Entanglement Entropy Research Articles

Quantum phase transition and entanglement entropy in the Dicke model with a squeezed light

Quantum phase transition and entanglement entropy in the Dicke model with a squeezed light

Non-Abelian braiding of Fibonacci anyons with a superconducting processor

Quantum many-body systems with a non-Abelian topological order can host anyonic quasiparticles. It has been proposed that anyons could be used to encode and manipulate information in a topologically protected manner that is immune to local noise, with quantum gates performed by braiding and fusing anyons. Unfortunately, realizing non-Abelian topologically ordered states is challenging, and it was not until recently that the signatures of non-Abelian statistics were observed through digital quantum simulation approaches. However, not all forms of topological order can be used to realize universal quantum computation. Here we use a superconducting quantum processor to simulate non-Abelian topologically ordered states of the Fibonacci string-net model and demonstrate braidings of Fibonacci anyons featuring universal computational power. We demonstrate the non-trivial topological nature of the quantum states by measuring the topological entanglement entropy. In addition, we create two pairs of Fibonacci anyons and demonstrate their fusion rule and non-Abelian braiding statistics by applying unitary gates on the underlying physical qubits. Our results establish a digital approach to explore non-Abelian topological states and their associated braiding statistics with current noisy intermediate-scale quantum processors.

Long-time properties of generic Floquet systems are approximately periodic with the driving period

A Floquet quantum system is governed by a Hamiltonian that is periodic in time. Consider the space of piecewise time-independent Floquet systems with (geometrically) local interactions. We prove that for all but a measure zero set of systems in this space, starting from a random product state, many properties (including expectation values of observables and the entanglement entropy of a macroscopically large subsystem) at long times are approximately periodic with the same period as the Hamiltonian. Thus, in almost every Floquet system of arbitrarily large but finite size, discrete time-crystalline behavior does not persist to strictly infinite time.

Entanglement Entropy of Free Fermions with a Random Matrix as a One-Body Hamiltonian.

We consider a quantum system of large size N and its subsystem of size L, assuming that N is much larger than L, which can also be sufficiently large, i.e., 1≪L≲N. A widely accepted mathematical version of this inequality is the asymptotic regime of successive limits: first the macroscopic limit N→∞, then an asymptotic analysis of the entanglement entropy as L→∞. In this paper, we consider another version of the above inequality: the regime of asymptotically proportional L and N, i.e., the simultaneous limits L→∞,N→∞,L/N→λ>0. Specifically, we consider a system of free fermions that is in its ground state, and such that its one-body Hamiltonian is a large random matrix, which is often used to model long-range hopping. By using random matrix theory, we show that in this case, the entanglement entropy obeys the volume law known for systems with short-range hopping but described either by a mixed state or a pure strongly excited state of the Hamiltonian. We also give streamlined proof of Page's formula for the entanglement entropy of black hole radiation for a wide class of typical ground states, thereby proving the universality and the typicality of the formula.

Entanglement entropy in string compactifications

We consider ℤN orbifolds of Type-II compactifications to four and six dimensions on several Calabi-Yau manifolds in the orbifold limit with the aim to compute the entanglement entropy. The spectrum can contain tachyons in the doubly-twisted sectors which can lead to new infrared divergences for the partition function that are not present in the orbifolds of the uncompactified ten-dimensional theory. We show that all tachyonic contributions in these models admit a resummation and analytic continuation that yields finite entropy in the physical region 0 < N ≤ 1 just as in ten dimensions.

