Large Subsystems Research Articles

Phase Thermalization: from Fermi Liquid to Incoherent Metal

When a system consists of a large subsystem (bath) and a small one (probe), thermalization implies induction of temperature of the bath onto the probe. If both the bath and the probe are described by same microscopic Hamiltonian, thermalisation further entails that the probe imbibes the phase of the bath. We refer to this phenomenon as phase thermalization. However, it is not clear whether this phenomenon is realizable when the probe and the bath are described by different microscopic Hamiltonians. We show phase thermalization is possible even when the microscopic Hamiltonians differ significantly. We provide an explicit example, where the probe is a Fermi liquid realised by a Majorana chain with n≫1 fermions per site interacting through random hopping and the bath is an incoherent metal described by another Majorana chain with N>n n$$\\end{document}]]> fermions per site interacting through arbitrarily long range random four-fermion interaction. In deep infrared (i.e. at very low energies), the probe turns into an incoherent metal, with Lyapunov spectrum and diffusion coefficient identical to the bath.

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 asymmetry in CFT and its relation to non-topological defects

The entanglement asymmetry is an information based observable that quantifies the degree of symmetry breaking in a region of an extended quantum system. We investigate this measure in the ground state of one dimensional critical systems described by a CFT. Employing the correspondence between global symmetries and defects, the analysis of the entanglement asymmetry can be formulated in terms of partition functions on Riemann surfaces with multiple non-topological defect lines inserted at their branch cuts. For large subsystems, these partition functions are determined by the scaling dimension of the defects. This leads to our first main observation: at criticality, the entanglement asymmetry acquires a subleading contribution scaling as log ℓ/ℓ for large subsystem length ℓ. Then, as an illustrative example, we consider the XY spin chain, which has a critical line described by the massless Majorana fermion theory and explicitly breaks the U(1) symmetry associated with rotations about the z-axis. In this situation the corresponding defect is marginal. Leveraging conformal invariance, we relate the scaling dimension of these defects to the ground state energy of the massless Majorana fermion on a circle with equally-spaced point defects. We exploit this mapping to derive our second main result: the exact expression for the scaling dimension associated with n defects of arbitrary strengths. Our result generalizes a known formula for the n = 1 case derived in several previous works. We then use this exact scaling dimension to derive our third main result: the exact prefactor of the log ℓ/ℓ term in the asymmetry of the critical XY chain.

Exact exponent for atypicality of random quantum states

We study the properties of the random quantum states induced from the uniformly random pure states on a bipartite quantum system by taking the partial trace over the larger subsystem. Most of the previous studies have adopted a viewpoint of ‘concentration of measure’ and have focused on the behavior of the states close to the average. In contrast, we investigate the large deviation regime, where the states may be far from the average. We prove the following results: first, the probability that the induced random state is within a given set decreases no slower or faster than exponential in the dimension of the subsystem traced out. Second, the exponent is equal to the quantum relative entropy of the maximally mixed state and the given set, multiplied by the dimension of the remaining subsystem. Third, the total probability of a given set strongly concentrates around the element closest to the maximally mixed state, a property that we call conditional concentration. Along the same line, we also investigate an asymptotic behavior of coherence of random pure states in a single system with large dimensions.

Exploring large-scale entanglement in quantum simulation.

Entanglement is a distinguishing feature of quantum many-body systems, and uncovering the entanglement structure for large particle numbers in quantum simulation experiments is a fundamental challenge in quantum information science1. Here we perform experimental investigations of entanglement on the basis of the entanglement Hamiltonian (EH)2 as an effective description of the reduceddensity operator for large subsystems. We prepare ground and excited states of a one-dimensional XXZ Heisenberg chain on a 51-ion programmable quantum simulator3 and perform sample-efficient 'learning' of the EH for subsystems of up to 20 lattice sites4. Our experiments provide compelling evidence for a local structure of the EH. To our knowledge, this observation marks the first instance of confirming the fundamental predictions of quantum field theory by Bisognano and Wichmann5,6, adapted to lattice models that represent correlated quantum matter. The reduced state takes the form of a Gibbs ensemble, with a spatially varying temperature profile as a signature of entanglement2. Our results also show the transition from area- to volume-law scaling7 of von Neumann entanglement entropies from ground to excited states. As we venture towards achieving quantum advantage, we anticipate that our findings and methods have wide-ranging applicability to revealing and understanding entanglement in many-body problems with local interactions including higher spatial dimensions.

