Hao Chen
Our previous work about algebraic-geometric invariants of the mixed states are extended and a stronger separability criterion is given. We also show that the Schmidt number of pure states in bipartite quantum systems, a classical concept, is actually an algebraic-geometric invariant.
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The Schmidt number is a fundamental parameter characterizing the properties of quantum states, and the local projections are a fundamental operation in quantum physics. We investigate the relation between the Schmidt numbers of bipartite states and their projected states. We show that there exist bipartite positive-partial-transpose (PPT) entangled states of any given Schmidt number. We further construct the notion of joint Schmidt number for multipartite states, and its relation with the Schmidt number of bipartite reduced density operators.
We prove a lower bound for Schmidt numbers of bipartite mixed states. This lower bound can be applied easily to low rank bipartite mixed states. From this lower bound it is known that generic low rank bipartite mixed states have relatively high Schmidt numbers and thus entangled. We can compute Schmidt numbers exactly for some mixed states by this lower bound as shown in Examples. This lower bound can also be used effectively to determine that some mixed states cannot be convertible to other mixed states by local operations and classical communication. The results in this paper are proved by O…
It is shown that for each mixed state there exists a Schmidt (super state vector) decomposition in terms of Hermitian operators. Its utilization for finding all twins is illustrated in full detail in the case of the two spin-one-half-particle states with maximally disordered subsystems (mixtures of Bell states).
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 …
We consider the Schmidt decomposition of a bipartite density operator induced by the Hilbert-Schmidt scalar product, and we study the relation between the Schmidt coefficients and entanglement. First, we define the Schmidt equivalence classes of bipartite states. Each class consists of all the density operators (in a given bipartite Hilbert space) sharing the same set of Schmidt coefficients. Next, we review the role played by the Schmidt coefficients with respect to the separability criterion known as the `realignment' or `computable cross norm' criterion; in particular, we highlight the fact…
We consider a bipartite mixed state of the form, $ρ=\sum_{α, β=1}^{l}a_{αβ} | ψ_α> < ψ_ β}| $, where $| ψ_α>$ are normalized bipartite state vectors, and matrix $(a_{αβ})$ is positive semidefinite. We provide a necessary and sufficient condition for the state $ρ$ taking the form of maximally correlated states by a local unitary transformation. More precisely, we give a criterion for simultaneous Schmidt decomposability of $| ψ_α>$ for $α=1,2,..., l$. Using this criterion, we can judge completely whether or not the state $ρ$ is equivalent to the maximally correlated state, in which the distilla…
Entanglement distillation in terms of Schmidt rank and matrix rank
本文研究给定施密特秩和矩阵秩的NPT二分态的纠缠蒸馏问题，证明施密特秩为二的态局域等价于经典-经典态、施密特秩为三的态1-不可蒸馏，并给出低秩B-不可约NPT态可蒸馏的充分条件及秩为max{M,N}+1的M×N二分态可蒸馏的等价条件。
We study the problem of transforming a tripartite pure state to a bipartite one using stochastic local operations and classical communication (SLOCC). It is known that the tripartite-to-bipartite SLOCC convertibility is characterized by the maximal Schmidt rank of the given tripartite state, i.e. the largest Schmidt rank over those bipartite states lying in the support of the reduced density operator. In this paper, we further study this problem and exhibit novel results in both multi-copy and asymptotic settings. In the multi-copy regime, we observe that the maximal Schmidt rank is strictly s…
We prove, using a new method based on map-state duality, lower bounds on entanglement resources needed to deterministically implement a bipartite unitary using separable (SEP) operations, which include LOCC (local operations and classical communication) as a particular case. It is known that the Schmidt rank of an entangled pure state resource cannot be less than the Schmidt rank of the unitary. We prove that if these ranks are equal the resource must be uniformly (maximally) entangled: equal nonzero Schmidt coefficients. Higher rank resources can have less entanglement: we have found numerica…
We investigate the differential geometry of bipartite quantum states. In particular the manifold structures of pure bipartite states are studied in detail. The manifolds with respect to all normalized pure states of arbitrarily given Schmidt ranks or Schmidt coefficients are explicitly presented. The dimensions of the related manifolds are calculated.
There are none.
Families of Schmidt-number witnesses for high dimensional quantum states
该研究利用 SIC-POVM 与相互无偏基构造 k-正映射及相应的 Schmidt 数见证，并据此量化两体态与低 Schmidt 数态集合的距离。
Restriction-Based Certificate of Bipartite Schmidt Rank in Hypergraph States
本文提出了一种基于限制的方法，用于在超图态中计算双分量辛格秩，并给出了无残留双线性核心的构造和验证方法。
We examine the application of Schmidt-mode analysis to pure state entanglement. Several examples permitting exact analytic calculation of Schmidt eigenvalues and eigenfunctions are included, as well as evaluation of the associated degree of entanglement.
