Entanglement Entropy Research Articles

Comparative microscopic study of entropies and their production

We study the time evolution of eleven microscopic entropy definitions (of Boltzmann-surface, Gibbs-volume, canonical, coarse-grained-observational, entanglement and diagonal type) and three microscopic temperature definitions (based on Boltzmann, Gibbs or canonical entropy). This is done for the archetypal nonequilibrium setup of two systems exchanging energy, modeled here with random matrix theory, based on numerical integration of the Schrödinger equation. We consider three types of pure initial states (local energy eigenstates, decorrelated and entangled microcanonical states) and three classes of systems: (A) two normal systems, (B) a normal and a negative temperature system and (C) a normal and a negative heat capacity system. We find: (1) All types of initial states give rise to the same macroscopic dynamics. (2) Entanglement and diagonal entropy sensitively depend on the microstate, in contrast to all other entropies. (3) For class B and C, Gibbs-volume entropies can violate the second law and the associated temperature becomes meaningless. (4) For class C, Boltzmann-surface entropies can violate the second law and the associated temperature becomes meaningless. (5) Canonical entropy has a tendency to remain almost constant. (6) For a Haar random initial state, entanglement or diagonal entropy behave similar or identical to coarse-grained-observational entropy.

Distillable entanglement under dually non-entangling operations

Computing the exact rate at which entanglement can be distilled from noisy quantum states is one of the longest-standing questions in quantum information. We give an exact solution for entanglement distillation under the set of dually non-entangling (DNE) operations—a relaxation of the typically considered local operations and classical communication, comprising all channels which preserve the sets of separable states and measurements. We show that the DNE distillable entanglement coincides with a modified version of the regularised relative entropy of entanglement in which the arguments are measured with a separable measurement. Ours is only the second known regularised formula for the distillable entanglement under any class of free operations in entanglement theory, after that given by Devetak and Winter for (one-way) local operations and classical communication. An immediate consequence of our finding is that, under DNE, entanglement can be distilled from any entangled state. As our second main result, we construct a general upper bound on the DNE distillable entanglement, using which we prove that the separably measured relative entropy of entanglement can be strictly smaller than the regularisation of the standard relative entropy of entanglement, solving an open problem posed by Li and Winter. Finally, we study also the reverse task of entanglement dilution and show that the restriction to DNE operations does not change the entanglement cost when compared with the larger class of non-entangling operations. This implies a strong form of irreversiblility of entanglement theory under DNE operations: even when asymptotically vanishing amounts of entanglement may be generated, entangled states cannot be converted reversibly.

Realizing the entanglement Hamiltonian of a topological quantum Hall system

Topological quantum many-body systems are characterized by a hidden order encoded in the entanglement between their constituents. While entanglement is often quantified using the entanglement entropy, its full description relies on the entanglement Hamiltonian, which is commonly used to identify complex phases arising in numerical simulations, but whose measurement remains an outstanding challenge. Here, we map entanglement to spectral properties by realizing a physical system whose single-particle dynamics is governed by the entanglement Hamiltonian of a quantum Hall system. We use a synthetic dimension, encoded in the electronic spin of dysprosium atoms, to implement spatially deformed dynamics, as suggested by the Bisognano-Wichmann prediction. The realized Hamiltonian, probed with bosonic atoms with negligible interactions, exhibits a chiral dispersion akin to a topological edge mode, revealing the fundamental link between entanglement and boundary physics. We numerically show that our protocol could be extended to interacting systems in fractional quantum Hall states.

A multiverse model in T2 dS wedge holography

We construct a multiverse model where empty AdSd+1 space is cut off by a pair of accelerated dSd space universes, at a finite AdS boundary cutoff which we treat as a T2 deformation in the holographic dual, and one in the AdS interior, the IR brane; and denote the construction as T2 dS wedge holography. We glue together several copies of this configuration along the UV cutoff and the IR branes in a periodic matter. To provide the model with dynamics similar to those of near Nariai black holes used in other multiverse toy models, we specialize to d = 2 and add dS JT gravity as an intrinsic gravity theory on the IR branes. We then study the entanglement entropy with respect to a finite cutoff observer, who finds a Page curve transition due to an entanglement island connecting the UV cutoff and IR brane. This process involves the coarse-graining of information outside the causally accessible region to the observer. Our model provides an explicit realization of entanglement between IR and UV degrees of freedom encoded in the multiverse.

