We study the Rényi entropy and subsystem distances on one interval for the finite size and thermal states in the critical XY chains, focusing on the critical Ising chain and XX chain with zero transverse field. We construct numerically the reduced density matrices and calculate the von Neumann entropy, Rényi entropy, subsystem trace distance, Schatten two-distance, and relative entropy. As the continuum limit of the critical Ising chain and XX chain with zero field are, respectively, the two-dimensional free massless Majorana and Dirac fermion theories, which are conformal field theories, we compare the spin chain numerical results with the analytical results in CFTs and find perfect matches in the continuum limit.
We study the excited state Rényi entropy and subsystem Schatten distance in the two-dimensional free massless non-compact bosonic field theory, which is a conformal field theory. The discretization of the free non-compact bosonic theory gives the harmonic chain with local couplings. We consider the field theory excited states that correspond to the harmonic chain states with excitations of more than one quasiparticle, which we call multi-particle states. This extends the previous work by the same authors to more general excited states. In the field theory we obtain the exact Rényi entropy and …
We investigate the Rényi entropy of the excited states produced by the current and its derivatives in the two-dimensional free massless non-compact bosonic theory, which is a two-dimensional conformal field theory. We also study the subsystem Schatten distance between these states. The two-dimensional free massless non-compact bosonic theory is the continuum limit of the finite periodic gapless harmonic chains with the local interactions. We identify the excited states produced by current and its derivatives in the massless bosonic theory as the single-particle excited states in the gapless ha…
Tianjin University（天津大学）；Sun Yat-Sen University（中山大学）；Freie Universität Berlin（柏林自由大学）；Julius-Maximilians-Universität Würzburg（维尔茨堡朱利叶斯-马克西米利安大学）；Shanghai Institute for Mathematics and Interdisciplinary Sciences (SIMIS)（上海数学与交叉学科研究院）；Ningbo University（宁波大学）；Peking University（北京大学）
Genuine Multi-Entropy of Fully Symmetric Gaussian States
National Center for Theoretical Sciences, National Taiwan University（台湾大学国家理论科学中心）；Department of Physics, National Sun Yat-sen University（国立中山大学物理系）；National Institute of Technology, Yuge College（国立技术研究所育智学院）
We investigate a weak version of subsystem eigenstate thermalization hypothesis (ETH) for a two-dimensional large central charge conformal field theory by comparing the local equivalence of high energy state and thermal state of canonical ensemble. We evaluate the single-interval Rényi entropy and entanglement entropy for a heavy primary state in short interval expansion. We verify the results of Rényi entropy by two different replica methods. We find nontrivial results at the eighth order of short interval expansion, which include an infinite number of higher order terms in the large central …
We numerically study quenches from a fully ordered state to the ferromagnetic regime of the chiral $\mathbb{Z}_3$ clock model, where the physics can be understood in terms of sparse domain walls of six flavors. As in the previously studied models, the spread of entangled domain wall pairs generated by the quench lead to a linear growth of entropy with time, upto a time $\ell/2v_g$ in size-$\ell$ subsystems in the bulk where $v_g$ is the maximal group velocity of domain walls. In small subsystems located in the bulk, the entropy continues to further grow towards $\ln 3$, as domain walls travers…
In a quantum system in a pure state, a subsystem generally has a nonzero entropy because of entanglement with the rest of the system. Is the average entanglement entropy of pure states also the typical entropy of the subsystem? We present a method to compute the exact formula of the momenta of the probability $P(S_A) \mathrm{d}S_A$ that a subsystem has entanglement entropy $S_A$. The method applies to subsystems defined by a subalgebra of observables with a center. In the case of a trivial center, we reobtain the well-known result for the average entropy and the formula for the variance. In th…
Santa Fe Institute（圣塔菲研究所）；Complexity Science Hub（复杂性科学中心）；Arizona State University（亚利桑那州立大学）
I consider multipartite processes in which there are constraints on each subsystem's rate matrix, restricting which other subsystems can directly affect its dynamics. I derive a strictly nonzero lower bound on the minimal achievable entropy production rate of the process in terms of these constraints on the rate matrices of its subsystems. The bound is based on constructing counterfactual rate matrices, in which some subsystems are held fixed while the others are allowed to evolve. This bound is related to the "learning rate" of stationary bipartite systems, and more generally to the "informat…
