Higher order Schmidt decompositions
Asher Peres
给出了具有两个以上子空间的广义Schmidt分解存在的充要条件。
Necessary and sufficient conditions are given for the existence of extended Schmidt decompositions, with more than two subspaces.
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Schmidt modes of non-collinear biphoton angular wave functions are found analytically. The experimentally realizable procedure is described for their separation. Parameters of the Schmidt decomposition are used for evaluation of the degree of biphoton's angular entanglement.
Existence of the Schmidt decomposition for tripartite systems
对于任何二分量子系统，施密特分解允许我们用单和而不是双和来表示状态向量。在一定条件下证明了三分系统Schmidt分解的存在性。如果一个基（属于一个较小维数的Hilbert空间）与复合系统的状态的部分内积给出一个解纠缠基，则三分系统的Schmidt分解存在。在这种情况下，每个子系统的约化密度矩阵在施密特基中具有相等的谱。
Note on Schmidt Decomposition in Infinite Dimensional Hilbert Spaces
这篇论文已被作者撤回。
Nonlocal two-qubit quantum gates are represented by canonical decomposition or equivalently by operator-Schmidt decomposition. The former decomposition results in geometrical representation such that all the two-qubit gates form tetrahedron within which perfect entanglers form a polyhedron. On the other hand, it is known from the later decomposition that Schmidt number of nonlocal gates can be either 2 or 4. In this work, some aspects of later decomposition are investigated. It is shown that two gates differing by local operations possess same set of Schmidt coefficients. Employing geometrical…
We study the mathematical structures and relations among some quantities in the theory of quantum entanglement, such as separability, weak Schmidt decompositions, Hadamard matrices etc.. We provide an operational method to identify the Schmidt-correlated states by using weak Schmidt decomposition. We show that a mixed state is Schmidt-correlated if and only if its spectral decomposition consists of a set of pure eigenstates which can be simultaneously diagonalized in weak Schmidt decomposition, i.e. allowing for complex-valued diagonal entries. For such states, the separability is reduced to t…
Purely azimuthal entanglement is analyzed for noncollinear frequency-degenerate biphoton states. The degree of azimuthal entanglement is found to be very high, with the Schmidt parameter $K$ on the order of the ratio of the pump waist to its wavelength. A scheme is suggested for partial realization of this high entanglement resource in the form of a multichannel Schmidt-type decomposition.
The Correlations of Greenberger-Horne-Zeilinger States Described by Hilbert-Schmidt decomposition
本文用简单的量子分析方法，用希尔伯特-施密特（HS）表象描述了Greenberger-Horne-Zeilinger（GHZ）态的关联。我们的结论是，虽然这些国家反驳本地现实主义，他们没有证明任何非定域性。
Conditions for entanglement transformation between a class of multipartite pure states with generalized Schmidt decompositions
本文将Nielsen关于二体纯态可转换性的Marjoization判据[Phys.Rev.Lett \textbf{83}，436（1999）]推广到一类具有广义Schmidt分解的多体纯态。
Detecting maximally entangled states without making the Schmidt decomposition
已知纯量子态的两体纠缠可以用Schmidt分解来表征。特别地，当所有施密特系数相等时，状态是最大纠缠的。我们指出了一个方便的方法，总是产生一个单一的分析条件的状态是最大的纠缠，在任何基础上的扩展系数。即使施密特系数不能解析计算，该方法也有效，并且不需要计算施密特系数。作为一个例子，这种技术被用来推导两个量子比特系统的贝尔基础。在第二个例子中，该技术显示了\textit{never}的特定状态是最大纠缠的，这是使用施密特分解无法达到的一般结论。
Hermitian Schmidt Decomposition and Twin Observables of Bipartite Mixed States
结果表明，对于每一个混合态，存在一个施密特（超态矢量）分解的厄米特算子。它的利用寻找所有的双胞胎充分详细说明的情况下，两个自旋半粒子状态与最大的无序子系统（贝尔态的混合物）。
It is well known that the Schmidt decomposition exists for all pure states of a two-party quantum system. We demonstrate that there are two ways to obtain an analogous decomposition for arbitrary rank-1 operators acting on states of a bipartite finite-dimensional Hilbert space. These methods amount to joint Schmidt-type decompositions of two pure states where the two sets of coefficients and local bases depend on the properties of either state, however, at the expense of the local bases not all being orthonormal and in one case the complex-valuedness of the coefficients. With these results we …
Entanglement distribution in pure non-Gaussian tripartite states: a Schmidt decomposition approach
本文利用施密特分解方法研究三个耦合量子谐振子中纠缠在不同二分划分间的分布，揭示施密特系数随相互作用强度的演化规律，为网络化量子系统中的多体纠缠及多节点量子信息处理提供基础理解。
Dealing with entanglement of continuous variables: Schmidt decomposition with discrete sets of orthogonal functions
提出了一种求连续变量二部系统Schmidt分解的方法。它近似的模式，以规定的精度由众所周知的正交函数。我们给出了一些控制误差的准则。我们说明的方法比较其结果与已经发表的双光子纠缠的分析。该协议非常好。
Pati (Physics Letters A, 2000) derived a sufficient condition for the existence of Schmidt decomposition in tripartite Hilbert spaces. In this paper, we show that the condition is erroneous by demonstrating some counter-examples. Moreover, we suitably modify the condition and provide a correctness proof. We also show for the first time how this can be generalized to $n$-partite Hilbert spaces for any $n\geq 2$. Finally, we prove that this condition is also a necessary condition.
