Abstract Quantum entanglement, crucial for understanding quantum many-body systems and quantum gravity, is commonly assessed through various measures such as von Neumann entropy, mutual information, and entanglement contour, each with its inherent limitations. In this work, we introduce the hyperfine structure of entanglement, which decomposes entanglement contours known as the fine structure into particle-number cumulants. This measure exhibits a set of universal properties with its significance in quantum information science. We apply it across diverse contexts: in Fermi gases, establishing connections to mutual information and interacting conformal field theory; in AdS 3 /CFT 2 holographic duality, unveiling finer subregion–subregion duality; and in Chern insulators, distinguishing between different quantum phases, especially topological gapped state and trivial gapped state. Our findings suggest experimental accessibility, offering fresh insights into quantum entanglement across physical systems.
Abstract Quantum entanglement, crucial for understanding quantum many-body systems and quantum gravity, is commonly assessed through various measures such as von Neumann entropy, mutual information, and entanglement contour, each with its inherent limitations. In this work, we introduce the hyperfine structure of entanglement, which decomposes entanglement contours known as the fine structure into particle-number cumulants. This measure exhibits a set of universal properties with its significance in quantum information science. We apply it across diverse contexts: in Fermi gases, establishing connections to mutual information and interacting conformal field theory; in AdS 3 /CFT 2 holographic duality, unveiling finer subregion–subregion duality; and in Chern insulators, distinguishing between different quantum phases, especially topological gapped state and trivial gapped state. Our findings suggest experimental accessibility, offering fresh insights into quantum entanglement across physical systems.
We study the structure and dynamics of entanglement in CFTs and black holes. We use a local entanglement measure, the entanglement contour, which is a spatial density function for von Neumann entropy with some additional properties. The entanglement contour can be calculated in many 1+1d condensed matter systems and simple models of black hole evaporation. We calculate the entanglement contour of a state excited by a splitting quench, and find universal results for the entanglement contours of low energy non-equilibrium states in 2d CFTs. We also calculate the contour of a non-gravitational bath coupled to an extremal AdS _2 2 black hole, and find that the contour only has finite support within the bath, due to an island phase transition. The particular entanglement contour proposal we use quantifies how well the bath’s state can be reconstructed from its marginals, through its connection to conditional mutual information, and the vanishing contour is a reflection of the protection of bulk island regions against erasures of the boundary state.
We study the structure and dynamics of entanglement in CFTs and black holes. We use a local entanglement measure, the entanglement contour, which is a spatial density function for von Neumann entropy with additional properties. We calculate the entanglement contour of a state excited by a splitting quench, and find universal results for the entanglement contours of low energy non-equilibrium states in 2d CFTs. We also calculate the contour of a non-gravitational bath coupled to an extremal AdS2 black hole, and find that the contour only has finite support within the bath, due to an island phase transition. The particular entanglement contour proposal we use quantifies how well the bath's state can be reconstructed from its marginals, through its connection to conditional mutual information, and the vanishing contour is a reflection of the protection of bulk island regions against erasures of the boundary state.
This paper explores the fundamental relationship between the geometry of entanglement and von Neumann entropy, shedding light on the intricate nature of quantum correlations. We provide a comprehensive overview of entanglement, highlighting its crucial role in quantum mechanics. Our focus centers on the connection between entanglement, von Neumann entropy, a measure of the information content within quantum systems and the geometry of composite Hilbert spaces. We discuss various methods for quantifying and characterizing entanglement through a geometric perspective and elucidate how this connection unveils the nature of quantum entanglement, offering valuable insights into the underlying structure of quantum systems. This study underscores the significance of geometry as a key tool for understanding the rich landscape of quantum correlations and their implications across various domains of physics and information theory. An example of entanglement as an indispensable resource for the task of state teleportation is presented at the end.
In the context of characterizing the structure of quantum entanglement in many-body systems, we introduce the entanglement contour, a tool to identify which real-space degrees of freedom contribute, and how much, to the entanglement of a region A with the rest of the system B . The entanglement contour provides a complementary, more refined approach to characterizing entanglement than just considering the entanglement entropy between A and B , with several concrete advantages. We illustrate this in the context of ground states and quantum quenches in fermionic quadratic systems. For instance, in a quantum critical system in D = 1 spatial dimensions, the entanglement contour allows us to determine the central charge of the underlying conformal field theory from just a single partition of the system into regions A and B (using the entanglement entropy for the same task requires considering several partitions). In D ⩾ 2 dimensions, the entanglement contour can distinguish between gapped and gapless phases that obey the same boundary law for entanglement entropy. During a local or global quantum quench, the time-dependent contour provides a detailed account of the dynamics of entanglement, including propagating entanglement waves, which offers a microscopic explanation of the behavior of the entanglement entropy as a function of time.