Relative entropy of entanglement (REE) is an entanglement measure of bipartite mixed states, defined by the minimum of the relative entropy $S(ρ_{AB}|| σ_{AB})$ between a given mixed state $ρ_{AB}$ and an arbitrary separable state $σ_{AB}$. The REE is always bounded by the mutual information $I_{AB}=S(ρ_{AB} || ρ_{A}\otimes ρ_{B})$ 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 perturbati…
We investigate the time evolution of the subsystem trace distance and Schatten distances after local operator quenches in two-dimensional conformal field theory (CFT) and in one-dimensional quantum spin chains. We focus on the case of a subsystem being an interval embedded in the infinite line. The initial state is prepared by inserting a local operator in the ground state of the theory. We only consider the cases in which the inserted local operator is a primary field or a sum of several primaries. While a nonchiral primary operator can excite both left-moving and right-moving quasiparticles,…
We introduce Rényi entropy of a subsystem energy as a natural quantity which closely mimics the behavior of the entanglement entropy and can be defined for all the quantum many body systems. For this purpose, consider a quantum chain in its ground state and then, take a subdomain of this system with natural truncated Hamiltonian. Since the total Hamiltonian does not commute with the truncated Hamiltonian, the subsystem can be in one of its eigenenergies with different probabilities. Using the fact that the global energy eigenstates are locally close to diagonal in the local energy eigenbasis, …
By numerically exact calculations of spin-1/2 antiferromagnetic Heisenberg models on small clusters, we demonstrate that quantum entanglement between subsystems $A$ and $B$ in a pure ground state of a whole system $A+B$ can induce thermal equilibrium in subsystem $A$. Here, the whole system is bipartitoned with the entanglement cut that covers the entire volume of subsystem $A$. Temperature ${\cal T}_{A}$ of subsystem $A$ is not a parameter but can be determined from the entanglement von Neumann entropy ${\cal S}_{A}$ and the total energy ${\cal E}_{A}$ of subsystem $A$ calculated for the grou…
The eigenstate entanglement entropy has been recently shown to be a powerful tool to distinguish integrable from generic quantum-chaotic models. In integrable models, a unique feature of the average eigenstate entanglement entropy (over all Hamiltonian eigenstates) is that the volume-law coefficient depends on the subsystem fraction. Hence, it deviates from the maximal (subsystem fraction independent) value encountered in quantum-chaotic models. Using random matrix theory for quadratic Hamiltonians, we obtain a closed-form expression for the average eigenstate entanglement entropy as a functio…
We investigate a total thermodynamic entropy production rate of an isolated quantum system. In particular, we consider a quantum model of coupled harmonic oscillators in a star configuration, where a central harmonic oscillator (system) is coupled to a finite number of surrounding harmonic oscillators (bath). In this model, when the initial state of the total system is given by the tensor product of the Gibbs states of the system and the bath, every harmonic oscillator is always in a Gibbs state with a time-dependent temperature. This enables us to define time-dependent thermodynamic entropy f…
We study the temporal evolution of the circuit complexity after the local quench where two harmonic chains are suddenly joined, choosing the initial state as the reference state. We discuss numerical results for the complexity for the entire chain and the subsystem complexity for a block of consecutive sites, obtained by exploiting the Fisher information geometry of the covariance matrices. The qualitative behaviour of the temporal evolutions of the subsystem complexity depends on whether the joining point is inside the subsystem. The revivals and a logarithmic growth observed during these tem…
Global quantum quench with a finite quench rate which crosses critical points is known to lead to universal scaling of correlation functions as functions of the quench rate. In this work, we explore scaling properties of the entanglement entropy of a subsystem in a harmonic chain during a mass quench which asymptotes to finite constant values at early and late times and for which the dynamics is exactly solvable. When the initial state is the ground state, we find that for large enough subsystem sizes the entanglement entropy becomes independent of size. This is consistent with Kibble-Zurek sc…
In a seminal paper [D. N. Page, Phys. Rev. Lett. 71, 1291 (1993)], Page proved that the average entanglement entropy of subsystems of random pure states is $S_{\rm ave}\simeq\ln{\cal D}_{\rm A} - (1/2) {\cal D}_{\rm A}^2/{\cal D}$ for $1\ll{\cal D}_{\rm A}\leq\sqrt{\cal D}$, where ${\cal D}_{\rm A}$ and ${\cal D}$ are the Hilbert space dimensions of the subsystem and the system, respectively. Hence, typical pure states are (nearly) maximally entangled. We develop tools to compute the average entanglement entropy $\langle S\rangle$ of all eigenstates of quadratic fermionic Hamiltonians. In part…