Schmidt number criterion via general symmetric informationally complete measurements
本研究提出一种基于通用对称信息完备测量所得关联矩阵迹范数的施密特数判据，可有效量化任意局域维度两体态的纠缠维度，且经示例验证其性能优于多种现有判据。
It is known that any bipartite unitary operator of Schmidt rank three is equivalent to a controlled unitary under local unitaries. We propose a standard form of such operators. Using the form we improve the upper bound for the entanglement cost to implement such operators under local operations and classical communications (LOCC), and provide a corresponding protocol. A part of our protocol is based on a recursive-control protocol which is helpful for implementing other unitary operators. We show that any bipartite permutation unitary of Schmidt rank three can be implemented using LOCC and two…
Quantum circuit model is the most popular paradigm for implementing complex quantum computation. Based on Cartan decomposition, we show that $2(N-1)$ generalized controlled-$X$ (GCX) gates, $6$ single-qubit rotations about the $y$- and $z$-axes, and $N+5$ single-partite $y$- and $z$-rotation-types which are defined in this paper are sufficient to simulate a controlled-unitary gate $\mathcal{U}_{cu(2\otimes N)}$ with $A$ controlling on $\mathbb{C}^2\otimes \mathbb{C}^N$. In the scenario of the unitary gate $\mathcal{U}_{cd(M\otimes N)}$ with $M\geq3$ that is locally equivalent to a diagonal uni…
In this note we generalize Nielsen's marjoization criterion for the convertibility of bipartite pure states [Phys. Rev. Lett \textbf{83}, 436(1999)] to a special class of multipartite pure states which have generalized Schmidt decompositions.
In this letter we have established the physical character of pure bipartite states with the same amount of entanglement in the same Schmidt rank that either they are local unitarily connected or they are incomparable. There exist infinite number of deterministically locally inequivalent classes of pure bipartite states in the same Schmidt rank (starting from three) having same amount of entanglement. Further, if there exists incomparable states with same entanglement in higher Schmidt ranks (greater than three), then they should differ in at least three Schmidt coefficients.
Extending Schmidt vector from pure to mixed states for characterizing entanglement
本研究利用优超格结构将Schmidt向量从纯态扩展至混合两体态，证明其在LOCC下具有单调性且能表征可分与最大纠缠态，并据此引入了一族纠缠单调量。
An entanglement measure for pure-state continuous-variable bi-partite problem, the Schmidt number, is analytically calculated for one simple model of atom-field scattering.
Unitary operations are physically implementable. We further the understanding of such operations by studying the possible forms of nonlocal unitary operators, which are bipartite or multipartite unitary operators that are not tensor product operators. They are of broad relevance in quantum information processing. We prove that any nonlocal unitary operator of Schmidt rank three on a $d_A\times d_B$ bipartite system is locally equivalent to a controlled unitary. This operator can be locally implemented assisted by a maximally entangled state of Schmidt rank $\min\{d_A^2,d_B\}$ when $d_A\le d_B$…
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 study Schmidt rank for a vector (i.e., a pure state) and Schmidt number for a mixed state which are entanglement measures. We show that if a subspace of a certain bipartite system contains no vector of Schmidt rank $\leqslant k$, then any state supported on that space has Schmidt number at least $k+1$. A construction of subspace of $\mathbb{C}^m \otimes \mathbb{C}^n$ of maximal dimension, which does not contain any vector of Schmidt rank less than $3$, is given here.
The Schmidt number is an entanglement measure whose logarithm quantifies the zero-error entanglement cost of generating a given quantum state using local operations and classical communication (LOCC). %However, the Schmidt number is a notoriously difficult quantity to compute, and its relationship to other entanglement measures is largely unknown. In this paper we show that the Schmidt number is highly non-multiplicative in the sense that for any integer $n$, there exists states whose Schmidt number remains constant when taking $n$ copies of the given state. These states also provide a rare in…
The Schmidt number is of crucial importance in characterizing the bipartite pure states. We explore and propose here a generalization of Schmidt number for states in multipartite systems. It is shown to be entanglement monotonic and valid for both pure and mixed states. In addition, the corresponding generalization of multipartite Schmidt coefficients is introduced. Our approach is applicable for systems with arbitrary number of parties and for arbitrary dimensions.
