Mutual information of excited states and relative entropy of two disjoint subsystems in CFT

In this paper, we first study mutual information of excited states in the small subsystem size limit in generic conformal field theory. We then discuss relative entropy of two disjoint subsystems in the same limit.Comment: 18 page

Modular Hamiltonians of excited states, OPE blocks and emergent bulk fields

We study the entanglement entropy and the modular Hamiltonian of slightly excited states reduced to a ball shaped region in generic conformal field theories. We set up a formal expansion in the one point functions of the state in which all orders are explicitly given in terms of integrals of multi-point functions along the vacuum modular flow, without a need for replica index analytic continuation. We show that the quadratic order contributions in this expansion can be calculated in a way expected from holography, namely via the bulk canonical energy for the entanglement entropy, and its variation for the modular Hamiltonian. The bulk fields contributing to the canonical energy are defined via the HKLL procedure. In terms of CFT variables, the contribution of each such bulk field to the modular Hamiltonian is given by the OPE block corresponding to the dual operator integrated along the vacuum modular flow. These results do not rely on assuming large $N$ or other special properties of the CFT and therefore they are purely kinematic.Comment: 40 pages, 2 figures. v3: some typos corrected, references added, extended discussion on convergence and holographic interpretatio

An Entropy Formula for Higher Spin Black Holes via Conical Singularities

We consider the entropy of higher spin black holes in 2+1 dimensions using the conical singularity approach. By introducing a conical singularity along a non contractible cycle and carefully evaluating its contribution to the Chern Simons action, we derive a simple expression for the entropy of a general stationary higher spin black hole. The resulting formula is shown to satisfy the first law of thermodynamics, and yields agreement with previous results based on integrating the first law.Comment: 18 pages, typos correcte

Chaos and relative entropy

One characterization of a chaotic system is the quick delocalization of quantum information (fast scrambling). One therefore expects that in such a system a state quickly becomes locally indistinguishable from its perturbations. In this paper we study the time dependence of the relative entropy between the reduced density matrices of the thermofield double state and its perturbations in two dimensional conformal field theories. We show that in a CFT with a gravity dual, this relative entropy exponentially decays until the scrambling time. This decay is not uniform. We argue that the early time exponent is universal while the late time exponent is sensitive to the butterfly effect. This large $c$ answer breaks down at the scrambling time, therefore we also study the relative entropy in a class of spin chain models numerically. We find a similar universal exponential decay at early times, while at later times we observe that the relative entropy has large revivals in integrable models, whereas there are no revivals in non-integrable models.Comment: 34+11 pages, 8 figure

Towards an Entanglement Measure for Mixed States in CFTs Based on Relative Entropy

Relative entropy of entanglement (REE) is an entanglement measure of bipartite mixed states, defined by the minimum of the relative entropy $S(\rho_{AB}|| \sigma_{AB})$ between a given mixed state $\rho_{AB}$ and an arbitrary separable state $\sigma_{AB}$. The REE is always bounded by the mutual information $I_{AB}=S(\rho_{AB} || \rho_{A}\otimes \rho_{B})$ because the latter measures not only quantum entanglement but also classical correlations. In this paper we address the question of to what extent REE can be small compared to the mutual information in conformal field theories (CFTs). For this purpose, we perturbatively compute the relative entropy between the vacuum reduced density matrix $\rho^{0}_{AB}$ on disjoint subsystems $A \cup B$ and arbitrarily separable state $\sigma_{AB}$ in the limit where two subsystems A and B are well separated, then minimize the relative entropy with respect to the separable states. We argue that the result highly depends on the spectrum of CFT on the subsystems. When we have a few low energy spectrum of operators as in the case where the subsystems consist of a finite number of spins in spin chain models, the REE is considerably smaller than the mutual information. However in general our perturbative scheme breaks down, and the REE can be as large as the mutual information.Comment: 35 pages, 2 figure

Moving islands [rafts]: a collective art conglomeration in second life

This chapter will discuss both the artistic processes as well as the background and the related theoretical premises of a collaborative art undertaking that emerged as an evolving combination of creative output which was displayed in Second Life® Fall 2013, and Spring 2014. Despite the idiosyncratic, highly individualized nature of its components, the project nevertheless achieved a remarkable state of cohesion. What may have contributed to this visual unity will be the subject under investigation at the core of this text. Since these constitute the background of the project, the text will commence with a survey of the creative mechanisms and strategies of the metaverse, also bringing in Axel Bruns's term 'produsage' and Brian Eno's notion of the 'unfinished artifact' as strongly related terms; after which a description of the project, its curatorial premises, including the usage of metaverse geography and climate as an agent of visual harmony will also be delivered. The chapter will then conclude with an examination of the collective art process within the context of the 'unfinished artifact' and John Dewey's deliberations on the experiential nature of artwork/art process as a potential framework for metaverse artistic collaborations