Universality in the tripartite information after global quenches: spin flip and semilocal charges

We study stationary states emerging after global quenches in which the time evolution is under local Hamiltonians that possess semilocal conserved operators. In particular, we study a model that is dual to quantum XY chain. We show that a localized perturbation in the initial state can turn an exponential decay of spatial correlations in the stationary state into an algebraic decay. We investigate the consequences on the behavior of the (Rényi-α) entanglement entropies, focusing on the tripartite information of three adjacent subsystems. In the limit of large subsystems, we show that in the stationary state with the algebraic decay of correlations the tripartite information exhibits a non-zero value with a universal dependency on the cross ratio, while it vanishes in the stationary state with the exponential decay of correlations.

Logarithmic negativity in out-of-equilibrium open free-fermion chains: An exactly solvable case

We derive the quasiparticle picture for the fermionic logarithmic negativity in a tight-binding chain subject to gain and loss dissipation. We focus on the dynamics after the quantum quench from the fermionic Néel state. We consider the negativity between both adjacent and disjoint intervals embedded in an infinite chain. Our result holds in the standard hydrodynamic limit of large subsystems and long times, with their ratio fixed. Additionally, we consider the weakly-dissipative limit, in which the dissipation rates are inversely proportional to the size of the intervals. We show that the negativity is proportional to the number of entangled pairs of quasiparticles that are shared between the two intervals, as is the case for the mutual information. Crucially, in contrast with the unitary case, the negativity content of quasiparticles is not given by the Rényi entropy with Rényi index 1/21/2, and it is in general not easily related to thermodynamic quantities.

Boundary chaos: Exact entanglement dynamics

We compute the dynamics of entanglement in the minimal setup producing ergodic and mixing quantum many-body dynamics, which we previously dubbed boundary chaos. This consists of a free, non-interacting brickwork quantum circuit, in which chaos and ergodicity is induced by an impurity interaction, i.e., an entangling two-qudit gate, placed at the system’s boundary. We compute both the conventional bipartite entanglement entropy with respect to a connected subsystem including the impurity interaction for initial product states as well as the so-called operator entanglement entropy of initial local operators. Thereby we provide exact results in a particular scaling limit of both time and system size going to infinity for either very small or very large subsystems. We show that different classes of impurity interactions lead to very distinct entanglement dynamics. For impurity gates preserving a local product state forming the bulk of the initial state, entanglement entropies of states show persistent spikes with period set by the system size and suppressed entanglement in between, contrary to the expected linear growth in ergodic systems. We observe similar dynamics of operator entanglement for generic impurities. In contrast, for T-dual impurities, which remain unitary under partial transposition, we find entanglement entropies of both states and operators to grow linearly in time with the maximum possible speed allowed by the geometry of the system. The intensive nature of interactions in all cases causes entanglement to grow on extensive time scales proportional to system size.

Parametrically Managed Activation Function for Fitting a Neural Network Potential with Physical Behavior Enforced by a Low-Dimensional Potential.

Machine-learned representations of potential energy surfaces generated in the output layer of a feedforward neural network are becoming increasingly popular. One difficulty with neural network output is that it is often unreliable in regions where training data is missing or sparse. Human-designed potentials often build in proper extrapolation behavior by choice of functional form. Because machine learning is very efficient, it is desirable to learn how to add human intelligence to machine-learned potentials in a convenient way. One example is the well-understood feature of interaction potentials that they vanish when subsystems are too far separated to interact. In this article, we present a way to add a new kind of activation function to a neural network to enforce low-dimensional constraints. In particular, the activation function depends parametrically on all of the input variables. We illustrate the use of this step by showing how it can force an interaction potential to go to zero at large subsystem separations without either inputting a specific functional form for the potential or adding data to the training set in the asymptotic region of geometries where the subsystems are separated. In the process of illustrating this, we present an improved set of potential energy surfaces for the 14 lowest 3A' states of O3. The method is more general than this example, and it may be used to add other low-dimensional knowledge or lower-level knowledge to machine-learned potentials. In addition to the O3 example, we present a greater-generality method called parametrically managed diabatization by deep neural network (PM-DDNN) that is an improvement on our previously presented permutationally restrained diabatization by deep neural network (PR-DDNN).