We consider polarization states of three photons, each in the same given spectral-angular mode. A general form of such states is a superposition of four basic three-photon polarization modes, to be referred to as three-photon polarization ququarts. All such states can be considered as consisting of one- and two-photon parts, which can be entangled with each other. The degrees of entanglement and polarization as well as the Schmidt decomposition and Stokes vectors of three-photon polarization ququarts are found and discussed.
Summary. A simple derivation of finite Schmidt decomposition of pure states describing finite dimensional systems interacting with the infinite dimensional ones is presented. In particular, maximally entangled pure states in such systems are being characterized. The concept of mixed-pure states has been introduced and some criterions for checking separability and entanglement of them are presented .The notion of LOCC equivalence and LOCC semi-order on the space of pure states of systems analyzed has been adopted suitably. In particular a Nielsen -like theorem has been extended to the pure (d, …
We investigate quantum properties of light in optical parametric oscillators (OPOs) based on four-wave mixing gain in media with third-order non-linearities. In spite of other competing $χ^{(3)}$ effects, such as phase modulation, bipartite and tripartite entanglement are predicted above threshold. These findings are relevant for recent implementations of CMOS-compatible on-chip OPOs.
Formulas for Mutually Orthogonal Quantum States in Two-Qubit Systems: Orthogonal Schmidt Decompositions
该研究针对两量子比特系统，推导相互正交纯态的施密特分解公式，基于纠缠结构分类正交集，分析含最大纠缠态的正交基，并证明不存在3个直积态加1个纠缠态构成的正交基。
The new method of multivariate data analysis based on the complements of classical probability distribution to quantum state and Schmidt decomposition is presented. We considered Schmidt formalism application to problems of statistical correlation analysis. Correlation of photons in the beam splitter output channels, when input photons statistics is given by compound Poisson distribution is examined. The developed formalism allows us to analyze multidimensional systems and we have obtained analytical formulas for Schmidt decomposition of multivariate Gaussian states. It is shown that mathemati…
We consider a system of two spins that are coupled via an isotropic Heisenberg Hamiltonian. For the first time, a two-step method for the preparation of an arbitrary quantum state of two qubits in the form of the Schmidt decomposition is proposed. The simplified version of this method is applied to the physical system of an atom having with a nuclear spin $1/2$ and one valence electron. As an example, the preparation of two-spin quantum states in the $^{31}$P system is considered.