The influence functional in open holography: entanglement and Rényi entropies

Open quantum systems are defined as ordinary unitary quantum theories coupled to a set of external degrees of freedom, which are introduced to take on the rôle of an unobserved environment. Here we study examples of open quantum field theories, with the aid of the so-called Feynman-Vernon Influence Functional (“IF”), including field theories that arise in holographic duality. We interpret the system in the presence of an IF as an open effective field theory, able to capture the effect of the unobserved environment. Our main focus is on computing Rényi and entanglement entropies in such systems, whose description from the IF, or “open EFT”, point of view we develop in this paper. The issue of computing the entanglement-Rényi entropies in open quantum systems is surprisingly rich, and we point out how different prescriptions for the IF may be appropriate depending on the application of choice. A striking application of our methods concerns the fine-grained entropy of subsystems when including gravity in the setup, for example when considering the Hawking radiation emitted by black holes. In this case we show that one prescription for the IF leads to answers consistent with unitary evolution, while the other merely reproduces standard EFT results, well known to be inconsistent with unitary global evolution. We establish these results for asymptotically AdS gravity in arbitrary dimensions, and illustrate them with explicit analytical expressions for the IF in the case of matter-coupled JT gravity in two dimensions.

Boundary-induced transitions in Möbius quenches of holographic BCFT

Boundary effects play an interesting role in finite-size physical systems. In this work, we study the boundary-induced properties of 1+1-dimensional critical systems driven by inhomogeneous Möbius-like quenches. We focus on the entanglement entropy in BCFTs with a large central charge and a sparse spectrum of low-dimensional operators. We find that the choice of boundary conditions leads to different scenarios of dynamical phase transitions. We also derive these results in a holographic description in terms of intersecting branes in AdS3, and find a precise match.

Monogamy of quantum correlations shared in a cavity-freeoptomechanical system

In this work, we discuss the monogamy property of quantum correlations (entanglement and steering) distributed in a cavity-free optomechanical system. In a tripartite pure Gaussian state, we verify the hierarchy of the studied quantum correlations. We investigate in detail the CKW-type monogamy inequalities, based on Gaussian Rényi-2 entropy entanglement, as well as, Gaussian steering. We quantify genuine tripartite entanglement by using the minimum residual Gaussian entanglement. In addition, we consider the minimum residual Gaussian steering as a quantifier of genuine tripartite steering. Our results indicate that, in the one-mode versus two-mode Gaussian state, the correlations hierarchy was respected; in the considered pure state, the monogamy of entanglement and steering distribution was proven and the studied genuine tripartite correlations have similar time evolution under the same conditions.

Entanglement and pseudo entanglement dynamics versus fusion in CFT

The fusion rules and operator product expansion (OPE) serve as crucial tools in the study of operator algebras within conformal field theory (CFT). Building upon the vision of using entanglement to explore the connections between fusion coefficients and OPE coefficients, we employ the replica method and Schmidt decomposition method to investigate the time evolution of entanglement entropy (EE) and pseudo entropy (PE) for linear combinations of operators in rational conformal field theory (RCFT). We obtain a formula that links fusion coefficients, quantum dimensions, and OPE coefficients. We also identify two definition schemes for linear combination operators. Under one scheme, the EE captures information solely for the heaviest operators, while the PE retains information for all operators, reflecting the phenomenon of pseudo entropy amplification. Irrespective of the scheme employed, the EE demonstrates a step-like evolution, illustrating the effectiveness of the quasiparticle propagation picture for the general superposition of locally excited states in RCFT. From the perspective of quasiparticle propagation, we observe spontaneous block-diagonalization of the reduced density matrix of a subsystem when quasiparticles enter the subsystem.

The Fermionic Entanglement Entropy of the Vacuum State of a Schwarzschild Black Hole Horizon

AbstractWe define and analyze the fermionic entanglement entropy of a Schwarzschild black hole horizon for the regularized vacuum state of an observer at infinity. Using separation of variables and an integral representation of the Dirac propagator, the entanglement entropy is computed to be a prefactor times the number of occupied angular momentum modes on the event horizon.

