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arXiv:1406.1471 · Journal of Statistical Mechanics Theory and Experiment

Entanglement contour

Yangang Chen, Guifré Vidal

Published Oct 9, 2014• 94 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/1742-5468/2014/10/p10011

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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.
We numerically explore the interplay of fractal geometry and quantum entanglement by analyzing the von Neumann entropy (known as entanglement entropy) and the entanglement contour in the scaling limit. Adopting quadratic fermionic models on Sierpinski carpet, we uncover intriguing findings. For gapless ground states exhibiting a finite density of states at the chemical potential, we reveal a super-area law characterized by the presence of a logarithmic correction for area law in the scaling of entanglement entropy. This extends the well-established super-area law observed on translationally invariant Euclidean lattices where the Gioev-Klich-Widom conjecture regarding the asymptotic behavior of Toeplitz matrices holds significant influence. Furthermore, different from the fractal structure of the lattice, we observe the emergence of a self-similar and universal pattern termed an “entanglement fractal” in the entanglement contour data as we approach the scaling limit. Remarkably, this pattern bears resemblance to intricate Chinese paper-cutting designs. We provide general rules to artificially generate this fractal, offering insights into the universal scaling of entanglement entropy. Meanwhile, as the direct consequence of the entanglement fractal and beyond a single scaling behavior of entanglement contour in translation-invariant systems, we identify two distinct scaling behaviors in the entanglement contour of fractal systems. For gapped ground states, we observe that the entanglement entropy adheres to a generalized area law, with its dependence on the Hausdorff dimension of the boundary between complementary subsystems. Published by the American Physical Society 2024
Abstract We study the entanglement contour, a quasi-local measure of entanglement, and propose a generic formula for the contour in 1 + 1d quantum systems. We use this formalism to investigate the real space entanglement structure of various static CFTs as well as local and global quantum quenches. The global quench elucidates the spatial distribution of entanglement entropy in strongly interacting CFTs and clarifies the interpretation of the entanglement tsunami picture. The entanglement tsunami effectively characterizes the non-local growth of entanglement entropy while the contour characterizes the local propagation of entanglement. We generalize the formula for the entanglement contour to arbitrary dimensions and entangling surface geometries using bit threads, and are able to realize a holographic contour for logarithmic negativity and the entanglement of purification by restricting the bulk spacetime to the entanglement wedge. Furthermore, we explore the connections between the entanglement contour, bit threads, and entanglement density in kinematic space.
In this paper, we study the entanglement structure of mixed states in quantum many-body systems using the negativity contour, a local measure of entanglement that determines which real-space degrees of freedom in a subregion are contributing to the logarithmic negativity and with what magnitude. We construct an explicit contour function for Gaussian states using the fermionic partial-transpose. We generalize this contour function to generic many-body systems using a natural combination of derivatives of the logarithmic negativity. Though the latter negativity contour function is not strictly positive for all quantum systems, it is simple to compute and produces reasonable and interesting results. In particular, it rigorously satisfies the positivity condition for all holographic states and those obeying the quasi-particle picture. We apply this formalism to quantum field theories with a Fermi surface, contrasting the entanglement structure of Fermi liquids and holographic (hyperscale violating) non-Fermi liquids. The analysis of non-Fermi liquids show anomalous temperature dependence of the negativity depending on the dynamical critical exponent. We further compute the negativity contour following a quantum quench and discuss how this may clarify certain aspects of thermalization.
The entanglement entropy of a pure quantum state of a bipartite system A union or logical sumB is defined as the von Neumann entropy of the reduced density matrix obtained by tracing over one of the two parts. In one dimension, the entanglement of critical ground states diverges logarithmically in the subsystem size, with a universal coefficient that for conformally invariant critical points is related to the central charge of the conformal field theory. We find that the entanglement entropy of a standard class of z=2 conformal quantum critical points in two spatial dimensions, in addition to a nonuniversal "area law" contribution linear in the size of the AB boundary, generically has a universal logarithmically divergent correction, which is completely determined by the geometry of the partition and by the central charge of the field theory that describes the critical wave function.
Entanglement contour characterizes the spatial structure of entanglement and quantifies the contribution from the degrees of freedom in any subset of the region A\mathcal{A}A to the total entanglement entropy SAS_{\mathcal{A}}SA​. Recently in \cite{Wen:2018whg} the author gave a simple proposal for the entanglement contour (or partial entanglement entropy) in two-dimensional theories, which involves the entanglement entropies of all the subsets inside A\mathcal{A}A. In this paper we explicitly study this proposal and show it satisfies many rational requirements for entanglement contour. Together with the holographic picture constructed with the modular planes \cite{Wen:2018whg}, we propose the correspondence between bulk geodesic chords and boundary partial entanglement entropies, which can be considered as a finer version of the Ryu-Takayanagi (RT) formula. As an example we calculate the fine correspondence between the points on A\mathcal{A}A and the points on the RT surface EA\mathcal{E}_{\mathcal{A}}EA​ for the BTZ black hole. We also give a strategy to extract the local modular flow from our proposal.
Entanglement and circuit complexity are fundamental concepts in quantum information theory which turned out to have considerable applications in various contexts ranging from many-body systems and condensed matter theory to quantum gravity and black hole physics. In order to find connections between these aspects, an insightful approach consists in comparing the complexity with some entanglement measures of a given bipartition, as for instance the Von Neumann entanglement entropy. Since, for a system in a given quantum state, the bipartite entanglement can be determined once a spatial subsystem and its complement are chosen, this comparison has to be made considering the complexity of reduced density matrices. This motivates the development of techniques for determining the circuit complexity of mixed states. In this thesis, focussing on free bosonic systems, we study the circuit complexity of mixed Gaussian states by employing the Fisher information geometry for the covariance matrices. This approach allows computing the complexity between reduced density matrices (subsystem complexity), as well as the complexity between thermal states. It also leads to a precise prescription for computing the spectrum and the basis complexity and the temporal evolution of the subsystem complexity after global and local quantum quenches. The study of the subsystem complexity of a given bipartition and a complete characterisation of its entanglement requires the detailed knowledge of the corresponding reduced density matrix, or, equivalently, of the entanglement Hamiltonian. A part of this thesis is devoted to determining the entanglement Hamiltonians in free theories. At equilibrium, we consider gapless systems, with the aim of recovering the underlying CFT results, and gapped lattice models. The evolution of entanglement Hamiltonians in quantum chains after a global quench is also studied. In addition, other entanglement quantifiers, as the entanglement spectrum and the contour for the entanglement entropies, are discussed in some of these settings. Finally, motivated by theoretical and experimental advances, we also address the issue of how the entanglement splits into the different charge sectors of a system endowed with a U(1) symmetry. We focus in particular on massive theories, either continuous or on the lattice, and we try to understand whether the entanglement equipartition, which is known to characterise gapless systems, survives away from criticality.