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置換不変密度行列が保持する量子情報量を用いた量子多体現象の解析

優希 宮﨑

Published Jan 1, 2024• 0 citations
Publisher

Abstract

It is becoming clear that the concepts originating from quantum information theory1are useful for the analysis of quantum many-body physics. For example, topological ordered phases which appear in quantum Hall effect can not be explained by conventional Ginzburg-Landau theory [1], because we cannot define any order parameter and no symmetry breaking occurs. However, it has been shown that it is possible to characterize quantum phases with high generality beyond the framework of the Ginzburg-Landau theory by considering entanglement entropy, which is one of quantum information stored in the quantum state itself (density matrix) [2, 3]. We can mathematically express a quantum state by a density matrix. Given the density matrix of a system, we can, in principle, obtain quantum information about the system, including its entanglement entropy, when the system is divided into two parts. However, in the field of solid state physics, experimentally obtaining the density matrix of a system is challenging. In artificial quantum systems including cold atomic gases in an optical lattice [4], trapped-ion systems [5], etc., the reconstruction of a density matrix of multi-qubit systems has been successfully achieved by preparing the identical state repeatedly and measuring the various microscopic observables, taking advantage of the cleanness and high controllability of such systems. This process is known as quantum state tomography (QST). However, scaling up QST is hard due to the exponentially increasing cost with respect to the system size. Moreover, especially in the experiments of cold atomic gases in an optical lattice, the local rotation of quantization axes is hard within currently-available optical techniques. In this work, we focus on the permutationally invariant part of the density matrix (PIDM), and investigate the role of PIDM in quantum many-body systems. PIDM is defined as the average over all possible site permutations of the density matrix. In the context of QST, it is recognized that PIDM requires only a polynomial number of measurements, in contrast to the exponential cost associated with reconstructing a full density matrix [6]. We examine how this experimentally tractable quantity reflects the property of the ground state of the system across a quantum phase transition. In this doctoral thesis, we develop a calculation method of second-order Rényi entropy of PIDM using density-matrix renormalization-group algorithm and tensor network. Applying this procedure, we compute the second-order Rényi entropy of PIDM for the ground state of the transverse-field Ising and XXZ chains. As a result, we find that the Rényi entropy of PIDM quantifies the uncertainness raised in discarding information regarding special structure by symmetrization, in other words, it quantifies the special information of the original quantum state. Moreover, we obtain the insight that the difference of the scaling law of the Rényi entropy of PIDM reflects the difference of quantum phases[7].