Imprints of phase transitions on Kasner singularities

Under the AdS/CFT correspondence, asymptotically anti–de Sitter geometries with backreaction can be viewed as conformal field theory states subject to a renormalization group (RG) flow from an ultraviolet (UV) description toward an infrared (IR) sector. For black holes, however, the IR point is the horizon, so one way to interpret the interior is as an analytic continuation to a “trans-IR” imaginary-energy regime. In this paper, we demonstrate that this analytic continuation preserves some imprints of the UV physics, particularly near its “end point” at the classical singularity. We focus on holographic phase transitions of geometric objects in round black holes. We first assert the consistency of interpreting such black holes, including their interiors, as RG flows by constructing a monotonic a function. We then explore how UV phase transitions of entanglement entropy and scalar two-point functions, each of which are encoded by bulk geometry under the holographic mapping, are related to the structure of the near-singularity geometry, which is quantified by Kasner exponents. Using 2D holographic flows triggered by relevant scalar deformations as test beds, we find that the 3D bulk’s near-singularity Kasner exponents can be viewed as functions of the UV physics precisely when the deformation is nonzero. Published by the American Physical Society 2024

New Additivity Properties of the Relative Entropy of Entanglement and Its Generalizations

New Additivity Properties of the Relative Entropy of Entanglement and Its Generalizations

Inhomogeneous quenches as state preparation in two-dimensional conformal field theories

The nonequilibrium process where the system does not evolve to the featureless state, that without quantum properties, is one of the new central objects in the nonequilibrium phenomena. In this paper, starting from the short-range entangled state in the two-dimensional conformal field theories (2D CFTs), the boundary state with a regularization, we evolve the system with the inhomogeneous Hamiltonians called Möbius/sine-square-deformed (SSD) ones. Regardless of the details of CFTs considered in this paper, during the Möbius evolution, the entanglement entropy exhibits the periodic motion called quantum revival. During SSD time evolution, except for some subsystems, in the large time regime, entanglement entropy and mutual information are approximated by those for the vacuum state. We argue the time regime for the subsystem to cool down to vacuum one is t1≫O(LlA), where t1, L, and lA are time, system, and subsystem sizes. This finding suggests the inhomogeneous quench induced by the SSD Hamiltonian may be used as the preparation for the approximately vacuum state. We propose the gravity dual of the systems considered in this paper, furthermore, and generalize it. In addition to them, we discuss the relation between the inhomogeneous quenches and continuous multiscale entanglement renormalization ansatz (cMERA). Published by the American Physical Society 2024

Quench dynamics in lattices above one dimension: The free fermionic case

We begin a systematic investigation of quench dynamics in higher-dimensional lattice systems considering the case of noninteracting fermions with conserved particle number. We prepare the system in a translational-invariant nonequilibrium initial state, the simplest example being a classical configuration with fermions at fixed positions on the lattice, and let it evolve in time. We characterize the system's dynamics by measuring the entanglement between a finite connected region and its complement. We observe the transmutation of entanglement entropy into thermodynamic entropy and investigate how this process depends on the shape and orientation of the region with respect to the underlying lattice. Interestingly, we find that irregular regions display a distinctive multislope entanglement growth, while the dependence on the orientation angle is generically fairly weak. This is particularly true for regions with a large (discrete) rotational symmetry group. The main tool of our analysis is the celebrated quasiparticle picture of Calabrese and Cardy, which we generalize to describe the case at hand. Specifically, we show that for generic initial configurations (even when restricting to classical ones) one has to allow for the production of multiplets involving n&gt;2 quasiparticles and carrying nondiagonal correlations. We obtain quantitatively accurate predictions, tested against exact numerics, and propose an efficient Monte Carlo based scheme to evaluate them for arbitrary connected regions of generic higher-dimensional lattices. Published by the American Physical Society 2024

Symmetry resolution in non-Lorentzian field theories

Starting from the computation of Symmetry Resolved Entanglement Entropy (SREE) for boosted intervals in a two dimensional Conformal Field Theory, we compute the same in various non-Lorentzian limits, viz, Galilean and Carrollian Conformal Field Theory in same number of dimensions. We approach the problem both from a limiting perspective and by using intrinsic symmetries of respective non-Lorentzian conformal algebras. In particular, we calculate the leading order terms, logarithmic terms, and the O1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{O}(1) $$\\end{document} terms and explicitly show exact compliance with equipartition of entanglement, even in the non-Lorentzian system. Keeping in mind the holographic origin of SREE for the Carrollian limit, we further compute SREE for BMS3-Kac-Moody, which couples a U(1) × U(1) theory with bulk gravity.