Hilbert-Schmidt (HS) decompositions and Frobenius norms are used to analyze biseparability of 3-qubit systems, with particular emphasis on density matrices with maximally disordered subsystems (MDS) and on the W state mixed with white noise. The biseparable form of a MDS density matrix is obtained by using the Bell states of a 2-qubit subsystem, multiplied by density matrices of the third qubit, which include the relevant HS parameters. Using our methods a sufficient condition and explicit biseparability of the W state mixed with white noise are given. They are compared with the sufficient con…
We show that there is a unique maximal decomposition of a pure multi-partite (N>2) quantum state into a sum of states which are "locally orthogonal" in the sense that the local reduced state for a term in the sum lives in its own orthogonal subspace for each subsystem. Observers can make local measurements on any subsystem and determine which "branch" they are on. The Shannon entropy of the resulting branch weights defines a new measure of global, GHZ-like entanglement, which is insensitive to local pairwise entangling operations and vanishes when there is no piece of information recorded at e…
Generalized Schmidt decomposition and classification of three-quantum-bit states
我们证明了任何纯三量子比特状态的存在的本地基地，允许建立一组五个正交的产品状态，其中状态可以写在一个独特的形式。这导致了一个规范形式，推广了两个量子位施密特分解。它的唯一特征是五个纠缠参数。它导致了一个完整的分类的三个量子比特的状态。它表明，适当的局部测量的正确结果总是消除其他两方之间的所有纠缠。
Multipartite generalisation of the Schmidt decomposition
我们发现一个典型的形式为纯态的一般多部系统，其中的坐标上的约束（相对于一个可分解的正交基）是简单的，某些消失和某些其他的是真实的。对于全同粒子，它们在粒子的排列下是不变的。作为一个应用，我们发现的通用局部等价类的维数。
Universal and Efficient Quantum State Verification via Schmidt Decomposition and Mutually Unbiased Bases
本文提出了一种基于施密特分解和互不偏正交基的通用量子态验证协议，实现了高效且无需依赖局部维度的样本复杂度上界。
We introduce an experimental procedure for the detection of quantum entanglement of an unknown quantum state with as few measurements as possible. The method requires neither a priori knowledge of the state nor a shared reference frame between the observers. The scheme starts with local measurements, possibly supplemented with suitable filtering, that can be regarded as calibration. Consecutive correlation measurements enable detection of the entanglement of the state. We utilize the fact that the calibration stage essentially establishes the Schmidt decomposition for pure states. Alternativel…
Operator-Schmidt decompositions and the Fourier transform, with applications to the operator-Schmidt numbers of unitaries
算符-施密特分解在量子信息理论中用于量化二分么正操作的非定域性。我们在C^n张量C^n上构造了一族酉算子，利用离散傅立叶变换计算了其算子-施密特分解。作为推论，对于{1，.，..}中的每个S，我们在C^3张量C^3上产生具有算符施密特数S的么正9}。这个推论是出乎意料的，因为它与尼尔森等人[物理学评论A 67（2003）052301]基于两量子比特情况下惊人结果的直觉的合理推论相矛盾。由杜尔，维达尔，和Cirac [物理学评论快报。89（2002）057901 quant-ph/0112124]，他也考虑了两个量子比特的情况，我们的结果意味着有九个等价类的酉C^3张量C^3，这是概率相互转换的（随机）本地操作和经典通信。作为另一个推论，处方是由双幺模函数构造最大纠缠算子。反过来，我们把量子傅里叶变换的广义算符-施密特分解看作算符C^M_1张量C^M_2 -> C^N_1张量C^N_2，其中M_1 × M_2 = N_1 × N_2.这种分解表明（由尼尔森的界限），QFT的通信成本保持最大时，净转移qudits是允许的。在附录中，给出了一个规范的过程，以消除基依赖的结果和证明依赖于“魔基”在[S。希尔和W. Wootters，“Entanglement of a pair of quantum bits，”Phys Rev. Lett 78（1997）5022-5025，quant-…
Separability and entanglement for n-qubits systems are quantified by using Hilbert-Schmidt (HS) decompositions in which the density matrices are decomposed into various terms representing certain one qubit, two-qubits,and larger qubits measurements. The present method is more general than previous methods for bipartite systems as it can be used for quantification of entanglement for large n-qubits systems.We demonstrate the use of such method by analyzing three qubits GHZ states, and three qubits general Bell states produced by a certain multiplications of Braid operators, operating on the com…