Musings on SVD and pseudo entanglement entropies

Pseudo-entropy and SVD entropy are generalizations of the entanglement entropy that involve post-selection. In this work we analyze their properties as measures on the spaces of quantum states and argue that their excess provides useful characterization of a difference between two (i.e. pre-selected and post-selected) states, which shares certain features and in certain cases can be identified as a metric. In particular, when applied to link complement states that are associated to topological links via Chern-Simons theory, these generalized entropies and their excess provide a novel quantification of a difference between corresponding links. We discuss the dependence of such entropy measures on the level of Chern-Simons theory and determine their asymptotic values for certain link states. We find that imaginary part of the pseudo-entropy is sensitive to, and can diagnose chirality of knots. We also consider properties of entropy measures for simpler quantum mechanical systems, such as generalized SU(2) and SU(1,1) coherent states, and tripartite GHZ and W states.

Fix three dimensional geometries from entanglement entropies of CFT$_2$

Abstract In this paper, we propose an approach to fix the leading behaviors of three dimensional geometries from the dual CFT$_2$ entanglement entropies. We only employ the holographic principle and do not use any assumption about the AdS/CFT correspondence and bulk geometry. Our strategy is to use both UV and IR-like CFT$_2$ entanglement entropies to fix the bulk geodesics. By a simple trick, then the metric can be extracted from the geodesics. As examples, we fix the leading behaviors of the pure AdS$_3$ metric from the entanglement entropies of free CFT$_2$, and more importantly the BTZ black hole from the entanglement entropies of finite temperature CFT$_2$. Consequently, CFT$_2$ with finite size or topological defects can be worked out by simple transformations. Following the same steps, in principle, it is possible to fix the leading behaviors of all three dimensional (topologically distinct) holographic classical geometries from the dual CFT$_2$ entanglement entropies.Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Article funded by SCOAP3 and published under licence by Chinese Physical Society and the Institute of High Energy Physics of the Chinese Academy of Science and the Institute of Modern Physics of the Chinese Academy of Sciences and IOP Publishing Ltd.

Entanglement fractalization

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

Quantum real-time evolution of entanglement and hadronization in jet production: Lessons from the massive Schwinger model

The possible link between entanglement and thermalization, and the dynamics of hadronization are addressed by studying the real-time response of the massive Schwinger model coupled to external sources. This setup mimics the production and fragmentation of quark jets, as the Schwinger model and quantum chromodynamics (QCD) share the properties of confinement and chiral symmetry breaking. By using simulations of quantum dynamics on classical hardware, we study the entanglement between the produced jets, and observe the growth of the corresponding entanglement entropy in time. This growth arises from the increased number of contributing eigenstates of the reduced density matrix with sufficiently large and close eigenvalues. We also investigate the physical nature of these eigenstates, and find that at early times they correspond to fermionic Fock states. We then observe the transition from these fermionic Fock states to mesonlike bound states as a function of time. In other words, we observe how hadronization develops in real time. At late times, the local observables at midrapidity (such as the fermion density and the electric field) approach approximately constant values, suggesting the onset of equilibrium and approach to thermalization. Published by the American Physical Society 2024

Mode entanglement and isospin pairing in two-nucleon systems

Abstract In this study, we explore the entanglement and correlation in two-nucleon systems using isospin formalism. With the help of Slater decomposition, we derive analytical expressions for various entanglement measures. Specifically, we analyse the one- and two-mode entropies, mutual informations, and a basis-independent characteristic known as the one-body entanglement entropy.

To understand the impact of pairing, we consider interactions involving isovector and isoscalar L=0 pairing terms. Our findings show that certain pairing interactions can maximize one-body entanglement entropy of ground states when both total angular momentum and total isospin have zero projections.

We provide numerical examples for the sd shell and explore the mutual informations in LS coupled and jj coupled single-particle bases. We find that the shell structure and angular momentum coupling significantly impact the measures of entanglement. We outline the implications of conserving angular momentum and isospin on one-mode entropies, irrespective of particle number.

Exactly solvable inhomogeneous fermion systems

Abstract 15 exactly solvable inhomogeneous (spinless) fermion systems on one-dimensional lattices are constructed explicitly based on the discrete orthogonal polynomials of Askey scheme, e.g. the Krawtchouk, Hahn, Racah, Meixner, q-Racah polynomials. The Schrödinger and Heisenberg equations are solved explicitly, as the entire set of the eigenvalues and eigenstates are known explicitly. The ground state two point correlation functions are derived explicitly. The multi point correlation functions are obtained by Wick’s Theorem. Corresponding 15 exactly solvable XX spin systems are also displayed. They all have nearest neighbour interactions. The exact solvability of Schrödinger equation means that of the corresponding Fokker-Planck equation. This leads to 15 exactly solvable Birth and Death fermions and 15 Birth and Death spin models. These provide plenty of materials for calculating interesting quantities, e.g. entanglement entropy, etc.