I study the mutual information between spatial subsystems in a variety of scale invariant quantum field theories. While it is derived from the bare entanglement entropy, the mutual information offers a more refined probe of the entanglement structure of quantum field theories because it remains finite in the continuum limit. I argue that the mutual information has certain universal singularities that are a manifestation of the idea of "entanglement per scale". Moreover, I propose a method, based on an ansatz for higher dimensional twist operators, to compute the entanglement entropy, Renyi entropy, and mutual information in a general quantum field theory. The relevance of these results to the search for renormalization group monotones, to holographic duality, and to entanglement based simulation methods for many body systems are all discussed.
In this thesis we will explore entanglement of various subsystems in quantum field theory, and its uses for describing underlying structure of states. We begin this journey by focusing on topological quantum field theory in chapters 1 and 2. These chapters are based heavily upon the papers [1] and [2, 3], respectively. In the introduction to chapter 1 we take the opportunity to recall the definition of bipartite entanglement entropy in quantum field theory and describe, briefly, how the entropy associated to connected spatial subregions provides a signature of topological order at the level of the wave-function. In particular, for a gapped phase with topological order, we recapitulate how the sub-leading correction to the “area-law” of the entanglement entropy contains coarse information about the topological phase while also being insensitive to the short-distance details of the underlying theory. We then present results that show that this sub-leading correction can also detect the class of interactions that glue a subregion to its complement. In principle, the subregion could be a different topological phase than its complement, and so we say that the entanglement entropy provides a signature of the interface between topological phases. In showing this, we discuss the classes of interactions between gapped topological phases and how they are represented as boundary conditions in the low-energy effective topological field theory, which in this case is an Abelian Chern-Simons theory. Using these boundary conditions, we perform path-integral calculations of the entanglement entropy using the replica trick and show that the universal contribution does indeed depend on the class of boundary conditions, and the gapped topological phases of the subsystem and its complement. To finish this chapter we reformulate this problem at the level of the Hilbert space and show how these universal contributions are related to resolving a generic ambiguity in defining subsystems of field theories with gauge invariance. We continue in the theme of topological field theories for chapter 2. Here we will shift focus from subregions of a connected spatial slice to disconnected spatial regions in an effort to explore, firstly, how long-range entanglement is encoded in topological field theory, and secondly, how that entanglement is encoded amongst multiple parties. We begin with a brief introduction of multi-party entanglement and recall the classification of three-party entanglement. Moving on to our specific setup, we describe the construction of states on multiple non-intersecting torus boundaries. Generically, these are states on link complements where the torus boundaries can be thought of as the edges of thickened closed strings entwined and knotted inside a compact manifold, which we will take to be the three-sphere. The wave-functions of these states are represented by various link invariants (such as the Gauss linking number for Abelian links, and Jones polynomial for SU(2)). We will exhibit several examples of multi-component links in both the Abelian and the SU(2) theories, and afterwards discuss generic features of the entanglement of link states in Chern-Simons theories with a compact group. In this discussion, we will conjecture a classification (albeit broad) of links via multi-partite entanglement of their corresponding states. We will end the chapter moving away from compact groups and discuss hyperbolic links in SL(2,C) Chern-Simons theory in the semi-classical limit. In particular, utilizing the generalized volume conjecture, we will show that the entanglement structure of hyperbolic links is greatly constrained in this semi-classical limit. Finally, in chapter 3 we switch gears and discuss renormalization of free bosonic field theory. The content of this chapter is drawn from the work of [4]. Motivated by work establishing duality between the singlet sector of O(N) vector models at large N and higher spin gauge theory on Anti de Sitter (AdS) space, we develop an exact renormalization group (ERG) framework for discussing the renormalization of a large class of O(N) singlet states. We will show that for the ground state and the excited states in this class the ERG is implemented by the action of unitary operators (that can be taken to be local). Consequently, the ERG framework can be regarded as a continuum tensor network. We contrast this tensor network with some well known tensor network renormalization schemes such as the muti-scale entanglement renormalization ansatz (MERA) and its proposed continuum version (cMERA). We also comment on the nature of this network with ordinary Wilsonian renormalization. One of the central features of the ERG network is how it acts on the momentum space entanglement of the field theory. In particular, we argue that for excited states, the ERG disentangles modes that lie just above and below the UV cutoff.