Generalized Schmidt decomposition based on injective tensor norm
本文给出了一个具有任意数目两层子系统的纯系统的广义Schmidt分解。该基础是对称的置换下的各方，并来自产品的状态定义的内射张量范数的状态。最大的系数量化了态的量子相关性。其他系数包含了很多信息，如未纠缠的粒子以及其约化态完全混合的粒子。该分解清楚地区分了以不等价方式纠缠的态，并提供了当给定的量子态作为量子信道时对隐形传态和超密集编码的适用性信息。
Schmidt decomposition is a widely employed tool of quantum theory which plays a key role for distinguishable particles in scenarios such as entanglement characterization, theory of measurement and state purification. Yet, it is held not to exist for identical particles, an open problem forbidding its application to analyze such many-body quantum systems. Here we prove, using a newly developed approach, that the Schmidt decomposition exists for identical particles and is thus universal. We find that it is affected by single-particle measurement localization and state overlap. We study paradigma…
Geometric measures of entanglement and the Schmidt decomposition
在测量纯态的纠缠的标准几何方法中，使用$\sin^2θ$，其中$θ$是状态与归一化量子比特状态的乘积的最接近的可分离状态之间的角度。我们认为在这里的一个概括，这一概念的可分离的国家组成的产品的非规范化状态的不同尺寸。在这样做时，发现纠缠度量$\sin^2θ$具有作为状态到最近的可分离状态之间的距离的解释。我们还发现，最接近的可分状态和它的范数的组件有一个解释，分别在施密特分解的状态矢量中产生的约化密度矩阵的本征向量和本征值。
In a previous paper we examined a geometric measure of entanglement based on the minimum distance between the entangled target state of interest and the space of unnormalized product states. Here we present a detailed study of this entanglement measure for target states with a large degree of symmetry. We obtain analytic solutions for the extrema of the distance function and solve for the Hessian to show that, up to the action of trivial symmetries, the solutions correspond to local minima of the distance function. In addition, we show that the conditions that determine the extremal solutions …
Separable decompositions of 2xn states with operator Schmidt rank three
本文证明算子Schmidt秩为三的2×n二分态可分解为rank(ρ)个纯乘积态或至多n+1个混合乘积态，后者紧致，并提供构造性方法，推广了已有结果。
Operator-Schmidt decomposition of the quantum Fourier transform on C^N1 tensor C^N2
计算C^N1张量C^N2上的量子傅里叶变换的算符施密特分解，对于所有N1，N2 > 1。当N1是N2的因子且N1>N2时，分解是完全退化的.第一个已知的特殊情况，N1=N2=2^n，是由尼尔森在他的研究中计算的，他研究了计算一组量子位在两方之间均匀分布的量子傅里叶变换的通信成本。[M. A. Nielsen，PhD Thesis，University of New Mexico（1998），Chapter 6，arXiv：quant-ph/0011036.]更一般地，N1=2^n1<2^n2=N2的特殊情况由Nielsen et.在他们的研究强度措施的量子操作。[硕士尼尔森等al，（接受在Phys Rev A中发表）; arXiv：quant-ph/0208077。]给定这里给出的施密特分解，可以得出，在所有情况下，量子傅里叶变换的精确计算的通信成本都是最大的。
Using the Schmidt Decomposition to Determine Quantum Entanglement
本文结合量子力学与信息理论，演示通过施密特分解判断系统是否纠缠的数学方法，展示其在量子隐形传态中的应用，并探讨扩展该分解用途的方向，助力理解量子信息理论核心要素。
Schmidt Decomposition of Multipartite States
本文研究了多粒子态的Schmidt分解条件，并提出了一种高效算法用于计算此类分解。
Analyzing quantum entanglement with the Schmidt decomposition in operator space
提出基于可观测量Schmidt分解构造纠缠见证的一般方法，适用于多体系统且严格强于保真度构造，能显著改善纠缠检测并量化纠缠维度。
In the standard geometric approach, the entanglement of a pure state is $\sin^2θ$, where $θ$ is the angle between the entangled state and the closest separable state of products of normalised qubit states. We consider here a generalisation of this notion by considering separable states that consist of products of unnormalised states of different dimension. The distance between the target entangled state and the closest unnormalised product state can be interpreted as a measure of the entanglement of the target state. The components of the closest product state and its norm have an interpretati…
The Schmidt-decomposition formalism is proposed to be used for evaluation of the degree of quadrature entanglement in two-mode multiphoton states.