Pseudoentropy sum rule by analytical continuation of the superposition parameter

In this paper, we establish a sum rule that connects the pseudoentropy and entanglement entropy of the superposition state. Through analytical continuation of the superposition parameter, we demonstrate that the transition matrix and density matrix of the superposition state can be treated in a unified manner. Within this framework, we naturally derive sum rules for the (reduced) transition matrix, pseudo-Rényi entropy, and pseudoentropy. Furthermore, we demonstrate the close relationship between the sum rule for pseudoentropy and the singularity structure of the entropy function for the superposition state after analytical continuation. We also explore potential applications of the sum rule, including its relevance to understanding the gravity dual of non-Hermitian transition matrices and establishing upper bounds for the absolute value of pseudoentropy.

Phenomenology of Many-Body Localization in Bond-Disordered Spin Chains.

Many-body localization (MBL) hinders the thermalization of quantum many-body systems in the presence of strong disorder. In this Letter, we study the MBL regime in bond-disordered spin-1/2 XXZ spin chain, finding the multimodal distribution of entanglement entropy in eigenstates, sub-Poissonian level statistics, and revealing a relation between operators and initial states required for examining the breakdown of thermalization in the time evolution of the system. We employ a real space renormalization group scheme to identify these phenomenological features of the MBL regime that extend beyond the standard picture of local integrals of motion relevant for systems with disorder coupled to on-site operators. Our results pave the way for experimental probing of MBL in bond-disordered spin chains.

Stretched horizon from conformal field theory

Recently, it has been observed that the Hartle-Hawking correlators, a signature of smooth horizon, can emerge from certain heavy excited state correlators in the (manifestly non-smooth) BTZ stretched horizon background, in the limit when the stretched horizon approaches the real horizon. In this note, we develop a framework of quantizing the CFT modular Hamiltonian, that explains the necessity of introducing a stretched horizon and the emergence of thermal features in the AdS-Rindler and (planar) BTZ backgrounds. In more detail, we quantize vacuum modular Hamiltonian on a spatial segment of S1, which can be written as a particular linear combination of sl(2,ℝ) generators. Unlike radial quantization, (Euclidean) time circles emerge naturally here which can be contracted smoothly to the ‘fixed points’(end points of the interval) of this quantization thus providing a direct link to thermal physics. To define a Hilbert space with discrete normalizable states and to construct a Virasoro algebra with finite central extension, a natural regulator (ϵ) is needed around the fixed points. Eventually, in the dual description the fixed points correspond to the horizons of AdS-Rindler patch or (planar) BTZ and the cut-off being the stretched horizon. We construct a (Lorentzian) highest weight representation of that Virasoro algebra where vacuum can be identified with certain boundary states on the cut-off surface. We further demonstrate that two point function in a (vacuum) descendant state of the regulated Hilbert space will reproduce thermal answer in ϵ → 0 limit which is analogous to the recent observation of emergent thermality in (planar) BTZ stretched horizon background. We also argue the thermal entropy of this quantization coincides with entanglement entropy of the subregion. Conversely, the microcanonical entropy corresponding to high energy density of states exactly reproduce the BTZ entropy. Quite remarkably, all these dominant high lying microstates are defined only at finite ϵ in the regulated Hilbert space. We expect that all our observations can be generalized to BTZ in stretched horizon background where the boundary spatial coordinate is compactified.

Non-Abelian symmetry-resolved entanglement entropy

We introduce a mathematical framework for symmetry-resolved entanglement entropy with a non-Abelian symmetry group. To obtain a reduced density matrix that is block-diagonal in the non-Abelian charges, we define subsystems operationally in terms of subalgebras of invariant observables. We derive exact formulas for the average and the variance of the typical entanglement entropy for the ensemble of random pure states with fixed non-Abelian charges. We focus on compact, semisimple Lie groups. We show that, compared to the Abelian case, new phenomena arise from the interplay of locality and non-Abelian symmetry, such as the asymmetry of the entanglement entropy under subsystem exchange, which we show in detail by computing the Page curve of a many-body system with SU(2)SU(2) symmetry.

