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arXiv:2411.10085 · Physical Review A

Entanglement entropy dynamics of non-Gaussian states in free boson systems: Random sampling approach

Ryui Kaneko, Daichi Kagamihara, Ippei Danshita

Published Mar 11, 2025• 0 citations
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Abstract

We develop a random sampling method for calculating the time evolution of the R\'enyi entanglement entropy after a quantum quench from an insulating state in free boson systems. Because of the non-Gaussian nature of the initial state, calculating the R\'enyi entanglement entropy calls for the exponential cost of computing a matrix permanent. We numerically demonstrate that a simple random sampling method reduces the computational cost of a permanent; for an ${N}_{\mathrm{s}}\ifmmode\times\else\texttimes\fi{}{N}_{\mathrm{s}}$ matrix corresponding to ${N}_{\mathrm{s}}$ sites at half filling, the sampling cost becomes $O({2}^{\ensuremath{\alpha}{N}_{\mathrm{s}}})$ with a constant $\ensuremath{\alpha}\ensuremath{\ll}1$, in contrast to the conventional algorithm with the $O({2}^{{N}_{\mathrm{s}}})$ number of summations requiring the exponential time cost. Although the computational cost is still exponential, this improvement allows us to obtain the entanglement entropy dynamics in free boson systems for more than 100 sites. We present several examples of the entanglement entropy dynamics in low-dimensional free boson systems.