We present a novel formulation of quantum game theory based on the Schmidt decomposition, which has the merit that the entanglement of quantum strategies is manifestly quantified. We apply this formulation to 2-player, 2-strategy symmetric games and obtain a complete set of quantum Nash equilibria. Apart from those available with the maximal entanglement, these quantum Nash equilibria are extensions of the Nash equilibria in classical game theory. The phase structure of the equilibria is determined for all values of entanglement, and thereby the possibility of resolving the dilemmas by entangl…
On properties of Schmidt Decomposition
本文回顾二部施密特分解性质并研究其向多部态的扩展，指出施密特数通过可分幺正变换定义等价类，证明最高施密特数多部态划分是NP完全问题，且复合系统密度矩阵纯化保持施密特可分解性。
Locally undetermined states, generalized Schmidt decomposition, and an application in distributed computing
多粒子量子态不能由它们的所有真子集的约化态唯一确定，它们表现出某种继承的“高阶”关联。本文通过给出一个纯多体态局部不可定的充分必要条件，并精确刻画了与它共享同一组约化态的所有纯态，详细阐述了这个问题.有趣的是，纯态的局部可定性与Schmidt分解的广义概念密切相关.此外，我们发现，局部待定状态有一些分布式计算中著名的一致性问题的应用。具体地说，给定一些物理上分离的智能体，当它们之间的通信是不可靠的，无论是经典的还是量子的，并且不允许它们使用局部辅助量子系统时，则存在一个完全正确且完全容错的协议，当且仅当它们共享一个先验的局部未确定的量子态。
Finding a maximally correlated state - Simultaneous Schmidt decomposition of bipartite pure states
我们考虑一个二分混合态的形式，$ρ=\sum_{α, β=1}^{l}a_{αβ} | ψ_α> < ψ_ β}| $，其中$| ψ_α>$是规范化的二分状态向量，和矩阵$(a_{αβ})$是半正定的。通过局域幺正变换，给出了态$ρ$为最大关联态的充分必要条件。更准确地说，我们给出了一个标准的同时施密特分解的$| ψ_α>$为$α=1,2,..., l$。利用这个判据，我们可以完全判断态$ρ$是否等价于最大关联态，其中的纠缠度由一个简单的公式给出。对于广义贝尔态，该判据被写成状态指数之间的简单代数关系。我们还讨论了广义贝尔态的局部可分解性，同时施密特分解。
The development of a self-consistent thermodynamic theory of quantum systems is of fundamental importance for modern physics. Still, despite its essential role in quantum science and technology, there is no unifying formalism for characterizing the thermodynamics within general autonomous quantum systems, and many fundamental open questions remain unanswered. Along these lines, most current efforts and approaches restrict the analysis to particular scenarios of approximative descriptions and semi-classical regimes. Here we propose a novel approach to describe the thermodynamics of arbitrary bi…
To easily calculate statistical properties of pairs correlated through Schmidt decomposition, as commonly used in Quantum Information, we propose a "commutator formalism" for these single-index pairs, somewhat simpler than the one we developed for double-index Wannier excitons. We use it here to get the pair number threshold for bosonic behavior of $N$ pairs through the requirement that their number operator mean value must stay close to $N$. While the main term of this mean value is controlled by the second moment of the Schmidt distribution, so that to increase this threshold, we must increa…
In singular value decomposition (SVD) of a complex matrix A, the singular vectors or the eigenvectors of AA† and A†A are unique up to complex phase factors. Thus, the two unitary matrices in SVD are unique up to diagonal matrices of phase factors, the phase-factor matrices. Also, the product of these two phase-factor matrices, or the product of phase factors of the corresponding singular vectors with the same singular value, is unique. In the Schmidt decomposition, a phase-factor matrix is a phase rotation operator acting on a subsystem alone. We summarize here three simple steps to consistent…
We experimentally study the properties of coherent mode decomposition for intensity correlation function of quasi-thermal light. We use the technique of spatial mode selection developed for studying transverse entanglement of photon pairs, and show that it can be extended to characterize classical spatial correlations. Our results demonstrate the existence of a unique for a given thermal source basis of coherent modes, correlated in a way much resembling the Schmidt modes of spatially entangled photons.
Analysis of localized Schmidt decomposition modes and of entanglement in atomic and optical quantum systems with continuous variables
研究了连续变量系统中Schmidt模的提取过程。利用奇异值矩阵分解算法研究了具有自发辐射的“原子-光子”系统的纠缠。此外，该算法被应用到研究的双光子系统的自发参量下转换与第二类相位匹配的宽带泵浦。我们证明了具有自发辐射的原子-光子系统中纠缠态的动力学性质是由精细结构常数和原子-电子质量比的乘积决定的。然后，我们考虑在辐射过程中的系统的演化，并表明原子和光子的自由度纠缠的时间相同的数量级的激发态的寿命。最后，我们研究了由非线性晶体性质引起的II型参量下转换中模间相干性损失的过程。
The thesis includes the original results of our articles [30, 37, 40, 42, 51, 53, 75]. A method is developed to compute analytically entanglement measures of three-qubit pure states. Owing to it closed-form expressions are presented for the geometric measure of entanglement for three-qubit states that are linear combinations of four orthogonal product states. It turns out that the geometric measure for these states has three different expressions depending on the range of definition in parameter space and the Hilbert space of three-qubits consists of three different entangled regions. The stat…