Regularized entanglement entropy of electron-positron scattering with a witness photon

Regularized quantum information metrics are calculated for the scattering process e−e+→γ,Z→μ−μ+ that has a witness photon entangled with the initial electron-positron state. Unitarity implies the correct regularization of divergences that appear in both the final density matrix and von Neumann entanglement entropies. The entropies are found to quantify uncertainty or randomness. The variation of information, entanglement entropy, and correlation between the muon’s and witness photon’s helicities are found to convey equivalent information. The magnitude of the muon’s expected helicity rises (falls) as the helicity entropy falls (rises). Area, or the scattering cross section, is a source of entropy for the muon’s helicity entropy and momentum entropy. The muon’s differential angular entropy distribution is similar to the differential angular cross section distribution, capturing the forward-backward asymmetry at high center-of-mass energies. Published by the American Physical Society 2024

Locally purified density operators for noisy quantum circuits

Abstract Simulating open quantum systems is essential for exploring novel quantum phenomena and evaluating noisy quantum circuits. In this Letter, we address the problem of whether mixed states generated from noisy quantum circuits can be efficiently represented by locally purified density operators (LPDOs). We map an LPDO of N qubits to a pure state of size 2 × N defined on a ladder and introduce a unified method for managing virtual and Kraus bonds. We numerically simulate noisy random quantum circuits with depths up to d = 40 using fidelity and entanglement entropy as accuracy measures. LPDO representation proves to be effective in describing mixed states in both quantum and classical regions but encounters significant challenges at the quantum-classical critical point, limiting its applicability to the quantum region exclusively. In contrast, the matrix product operator (MPO) successfully characterizes the entanglement trend throughout the simulation, while truncation in MPOs breaks the positivity condition required for a physical density matrix. This work advances our understanding of efficient mixed-state representation in open quantum systems and provides insights into the entanglement structure of noisy quantum circuits.

Quantum entanglement and non-Hermiticity in free-fermion systems

Abstract This topical review article reports rapid progress on the generalization and application of entanglement in non-Hermitian free-fermion quantum systems. We begin by examining the realization of non-Hermitian quantum systems through the Lindblad master equation, alongside a review of typical non-Hermitian free-fermion systems that exhibit unique features. A pedagogical discussion is provided on the relationship between entanglement quantities and the correlation matrix in Hermitian systems. Building on this foundation, we focus on how entanglement concepts are extended to non-Hermitian systems from their Hermitian free-fermion counterparts, with a review of the general properties that emerge. Finally, we highlight various concrete studies, demonstrating that entanglement entropy remains a powerful diagnostic tool for characterizing non-Hermitian physics. The entanglement spectrum also reflects the topological characteristics of non-Hermitian topological systems, while unique non-Hermitian entanglement behaviors are also discussed. The review is concluded with several future directions. Through this review, we hope to provide a useful guide for researchers who are interested in entanglement in non-Hermitian quantum systems.

Local probe of bulk and edge states in a fractional Chern insulator.

The fractional quantum Hall effect is a key example of topological quantum many-body phenomena, arising from the interplay between strong electron correlation, topological order and time-reversalsymmetry breaking. Recently, a lattice analogue of the fractional quantum Hall effect at zero magnetic field has been observed, confirming the existence of a zero-field fractional Chern insulator (FCI). Despite this, the bulk-edge correspondence-a hallmark of a FCI featuring an insulating bulk with conductive edges-has not been directly observed. In fact, this correspondence has not been visualized in any system for fractional states owing to experimental challenges. Here we report the imaging of FCI edge states in twisted MoTe2 (t-MoTe2) using microwave impedance microscopy1. By tuning the carrier density, we observe the system evolving between metallic and FCI states, the latter of which exhibits insulating bulk and conductive edges, as expected from the bulk-boundary correspondence. Further analysis suggests the composite nature of the FCI edge states. We also observe the evolution of edge states across the topological phase transition as a function of interlayer electric field and reveal exciting prospects of neighbouring domains with different fractional orders. These findings pave the way for research into topologically protected one-dimensional interfaces between various anyonic states at zero magnetic field, such as gapped one-dimensional symmetry-protected phases with non-zero topological entanglement entropy, Halperin-Laughlin interfaces and the creation of non-abelian anyons.

Entanglement production through a cosmological bounce

In quantum cosmology, it is expected that the Big Bang singularity is resolved and the universe undergoes a bounce. We find that for Gaussian initial states, matter-gravity entanglement entropy rises rapidly during the bounce, declines, and then approaches a steady-state value following the bounce. These observations suggest that matter-gravity entanglement is a feature of the macroscopic universe and that there is no Second Law of entanglement entropy.

Transmission coefficient of super-Janus solution

We calculate the transmission coefficient of the super-Janus interface conformal field theory, both at weak and at strong coupling, where latter is described holographically as a domain-wall solution on AdS2 × S2 × M4 × Σ. Surprisingly we find perfect agreement between the free and strong coupling answer, mirroring a similar unexpected equivalence previously found for the entanglement entropy.