variations. We argue that, in Einstein gravity, the extremality condition can be understood from the variational principle at the level of the action, without having to solve explicitly the equations of motion. This set-up is then generalized to arbitrary theories of gravity, where we show that the respective entanglement entropy functional needs to be extremized. We also extend this result to all orders in Newton’s constant
G
N
, providing a derivation of quantum extremality. Understanding quantum extremality for mixtures of states provides a generalization of the dual of the boundary modular Hamiltonian which is given by the bulk modular Hamiltonian plus the area operator, evaluated on the so-called modular extremal surface. This gives a bulk prescription for computing the relative entropies to all orders in
G
N
. We also comment on how these ideas can be used to derive an integrated version of the equations of motion, linearized around arbitrary states.
Journal Article
Enhanced corrections near holographic entanglement transitions: a chaotic case study
A
bstract
Recent work found an enhanced correction to the entanglement entropy of a subsystem in a chaotic energy eigenstate. The enhanced correction appears near a phase transition in the entanglement entropy that happens when the subsystem size is half of the entire system size. Here we study the appearance of such enhanced corrections holo-graphically. We show explicitly how to find these corrections in the example of chaotic eigenstates by summing over contributions of all bulk saddle point solutions, including those that break the replica symmetry. With the help of an emergent rotational symmetry, the sum over all saddle points is written in terms of an effective action for cosmic branes. The resulting Renyi and entanglement entropies are then naturally organized in a basis of fixed-area states and can be evaluated directly, showing an enhanced correction near holographic entanglement transitions. We comment on several intriguing features of our tractable example and discuss the implications for finding a convincing derivation of the enhanced corrections in other, more general holographic examples.
Journal Article
Holographic entanglement negativity and replica symmetry breaking
A
bstract
Since the work of Ryu and Takayanagi, deep connections between quantum entanglement and spacetime geometry have been revealed. The negative eigenvalues of the partial transpose of a bipartite density operator is a useful diagnostic of entanglement. In this paper, we discuss the properties of the associated
entanglement negativity
and its Rényi generalizations in holographic duality. We first review the definition of the Rényi negativities, which contain the familiar logarithmic negativity as a special case. We then study these quantities in the random tensor network model and rigorously derive their large bond dimension asymptotics. Finally, we study entanglement negativity in holographic theories with a gravity dual, where we find that Rényi negativities are often dominated by bulk solutions that break the replica symmetry. From these replica symmetry breaking solutions, we derive general expressions for Rényi negativities and their special limits including the logarithmic negativity. In fixed-area states, these general expressions simplify dramatically and agree precisely with our results in the random tensor network model. This provides a concrete setting for further studying the implications of replica symmetry breaking in holography.
Journal Article
Bulk locality and quantum error correction in AdS/CFT
A
bstract
We point out a connection between the emergence of bulk locality in AdS/CFT and the theory of quantum error correction. Bulk notions such as Bogoliubov transformations, location in the radial direction, and the holographic entropy bound all have natural CFT interpretations in the language of quantum error correction. We also show that the question of whether bulk operator reconstruction works only in the causal wedge or all the way to the extremal surface is related to the question of whether or not the quantum error correcting code realized by AdS/CFT is also a “quantum secret sharing scheme”, and suggest a tensor network calculation that may settle the issue. Interestingly, the version of quantum error correction which is best suited to our analysis is the somewhat nonstandard “operator algebra quantum error correction” of Beny, Kempf, and Kribs. Our proposal gives a precise formulation of the idea of “subregion-subregion” duality in AdS/CFT, and clarifies the limits of its validity.
Journal Article
Flat entanglement spectra in fixed-area states of quantum gravity
A
bstract
We use the Einstein-Hilbert gravitational path integral to investigate gravita- tional entanglement at leading order
O
(1
/G
). We argue that semiclassical states prepared by a Euclidean path integral have the property that projecting them onto a subspace in which the Ryu-Takayanagi or Hubeny-Rangamani-Takayanagi surface has definite area gives a state with a flat entanglement spectrum at this order in gravitational perturbation theory. This means that the reduced density matrix can be approximated as proportional to the identity to the extent that its Renyi entropies
Sn
are independent of
n
at this order. The
n
-dependence of
Sn
in more general states then arises from sums over the RT/HRT- area, which are generally dominated by different values of this area for each
n
. This provides a simple picture of gravitational entanglement, bolsters the connection between holographic systems and tensor network models, clarifies the bulk interpretation of alge- braic centers which arise in the quantum error-correcting description of holography, and strengthens the connection between bulk and boundary modular Hamiltonians described by Jafferis, Lewkowycz, Maldacena, and Suh.
Journal Article
Circularly polarized luminescence of agglomerate emitters
Circularly polarized luminescence (CPL) originates from the chiral emissive excited states. CPL materials have promising applications in 3D optical displays, encryptions, biological probes, chiral photoelectric devices, and CPL switches, most of which require excellent CPL performances including bright luminescence and high luminescence dissymmetry factor (glum) in the agglomerate state. This review systematically summarizes the progress about CPL of aggregate and solid materials, such as organic materials, metal‐organic materials (such as coordination polymers, organic‐inorganic metal halides, metal clusters, and cluster‐assembled materials), the assembled materials by supramolecular interactions, and liquid crystals. We also present the current challenges and a future perspective of chiral emitting agglomerate materials. In this review, besides the fundamental principles of circularly polarized luminescence (CPL), CPL measurement methods for solid‐state samples and the recent progress of the diverse agglomerate CPL‐emitters, including organic materials, crystalline metal‐organic materials (coordination polymers, organic–inorganic metal halides, metal clusters, and cluster‐assembled materials), supramolecular ensembles, and liquid crystals are summarized. The opportunities and challenges of CPL‐emitters in agglomerate state are presented.
Journal Article