The entanglement entropy of subsystems of typical eigenstates of quantum many-body Hamiltonians has been recently conjectured to be a diagnostic of quantum chaos and integrability. In quantum chaotic systems it has been found to behave as in typical pure states, while in integrable systems it has been found to behave as in typical pure Gaussian states. In this tutorial, we provide a pedagogical introduction to known results about the entanglement entropy of subsystems of typical pure states and of typical pure Gaussian states. They both exhibit a leading term that scales with the volume of the…
Using arguments built on ergodicity, we derive an analytical expression for the Renyi entanglement entropies corresponding to the finite-energy density eigenstates of chaotic many-body Hamiltonians. The expression is a universal function of the density of states and is valid even when the subsystem is a finite fraction of the total system - a regime in which the reduced density matrix is not thermal. We find that in the thermodynamic limit, only the von Neumann entropy density is independent of the subsystem to the total system ratio $V_A/V$, while the Renyi entropy densities depend non-linear…
We study the temporal evolution of the circuit complexity for a subsystem in harmonic lattices after a global quantum quench of the mass parameter, choosing the initial reduced density matrix as the reference state. Upper and lower bounds are derived for the temporal evolution of the complexity for the entire system. The subsystem complexity is evaluated by employing the Fisher information geometry for the covariance matrices. We discuss numerical results for the temporal evolutions of the subsystem complexity for a block of consecutive sites in harmonic chains with either periodic or Dirichle…
We study the fluctuations of subsystem entropies in closed quantum many-body systems after thermalization. Using a combination of analytics and numerics for both random quantum circuits and Hamiltonian dynamics, we find that the statistics of such entropy fluctuations is drastically different than in the classical setting. For instance, shortly after a system thermalizes, the probability of entropy fluctuations for a subregion is suppressed in the dimension of the Hilbert space of the complementary subregion. This suppression becomes increasingly stringent as a function of time, ultimately dep…
We formulate a new ``Wigner characteristics'' based method to calculate entanglement entropies of subsystems of Fermions using Keldysh field theory. This bypasses the requirements of working with complicated manifolds for calculating Rényi entropies for many body systems. We provide an exact analytic formula for Rényi and von-Neumann entanglement entropies of non-interacting open quantum systems, which are initialised in arbitrary Fock states. We use this formalism to look at entanglement entropies of momentum Fock states of one-dimensional Fermions. We show that the entanglement entropy of a …
Defects in two-dimensional conformal field theories (CFTs) contain signatures of their characteristics. In this work, we compute the entanglement entropy (EE) and the entanglement negativity (EN) of subsystems in the presence of energy and duality defects in the Ising CFT using the density matrix renormalization group (DMRG) technique. We show that the EE for the duality defect exhibits fundamentally different characteristics compared to the energy defect due to the existence of localized and delocalized zero energy modes. Of special interest is the nontrivial `finite-size correction' in the E…
Systems driven away from thermal equilibrium constantly deliver entropy to their environment. Determining this entropy production requires detailed information about the system's internal states and dynamics. However, in most practical scenarios, only a part of a complex experimental system is accessible to an external observer. In order to address this challenge, two notions of partial entropy production have been introduced in the literature as a way to assign an entropy production to an observed subsystem: one due to Shiraishi and Sagawa [Phys. Rev. E 91, 012130 (2015)] and another due to P…
Strict Entropy Decrease of Clausius Entropy in an Isolated System with Energy-Form Conversion: Theoretical Proof, Numerical Illustration, and Critical Examination
Key Laboratory for Special Area Highway Engineering of Ministry of Education（教育部特殊地区公路工程重点实验室）；Chang’an University（长安大学）
Using a Wigner function based approach, we study the Renyi entropy of a subsystem $A$ of a system of Bosons interacting with a local repulsive potential. The full system is assumed to be in thermal equilibrium at a temperature $T$ and density $ρ$. For a ${\cal U}(N)$ symmetric model, we show that the Renyi entropy of the system in the large $N$ limit can be understood in terms of an effective non-interacting system with a spatially varying mean field potential, which has to be determined self consistently. The Renyi entropy is the sum of two terms: (a) Renyi entropy of this effective system an…