Logarithmic, fractal and volume-law entanglement in a Kitaev chain with long-range hopping and pairing

Thanks to their prominent collective character, long-range interactions promote information spreading and generate forms of entanglement scaling, which cannot be observed in traditional systems with local interactions. In this work, we study the asymptotic behavior of the entanglement entropy for Kitaev chains with long-range hopping and pairing couplings decaying with a power law of the distance. We provide a fully-fledged analytical and numerical characterization of the asymptotic growth of the ground state entanglement in the large subsystem size limit, finding that the truly non-local nature of the model leads to an extremely rich phenomenology. Most significantly, in the strong long-range regime, we discovered that the system ground state may have a logarithmic, fractal, or volume-law entanglement scaling, depending on the value of the chemical potential and on the strength of the power law decay.

Entanglement negativity in a fermionic chain with dissipative defects: exact results

We investigate the dynamics of the fermionic logarithmic negativity in a free-fermion chain with a localized loss, which acts as a dissipative impurity. The chain is initially prepared in a generic Fermi sea. In the standard hydrodynamic limit of large subsystems and long times, with their ratio fixed, the negativity between two subsystems is described by a simple formula, which depends only on the effective absorption coefficient of the impurity. The negativity grows linearly at short times, then saturating to a volume-law scaling. Physically, this reflects the continuous production with time of entangling pairs of excitations at the impurity site. Interestingly, the negativity is not the same as the Rényi mutual information with Rényi index , in contrast with the case of unitary dynamics. This reflects the interplay between dissipative and unitary processes. The negativity content of the entangling pairs is obtained in terms of an effective two-state mixed density matrix for the subsystems. Criticality in the initial Fermi sea is reflected in the presence of logarithmic corrections. The prefactor of the logarithmic scaling depends on the loss rate, suggesting a nontrivial interplay between dissipation and criticality.

Architectural solutions and optimization methods to improve the performance of node.js and vue.js applications

Software architecture includes a number of important decisions about the organization of a software system, including the choice of structural elements and their interfaces that make up and unite the system into a single whole; the behavior provided by the joint work of these elements; the organization of these structural and behavioral elements into larger subsystems, as well as the architectural style that this organization adheres to. The performance of a web application is an objective measurement and user experience associated with the loading and operation of the program. Performance is about how long a web application takes to load, becomes interactive and responsive, and how smoothlythe interaction with content takes place. The architecture and the steps that need to be taken to optimize the program have always been and will be relevant. Any application has its own architecture, but not every application adheres to the rules for building a good architecture, the same applies to optimization. A program with a good architecture is easier to extend and change, as well as to test, configure and understand. As practice shows, people do not like to wait, and even a three-second delay can force a user to close a tab with a slow resource. Therefore, using a number of optimization methods to improve performance will lead to increased usability and will not force the user to leave the resource due to the fact that it is running slowly. Search engines also pay attention to hundreds of parameters when ranking pages in the search. And one of the most important is the speed of data transfer from the server to the client.

Overgroups of subsystem subgroups in exceptional groups: nonideal levels

In the present paper, a description of overgroups for the subsystem subgroups E ( Δ , R ) E(\Delta ,R) of the Chevalley groups G ( Φ , R ) G(\Phi ,R) over the ring R R , where Φ \Phi is a simply laced root system and Δ \Delta is its sufficiently large subsystem, is almost entirely finished. Namely, objects called levels are defined and it is shown that for any such overgroup H H there exists a unique level σ \sigma with E ( σ ) ≤ H ≤ Stab G ( Φ , R ) ⁡ ( L max ( σ ) ) E(\sigma )\le H\le \operatorname {Stab}_{G(\Phi ,R)}(L_{\max }(\sigma )) , where E ( σ ) E(\sigma ) is an elementary subgroup associated with the level σ \sigma and L max ( σ ) L_{\max }(\sigma ) is the corresponding subalgebra of the Chevalley algebra. Unlike the previous papers, here levels can be more complicated than nets of ideals.