Entanglement entropy and the boundary action of edge modes

We consider an antisymmetric gauge field in the Minkowski space of d-dimension and decompose it in terms of the antisymmetric tensor harmonics and fix the gauge. The Gauss law implies that the normal component of the field strength on the spherical entangling surface will label the superselection sectors. From the two-point function of the field strength on the sphere, we evaluate the logarithmic divergent term of the entanglement entropy of edge modes of p-form field. We observe that the logarithmic divergent term in entanglement entropy of edge modes coincides with the edge partition function of co-exact p-form on the sphere when expressed in terms of the Harish-Chandra characters. We also develop a boundary path integral of the antisymmetric p-form gauge field. From the boundary path integral, we show that the edge mode partition function corresponds to the co-exact (p − 1)-forms on the boundary. This boundary path integral agrees with the direct evaluation of the entanglement entropy of edge modes extracted from the two-point function of the normal component of the field strength on the entangling surface.

Anyon quantum dimensions from an arbitrary ground state wave function

Realizing topological orders and topological quantum computation is a central task of modern physics. An important but notoriously hard question in this endeavor is how to diagnose topological orders that lack conventional order parameters. A breakthrough in this problem is the discovery of topological entanglement entropy, which can be used to detect nontrivial topological order from a ground state wave function, but is far from enough for fully determining the topological order. In this work, we take a key step further in this direction: We propose a simple entanglement-based protocol for extracting the quantum dimensions of all anyons from a single ground state wave function in two dimensions. The choice of the space manifold and the ground state is arbitrary. This protocol is both validated in the continuum and verified on lattices, and we anticipate it to be realizable in various quantum simulation platforms.

The quantum Ising chain for beginners

We present here various techniques to work with clean and disordered quantum Ising chains, for the benefit of students and non-experts. Starting from the Jordan-Wigner transformation, which maps spin-1/2 systems into fermionic ones, we review some of the basic approaches to deal with the superconducting correlations that naturally emerge in this context. In particular, we analyze the form of the ground state and excitations of the model, relating them to the symmetry-breaking physics, and illustrate aspects connected to calculating dynamical quantities, thermal averages, correlation functions, and entanglement entropy. A few problems provide simple applications of the techniques.

Casimir effect and holographic dual of wedges

This paper investigates the Casimir effect of a wedge and its holographic dual. We prove that the displacement operator universally determines the wedge Casimir effect in the smooth limit. Besides, we argue that the wedge Casimir energy increases with the opening angle and test it with several examples. Furthermore, we construct the holographic dual of wedges in AdS/BCFT in general dimensions. We verify that our proposal can produce the expected Casimir effect within smooth and singular limits. We observe that the Casimir energy density of a wedge increases with the brane tension. Next, we discuss the wedge contribution to holographic entanglement entropy and find it increases with the opening angle, similar to the wedge Casimir energy. Finally, we briefly discuss the holographic polygon in AdS3/BCFT2 and its generalization to higher dimensions.

Holographic coarse-grained states and the necessity of perfect entanglement

In the framework of the holographic principle, focusing on a central concept, conditional mutual information, we construct a class of coarse-grained states, which are intuitively connected to a family of thread configurations. These coarse-grained states characterize the entanglement structure of holographic systems at a coarse-grained level. Importantly, these coarse-grained states can be used to further reveal nontrivial requirements for the holographic entanglement structure. Specifically, we employ these coarse-grained states to probe the entanglement entropies of disconnected regions and the entanglement wedge cross section dual to the inherent correlation in a bipartite mixed state. The investigations demonstrate the necessity of perfect tensor state entanglement. Moreover, in a certain sense, our work establishes the equivalence between the holographic entanglement of purification and the holographic balanced partial entropy. We also construct a thread configuration with the multiscale entanglement renormalization ansatz (MERA) structure, reexamining the connection between the MERA structure and kinematic space. Published by the American Physical Society 2024