Simplifying Schmidt number witnesses via higher-dimensional embeddings
我们适用于广义的概念证人运营商的任意凸集，并审查这些一般证人的优化标准。然后，我们定义一个嵌入的状态向量和运营商到一个高维希尔伯特空间。这种嵌入导致在原始希尔伯特空间中的任何施密特数见证与适当扩大的希尔伯特空间中的施密特数2的见证（即最一般的纠缠见证）之间的联系。利用这个关系，我们得到了一个概念上简单的方法，在二分系统中的施密特数证人的建设。
General Depolarized Pure States: Identification and Properties
Schmidt分解是量子系统研究中的一个重要工具，特别是对于纯态纠缠的量化。然而，Schmidt分解仅对二体纯态和某些多体纯态是唯一的.本文给出了与去极化纯态等价的态的广义Schmidt分解。这类混合状态的识别实验方法提供和一些例子进行了讨论，这表明这种描述的效用。一个特别有趣的例子，第一次提供了一个解释的数量负特征值的密度矩阵。
Hilbert-Schmidt (HS) decompositions are employed for analyzing systems of n-qubits, and a qubit with a qudit. Negative eigenvalues, obtained by partial-transpose (PT) plus local unitary transformations (PTU) for one qubit from the whole system, are used for indicating inseparability. A sufficient criterion for full separability of the n-qubits and qubit-qudit systems is given. We use the singular value decomposition (SVD) for improving the criterion for full separability. General properties of entanglement and separability are analyzed for a system of a qubit and a qudit and n-qubits systems, …
Tensor Network Efficiently Representing Schmidt Decomposition of Quantum Many-Body States
本文提出 Schmidt TNS，用线性复杂度的张量网络和正定 MPS 表示有限及无限量子多体态的 Schmidt 分解，并在阻挫准一维自旋模型中验证其高效性。
The present work studies quantum and classical correlations in three qubits and four qubits general Bell states, produced by operating with braid operators on the computational basis of states. The analogies between the general three qubits and four qubits Bell states and that of two qubits Bell states are discussed. The general Bell states are shown to be maximal entangled, i.e., with quantum correlations which are lost by tracing these states over one qubit, remaining only with classical correlations, as shown by HS decomposition.
This paper presents the general Schmidt decomposition of two-photon fields generated in spontaneous parametric down-conversion (SPDC). It discusses in particular the separation of the radial and azimuthal degrees of freedom, the role of projection in modal analysis, and the benefits of collinear phase mismatch. The paper is written in a review style and presents a wealth of numerical results. It aims at emphasising the physics beyond the mathematics, through discussions and graphical representations of key results.
Minimum error state discrimination between two mixed states ρand σcan be aided by the receipt of "classical side information" specifying which states from some convex decompositions of ρand σapply in each run. We quantify this phenomena by the average trace distance, and give lower and upper bounds on this quantity as functions of ρand σ. The lower bound is simply the trace distance between ρand σ, trivially seen to be tight. The upper bound is \sqrt{1 - tr(ρσ)}, and we conjecture that this is also tight. We reformulate this conjecture in terms of the existence of a pair of "unbiased decomposi…
In this work, we present an investigation on the spatial entanglement entropies in the helium atom by using highly correlated Hylleraas functions to represent the S-wave states. Singlet-spin 1sns 1Se states (with n = 1 to 6) and triplet-spin 1sns 3Se states (with n = 2 to 6) are investigated. As a measure on the spatial entanglement, von Neumann entropy and linear entropy are calculated. Furthermore, we apply the Schmidt-Slater decomposition method on the two-electron wave functions, and obtain eigenvalues of the one-particle reduced density matrix, from which the linear entropy and von Neuman…
Entanglement properties of quantum spin chains
我们研究了有限大小的1+1维Ising自旋链的纠缠特性，并展示了这些特性如何标度以及如何利用它们来重构基态波函数.即使在临界点，施密特分解中也很少有项对精确的基态和熵等物理性质有贡献。然而，由于施密特分解中的低位态，这里的纠缠是突出的。
Decomposition of (finite-dimensional) operators in terms of orthogonal bases of matrices has been a standard method in quantum physics for decades. In recent years, it has become increasingly popular because of various methodologies applied in quantum information, such as the graph state formalism and the theory of quantum error correcting codes, but also due to the intensified research on the Bloch representation of quantum states. In this contribution we collect various interesting facts and identities that hold for finite-dimensional orthogonal matrix bases.
Higher Berry Curvature from the Wave Function I: Schmidt Decomposition and Matrix Product States
本文提出利用 Schmidt 分解计算一维无限延伸系统的高阶 Berry 曲率公式，并给出矩阵乘积态的对应形式，证明平移不变 MPS 产生量子化不变量，通过精确可解模型和 iDMRG 数值验证。
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