Entanglement Estimation in Tensor Network States via Sampling

We introduce a method for extracting meaningful entanglement measures of tensor network states in general dimensions. Current methods require the explicit reconstruction of the density matrix, which is highly demanding, or the contraction of replicas, which requires an effort exponential in the number of replicas and which is costly in terms of memory. In contrast, our method requires the stochastic sampling of matrix elements of the classically represented reduced states with respect to random states drawn from simple product probability measures constituting frames. Even though not corresponding to physical operations, such matrix elements are straightforward to calculate for tensor network states, and their moments provide the Rényi entropies and negativities as well as their symmetry-resolved components. We test our method on the one-dimensional critical XX chain and the two-dimensional toric code in a checkerboard geometry. Although the cost is exponential in the subsystem size, it is sufficiently moderate so that—in contrast with other approaches—accurate results can be obtained on a personal computer for relatively large subsystem sizes.4 MoreReceived 21 March 2022Accepted 15 June 2022DOI:https://doi.org/10.1103/PRXQuantum.3.030312Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasEntanglement detectionEntanglement measuresQuantum information theoryTechniquesProjected entangled pair statesReplica methodsTensor network methodsQuantum InformationCondensed Matter, Materials & Applied Physics

Adaptive procedures for discriminating between arbitrary tensor-product quantum states

Discriminating between quantum states is a fundamental task in quantum information theory. Given two quantum states ${\ensuremath{\rho}}_{+}$ and ${\ensuremath{\rho}}_{\ensuremath{-}}$, the Helstrom measurement distinguishes between them with minimal probability of error. However, finding and experimentally implementing the Helstrom measurement can be challenging for quantum states on many qubits. Due to this difficulty, there is great interest in identifying local measurement schemes which are close to optimal. In the first part of this work, we generalize previous work by Acin et al. [Phys. Rev. A 71, 032338 (2005)] and show that a locally greedy scheme using Bayesian updating can optimally distinguish between any two states that can be written as a tensor product of arbitrary pure states. We then show that the same algorithm cannot distinguish tensor products of mixed states with vanishing error probability (even in a large subsystem limit), and introduce a modified locally greedy scheme with strictly better performance. In the second part of this work, we compare these simple local schemes with a general dynamic programming approach which finds both the optimal series of local measurements as well as the optimal order in which subsystems are measured.

Quantum statistical fluctuation of energy and its novel pseudo-gauge dependence

We discuss the quantum statistical fluctuations of energy in subsystems of hot relativistic gas for both spin-zero and spin-half particles. We explicitly show the system size dependence of the quantum statistical fluctuation of energy. Our results show that with decreasing system size quantum statistical fluctuations increase substantially. As the consistency of the framework, we also argue that the quantum statistical fluctuations give rise to the known result for statistical fluctuation of energy in the canonical ensemble if we consider the size of the subsystem to be sufficiently large. For a spin-half particle, quantum fluctuations show some interesting novel features. We show that within a small sub-system quantum statistical fluctuation of energy for spin-half particles depends on the various pseudo-gauge choices of the energy-momentum tensor. Interestingly, for sufficiently large subsystems quantum fluctuations obtained for different pseudo-gauge choices converge and we recover the canonical-ensemble formula known for statistical fluctuations of energy. Our calculation is very general and can be applied to any branch of physics whenever one deals with a thermal system. As a practical application, we argue that our results can be used to determine a coarse-graining scale to introduce the concept of classical energy density or fluid element relevant for the strongly interacting matter, in particular for small systems produced in heavy-ion collisions.