DOI
10.1103/physreva.111.032412

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We develop a random sampling method for calculating the time evolution of the R'enyi entanglement entropy after a quantum quench from an insulating state in free boson systems. Because of the non-Gaussian nature of the initial state, calculating the R'enyi entanglement entropy calls for the exponential cost of computing a matrix permanent. We numerically demonstrate that a simple random sampling method reduces the computational cost of a permanent; for an Ns\ifmmode×\else\texttimes\fiNs{N}_{\mathrm{s}}\ifmmode\times\else\texttimes\fi{}{N}_{\mathrm{s}}Ns​\ifmmode×\else\texttimes\fiNs​ matrix corresponding to Ns{N}_{\mathrm{s}}Ns​ sites at half filling, the sampling cost becomes O(2\ensuremathαNs)O({2}^{\ensuremath{\alpha}{N}_{\mathrm{s}}})O(2\ensuremathαNs​) with a constant \ensuremathα\ensuremath≪1\ensuremath{\alpha}\ensuremath{\ll}1\ensuremathα\ensuremath≪1, in contrast to the conventional algorithm with the O(2Ns)O({2}^{{N}_{\mathrm{s}}})O(2Ns​) number of summations requiring the exponential time cost. Although the computational cost is still exponential, this improvement allows us to obtain the entanglement entropy dynamics in free boson systems for more than 100 sites. We present several examples of the entanglement entropy dynamics in low-dimensional free boson systems.
We investigate the time-dependent R\'enyi entanglement entropy after a quantum quench starting from the Mott-insulating and charge-density-wave states in a one-dimensional free boson system. The second R\'enyi entanglement entropy is found to be the negative of the logarithm of the permanent of a matrix consisting of time-dependent single-particle correlation functions. From this relation and a permanent inequality, we obtain rigorous conditions for satisfying the volume-law entanglement growth. We also succeed in calculating the time evolution of the R\'enyi entanglement entropy in extremely large systems by brute-force computations of the permanent. We discuss possible applications of our findings to the real-time dynamics of noninteracting bosonic systems.
Fermionic non-Gaussianity is a key resource for universal quantum computation, and its behavior in quantum many-body systems has attracted growing interest. Recently, a convolution-based measure, which we call the magic Rényi entropy (MRE), has been proposed as a resource measure of non-Gaussianity, but its evaluation is limited to small systems due to the computational cost growing exponentially with system size. In this work, we develop a perfect-sampling method to calculate the second-order MRE from matrix product states (MPSs). Our method uses a bounded estimator to control sampling fluctuations and a recursive sweep to draw the samples directly from the input MPS, which allows us to generate each sample with O(nD3)\mathcal O(nD^3)O(nD3) time, where nnn and DDD denote the number of modes and MPS bond dimension, respectively. We benchmark our method on the XXZ chain with up to 128128128 sites, demonstrating that the MRE captures intricate higher-order correlations at large scales, which are inaccessible to covariance-based measures. These results provide a computational tool that quantitatively evaluates many-body fermionic non-Gaussianity by accounting for higher-order correlations.
Entanglement in a pure state of a many-body system can be characterized by the Rényi entropies S^{(\alpha)}=\ln\textrm{tr}(\rho^\alpha)/(1-\alpha) S ( α ) = ln tr ( ρ α ) / ( 1 − α ) of the reduced density matrix \rho ρ of a subsystem. These entropies are, however, difficult to access experimentally and can typically be determined for small systems only. Here we show that for free fermionic systems in a Gaussian state and with particle number conservation, S^{(2)} S ( 2 ) can be tightly bound—from above and below—by the much easier accessible Rényi number entropy S^{(2)}_N=-\ln \sum_n p^2(n) S N ( 2 ) = − ln ∑ n p 2 ( n ) which is a function of the probability distribution p(n) p ( n ) of the total particle number in the considered subsystem only. A dynamical growth in entanglement, in particular, is therefore always accompanied by a growth—albeit logarithmically slower—of the number entropy. We illustrate this relation by presenting numerical results for quenches in non-interacting one-dimensional lattice models including disorder-free, Anderson-localized, and critical systems with off-diagonal (bond) disorder.
We calculate the quantum Renyi entropy in a phase-space representation for either fermions or bosons. This can also be used to calculate purity and fidelity, or the entanglement between two systems. We show that it is possible to calculate the entropy from sampled phase-space distributions in normally ordered representations, although this is not possible for all quantum states. We give an example of the use of this method in an exactly soluble thermal case. The quantum entropy cannot be calculated at all using sampling methods in classical symmetric (Wigner) or antinormally ordered (Husimi) phase spaces, due to inner-product divergences. The preferred method is to use generalized Gaussian phase-space methods, which utilize a distribution over stochastic Green's functions. We illustrate this approach by calculating the reduced entropy and entanglement of bosonic or fermionic modes coupled to a time-evolving, non-Markovian reservoir.
In contrast to classical many-particle systems for which the number of degrees of freedom grows linearly with the system size, it grows exponentially for quantum many-particle systems. On the one hand, the resulting complexity of the quantum systems has a high potential for technological advances if it can be exploited directly. For example, in recent years, the fundamentals for the realization of quantum computers and their possible applications have been investigated intensely. On the other hand, the high complexity, especially in strongly correlated regimes, often makes the analytical or numerical analysis of underlying physical phenomena very difficult. This is why some systems of high technological relevance, like the high-temperature superconductors, are still unsatisfactorily understood. A main topic of this thesis is the development of efficient numerical methods for the simulation of strongly correlated quantum lattice models. For one-dimensional systems, the density-matrix renormalization-group (DMRG) is such a very successful method. The physical states of interest are approximated within a certain class of ansatz states. These ansatz states are designed in a way that the number of degrees of freedom are prevented from growing exponentially. With the DMRG method, ground states of interacting one-dimensional systems can often be determined with very high precision. This observation indicates already that ground states actually do not exhaust the available exponential number of degrees of freedom but can be described efficiently with much fewer parameters. The ansatz states used in DMRG are the so-called matrix product states. A very useful criterion for the construction of corresponding appropriate ansatz states for higher-dimensional systems is the quantum nonlocality as for example quantified by the entanglement entropy. The ansatz states should be able to reproduce the scaling properties of the quantum nonlocality in the physical states of interest with as few parameters as possible. The first part of the thesis, therefore, provides analytical and numerical analysis of the scaling of quantum nonlocality with the system size or time in different, physically relevant scenarios. For example, the scaling of Renyi entropies and their dependence on boundary conditions is derived within the 1+1-dimensional conformal field theory. Conjectures and analytical indications concerning the properties of entanglement entropy in critical fermionic and bosonic systems are confirmed numerically with high precision. For integrable models in the thermodynamic limit, general preconditions are derived under which subsystems converge to steady states. These steady states are non-thermal and retain information about the initial state. It is shown that the entanglement entropy in such steady states is extensive. For short times, the entanglement entropy grows typically linearly with time, causing an exponential increase in computation costs for the DMRG method. Consequently, time-dependent DMRG simulations usually have no direct access to the long time regime. The second part of the thesis focuses on the development and improvement of the above-mentioned numerical techniques. The time-dependent DMRG is complemented with an extrapolation technique for the evaluated observables. In this way, the problem of the entropy increase can be circumvented, allowing for a precise determination of spectral functions. The method is demonstrated using the example of the Heisenberg antiferromagnet and results are compared to Bethe-Ansatz data for T=0 and quantum Monte Carlo data for T>0. For the simulation of higher-dimensional systems, projected-entangled pair-states (PEPS) and the multiscale entanglement renormalization ansatz (MERA) turn out to be appropriate wavefunction classes. Initially, they could only be used for spin systems. Within this thesis, variants of those state classes are being suggested for the fermionic case. It is shown how the known algorithms for spin systems can be translated into corresponding algorithms for the fermionic systems. The occurring computation costs increase only by a marginal overhead. First numerical benchmarking tests proceeded successfully. Thus, new techniques are available which can for example be used to examine the two-dimensional Hubbard model. This model is a candidate for the description of high-temperature superconductivity.
We investigate the time evolution of the second-order R\'enyi entropy (RE) for bosons in a one-dimensional optical lattice following a sudden quench of the hopping amplitude JJJ. Specifically, we examine systems that are quenched into the strongly correlated Mott-insulating (MI) regime with $J/U\ensuremath{\ll}1$ (UUU denotes the strength of the on-site repulsive interaction) from the MI limit with J=0J=0J=0. In this regime, the low-energy excited states can be effectively described by fermionic quasiparticles known as doublons and holons. They are excited in entangled pairs through the quench dynamics. By developing an effective theory, we derive a direct relation between the RE and correlation functions associated with doublons and holons. This relation allows us to analytically calculate the RE and obtain a physical picture for the RE, both in the ground state and during time evolution through the quench dynamics, in terms of doublon-holon pairs. In particular, we show that the RE is proportional to the population of doublon-holon pairs that span the boundary of the subsystem. Our quasiparticle picture introduces some remarkable features that are absent in previous studies on the dynamics of entanglement entropy in free-fermion models. It provides with valuable insights into the dynamics of entanglement entropy in strongly-correlated systems.