DOI
10.34321/tf02037564

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It is becoming clear that the concepts originating from quantum information theory1are useful for the analysis of quantum many-body physics. For example, topological ordered phases which appear in quantum Hall effect can not be explained by conventional Ginzburg-Landau theory [1], because we cannot define any order parameter and no symmetry breaking occurs. However, it has been shown that it is possible to characterize quantum phases with high generality beyond the framework of the Ginzburg-Landau theory by considering entanglement entropy, which is one of quantum information stored in the quantum state itself (density matrix) [2, 3]. We can mathematically express a quantum state by a density matrix. Given the density matrix of a system, we can, in principle, obtain quantum information about the system, including its entanglement entropy, when the system is divided into two parts. However, in the field of solid state physics, experimentally obtaining the density matrix of a system is challenging. In artificial quantum systems including cold atomic gases in an optical lattice [4], trapped-ion systems [5], etc., the reconstruction of a density matrix of multi-qubit systems has been successfully achieved by preparing the identical state repeatedly and measuring the various microscopic observables, taking advantage of the cleanness and high controllability of such systems. This process is known as quantum state tomography (QST). However, scaling up QST is hard due to the exponentially increasing cost with respect to the system size. Moreover, especially in the experiments of cold atomic gases in an optical lattice, the local rotation of quantization axes is hard within currently-available optical techniques. In this work, we focus on the permutationally invariant part of the density matrix (PIDM), and investigate the role of PIDM in quantum many-body systems. PIDM is defined as the average over all possible site permutations of the density matrix. In the context of QST, it is recognized that PIDM requires only a polynomial number of measurements, in contrast to the exponential cost associated with reconstructing a full density matrix [6]. We examine how this experimentally tractable quantity reflects the property of the ground state of the system across a quantum phase transition. In this doctoral thesis, we develop a calculation method of second-order Rényi entropy of PIDM using density-matrix renormalization-group algorithm and tensor network. Applying this procedure, we compute the second-order Rényi entropy of PIDM for the ground state of the transverse-field Ising and XXZ chains. As a result, we find that the Rényi entropy of PIDM quantifies the uncertainness raised in discarding information regarding special structure by symmetrization, in other words, it quantifies the special information of the original quantum state. Moreover, we obtain the insight that the difference of the scaling law of the Rényi entropy of PIDM reflects the difference of quantum phases[7].
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We investigate the role of the permutationally invariant part of the density matrix (PIDM) in capturing the properties of the ground state of the system during a quantum phase transition. In the context of quantum state tomography, PIDM is known to be obtainable with only a low number of measurement settings, namely O(L2)O({L}^{2})O(L2), where LLL is the system size. Considering the transverse-field Ising chain and the XXZXXZXXZ chain as examples, we compute the second-order R\'enyi entropy of PIDM for the ground state by using the density-matrix renormalization-group algorithm. In the transverse-field Ising chain with ferromagnetic interaction, the ground state is permutationally invariant both in the limits of zero and infinite field, leading to a vanishing R\'enyi entropy of PIDM. The latter exhibits a broad peak as a function of the transverse field around the quantum critical point, which gets more pronounced for larger system size. In the antiferromagnetic case, the peak structure disappears and the R\'enyi entropy diverges like O(L)O(L)O(L) in the whole field range of the ordered phase. In the XXZXXZXXZ model with ferromagnetic interaction, the ground state is permutationally invariant in the full parameter regime of easy-axis anisotropy, causing the R\'enyi entropy of PIDM to vanish. In the easy-plane region, the R\'enyi entropy of PIDM grows almost linearly as a function of the system size. In the antiferromagnetic case, the R\'enyi entropy of PIDM takes finite values at any point in parameter space; importantly, its minimum appears at the phase-transition point. We discuss the cause of these behaviors of the R\'enyi entropy of PIDM, examining the possible application of this experimentally tractable quantity to the analysis of phase-transition phenomena.
摘 要:信息不对称是现代经济学解释市场失灵、契约设计、金融摩擦、逆向选择与道德风险的核心理论基石。经典信息不对称模型建立在知识离散二分、状态确定唯一、概率经典可加、信息静态外生与观测中立的经典世界观之上,将信息差异定义为“知道”与“不知道”的二元鸿沟,无法刻画现实中普遍存在的信念模糊、预期叠加、信息干涉、信心差异与观测改变信息结构等深层信息特征。本文严格立足于经济主体信息处理的本体逻辑,规避物理隐喻类比,引入量子叠加、量子相干、观测坍缩与量子信息熵构造公理体系,对经典信息不对称理论进行范式升级,构建一般性的量子信息不对称理论(Quantum Information Asymmetry,QIA)。本文严格证明,经典信息不对称是量子信息不对称在无叠加、无相干、无信念冲突与观测中立条件下的退化特例;通过推导量子信息不对称的数学结构、测度公式与均衡特征,系统阐释信息叠加、信念相干、观测坍缩与熵差演化四大微观作用机制。结合信贷市场、股票市场、委托—代理关系与政策传导等典型场景开展数值模拟分析,结果表明:量子信息不对称能够统一刻画信念混乱度差异、预期干涉效应、信息透明度内生与观测驱动信息跃迁等经典理论难以系统容纳的信息结构。本研究完善了信息经济学的基础公理体系,为契约理论、金融中介、公司治理、市场监管与宏观政策传导提供全新的基础性分析范式。 关键词:量子信息不对称;逆向选择;道德风险;信念叠加;观测坍缩;契约理论
This thesis is divided into two parts. In the first part we study quantum phases in bosonic systems in an optical lattice by using numerical unbiased Quantum Monte Carlo (QMC) method. Ultracold atoms experiments in optical lattices provide a possibility to simulate the materials of condensed matter in a clean and well-controled way. We focus on two components bosonic systems where exotic phases such as pair-superfluid and pair-supersolid appear. In the second part we provide numerical methods based on QMC simulations to study the entanglement properties of strong correlated systems. By employing the replica trick, we reconstruct the entanglement spectrum from the Renyi entropies. Furthermore, we study the trace of the power of the partial transposed reduced density matrix, which is related to the entanglement measurement negativity for mixed states. In the following we briefly summarize each chapter. In Chapter 1 we very briefly introduce the cold atom systems and the connection with QMC simulations. In Chapter 2 we review the QMC method based on the path integral (world line) representation. We focus on the directed worm algorithm which is an efficient algorithm with global updtae for bosonic and spin systems. All the following methods in this thesis – the method of two component bosonic systems, the reconstruction of entanglement spectrum and the measuring of partial transposed quantities, are based on directed worm algorithm introduced in this chapter. In Chapter 3 we discuss two component bosons in a square lattice. We show that the interspecies attraction and nearest-neighbor intraspecies repulsion result in the pair-supersolid phase, where a diagonal solid order coexists with an off-diagonal pair-superfluid order. The quantum and thermal transitions out of the pair-supersolid phase are characterized. It is found that there is a direct first-order transition from the pair-supersolid phase to the double-superfluid phase without an intermediate region. Furthermore, the melting of the pair-supersolid occurs in two steps. Upon heating, first the pair-superfluid is destroyed via a Kosterlitz-Thouless transition, then the solid order melts via an Ising transition. In Chapter 4 we represent a new method to reconstruct a subset of the entanglement spectrum of quantum many body systems by QMC, where the method can in principle be applied to two or higher dimension. The approach builds on the replica trick to evaluate particle number resolved traces of the first n of powers of a reduced density matrix. From this information we reconstruct first n entanglement spectrum levels using a polynomial root solver. We illustrate the power and limitations of the method by an application to the extended Bose-Hubbard model in one dimension where we are able to resolve the quasidegeneracy of the entanglement spectrum in the Haldane-insulator phase. In general, the method is able to reconstruct the largest few eigenvalues in each symmetry sector. In Chapter 5 we devise a Quantum Monte Carlo (QMC) method to calculate the moments of the partially transposed reduced density matrix at finite temperature. These are used to construct scale invariant combinations that are related to the negativity, a true measure of entanglement for two intervals embedded in a chain. In particular, we study several scale invariant combinations of the moments for the 1D hard-core boson model. For two adjacent intervals unusual finite size corrections are present, showing parity effects that oscillate with a filling dependent period. For large chains we find perfect agreement with conformal field theory (CFT) calculations. Oppositely, for disjoint intervals corrections are more severe and CFT is recovered only asymptotically. Furthermore, we provide evidence that their exponent is the same as that governing the corrections of the mutual information. Additionally we study the 1D Bose-Hubbard model in the superfluid phase. The finite-size effects are smaller and QMC data are already in impressive agreement with CFT at moderate large sizes.