Symmetry-resolved entanglement in a long-range free-fermion chain

We investigate the symmetry resolution of entanglement in the presence of long-range couplings. To this end, we study the symmetry-resolved entanglement entropy in the ground state of a fermionic chain that has dimerised long-range hoppings with power-like decaying amplitude—a long-range generalisation of the Su–Schrieffer–Heeger model. This is a system that preserves the number of particles. The entropy of each symmetry sector is calculated via the charged moments of the reduced density matrix. We exploit some recent results on block Toeplitz determinants generated by a discontinuous symbol to obtain analytically the asymptotic behaviour of the charged moments and of the symmetry-resolved entropies for a large subsystem. At leading order we find entanglement equipartition, but comparing with the short-range counterpart its breaking occurs at a different order and it does depend on the hopping amplitudes.

Overgroups of subsystem subgroups in exceptional groups: A 2𝐴₁-proof

In the present paper, a weak form of sandwich classification for the overgroups of the subsystem subgroup E ( Δ , R ) E(\Delta ,R) of the Chevalley group G ( Φ , R ) G(\Phi ,R) is proved in the case where Φ \Phi is a simply laced root system and Δ \Delta is its sufficiently large subsystem. Namely, it is shown that, for such an overgroup H H , there exists a unique net of ideals σ \sigma of the ring R R such that E ( Φ , Δ , R , σ ) ≤ H ≤ Stab G ( Φ , R ) ⁡ ( L ( σ ) ) E(\Phi ,\Delta ,R,\sigma )\le H\le \operatorname {Stab}_{G(\Phi ,R)}(L(\sigma )) , where E ( Φ , Δ , R , σ ) E(\Phi ,\Delta ,R,\sigma ) is an elementary subgroup associated with the net and L ( σ ) L(\sigma ) is the corresponding subalgebra of the Chevalley Lie algebra.

Two Sides of Quantum-Based Modeling of Enzyme-Catalyzed Reactions: Mechanistic and Electronic Structure Aspects of the Hydrolysis by Glutamate Carboxypeptidase.

We report the results of a computational study of the hydrolysis reaction mechanism of N-acetyl-l-aspartyl-l-glutamate (NAAG) catalyzed by glutamate carboxypeptidase II. Analysis of both mechanistic and electronic structure aspects of this multistep reaction is in the focus of this work. In these simulations, model systems are constructed using the relevant crystal structure of the mutated inactive enzyme. After selection of reaction coordinates, the Gibbs energy profiles of elementary steps of the reaction are computed using molecular dynamics simulations with ab initio type QM/MM potentials (QM/MM MD). Energies and forces in the large QM subsystem are estimated in the DFT(PBE0-D3/6-31G**) approximation. The established mechanism includes four elementary steps with the activation energy barriers not exceeding 7 kcal/mol. The models explain the role of point mutations in the enzyme observed in the experimental kinetic studies; namely, the Tyr552Ile substitution disturbs the “oxyanion hole”, and the Glu424Gln replacement increases the distance of the nucleophilic attack. Both issues diminish the substrate activation in the enzyme active site. To quantify the substrate activation, we apply the QTAIM-based approaches and the NBO analysis of dynamic features of the corresponding enzyme-substrate complexes. Analysis of the 2D Laplacian of electron density maps allows one to define structures with the electron density deconcentration on the substrate carbon atom, i.e., at the electrophilic site of reactants. The similar electronic structure element in the NBO approach is a lone vacancy on the carbonyl carbon atom in the reactive species. The electronic structure patterns revealed in the NBO and QTAIM-based analyses consistently clarify the reactivity issues in this system.

Entanglement spreading after local and extended excitations in a free-fermion chain

We study the time evolution of entanglement created by local or extended excitations upon the ground state of a free-fermion chain. A single particle or hole excitation produces a single bit of excess entropy for large times and subsystem lengths. In case of a double hole, some of the coherence between the excitations is preserved and the excess entropy becomes additive only for large hole separations. In contrast, the coherence is always lost for particle–hole excitations. Multiple hole excitations on a completely filled chain are also investigated. We find that for an extended contiguous hole the excess entropy scales logarithmically with the size, whereas the increase is linear for finite separations between the holes.