Quantum null-hypothesis device-independent Schmidt rank witness
本研究构建设备无关零见证测试以判定两体量子系统维度、Schmidt数及空间类型，经IBM Quantum设备实验验证其准确性，发现一项测试符合理论、一项存在超6标准差偏差，可用于量子系统诊断。
A quantum state's entanglement across a bipartite cut can be quantified with entanglement entropy or, more generally, Schmidt norms. Using only Schmidt decompositions, we present a simple iterative algorithm to maximize Schmidt norms. Depending on the choice of norm, the optimizing states maximize or minimize entanglement, possibly across several bipartite cuts at the same time and possibly only among states in a specified subspace. Recognizing that convergence but not success is certain, we use the algorithm to explore topics ranging from fermionic reduced density matrices and varieties of pu…
Inspired by the `computable cross norm' or `realignment' criterion, we propose a new point of view about the characterization of the states of bipartite quantum systems. We consider a Schmidt decomposition of a bipartite density operator. The corresponding Schmidt coefficients, or the associated symmetric polynomials, are regarded as quantities that can be used to characterize bipartite quantum states. In particular, starting from the realignment criterion, a family of necessary conditions for the separability of bipartite quantum states is derived. We conjecture that these conditions, which a…
For any bipartite quantum system the Schmidt decomposition allows us to express the state vector in terms of a single sum instead of double sums. We show the existence of the Schmidt decomposition for tripartite system under certain condition. If the partial inner product of a basis (belonging to a Hilbert space of smaller dimension) with the state of the composite system gives a disentangled basis, then the Schmidt decomposition for a tripartite system exists. In this case the reduced density matrix of each of the subsystem has equal spectrum in the Schmidt basis.
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.
Schmidt-number robustness as a unified quantifier of high dimensional entanglement in Buscemi nonlocality
本文通过Schmidt数鲁棒性统一量化高维纠缠，揭示了不同Buscemi型操作对象间的资源关系。
The Schmidt coefficients capture all entanglement properties of a pure bipartite state and therefore determine its usefulness for quantum information processing. While the quantification of the corresponding properties in mixed states is important both from a theoretical and a practical point of view, it is considerably more difficult, and methods beyond estimates for the concurrence are elusive. In particular this holds for a quantitative assessment of the most valuable resource, the maximum possible Schmidt number of an arbitrary mixed state. We derive a framework for lower bounding the appr…
In this work we estimate the transverse Schmidt number for the bipartite bright squeezed vacuum state by means of second-order intensity correlation function measurement. Assuming that the number of modes is equal in both beams we determine the Schmidt number considering only one of the subsystems. The obtained results demonstrate that this approach is equally efficient over the whole propagation of the state from the near field to the far field regions of its emitter.
We propose a method for obtaining the Schmidt decomposition of bipartite systems with continuous variables. It approximates the modes to the prescribed accuracy by well known orthogonal functions. We give some criteria for the control of errors. We illustrate the method comparing its results with the already published analysis for entanglement of biphotons. The agreement is excellent.
Measurement-device-independent Schmidt number certification of all entangled states
本文提出了一种基于半量子非本地游戏的测量设备无关认证方法，用于认证具有高Schmidt数的纠缠态，解决了传统贝尔非本地游戏无法认证所有纠缠态的问题。
The Schmidt rank for the commuting operator framework
本研究将量子信息中的施密特秩推广到对易算子框架，给出其代数与操作定义并证明等价，计算了多种典型两体态的施密特秩，最后提出该框架下的开放问题。
Entanglement measures quantify the amount of quantum entanglement that is contained in quantum states. Typically, different entanglement measures do not have to be partially ordered. The presence of a definite partial order between two entanglement measures for all quantum states, however, allows for meaningful conceptualization of sensitivity to entanglement, which will be greater for the entanglement measure that produces the larger numerical values. Here, we have investigated the partial order between the normalized versions of four entanglement measures based on Schmidt decomposition of bi…
Enhanced Schmidt number criteria based on correlation trace norms
本文提出基于关联迹范数的施密特数判据，适用于任意局域维度，且严格强于现有保真度见证和重排判据。
We apply the generalised concept of witness operators to arbitrary convex sets, and review the criteria for the optimisation of these general witnesses. We then define an embedding of state vectors and operators into a higher-dimensional Hilbert space. This embedding leads to a connection between any Schmidt number witness in the original Hilbert space and a witness for Schmidt number two (i.e. the most general entanglement witness) in the appropriate enlarged Hilbert space. Using this relation we arrive at a conceptually simple method for the construction of Schmidt number witnesses in bipart…
We consider a four-state pure bipartite system consisting of four qubits shared among two parties using the same Schmidt basis, two qubits per party. In some cases, transformation between two known pure states may not be possible using LOCC transformations but may be possible with the addition of a two-qubit catalyst. We provide a necessary and sufficient condition for this to occur.
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。
SU(N) algebras and new thumbrules for entanglement of bipartite qubit and qutrit systems
本文基于施密特分解提出检验二分量子比特与量子三态系统纠缠的新经验法则，并揭示其分别由SU(2)和SU(3)代数结构支撑。
Automated modal analysis of entanglement with bipartite self-configuring optics
提出一种基于双部分自配置光学的方法，通过变分优化输出功率或符合计数自动学习任意纯量子态的Schmidt分解，为双部分集成量子光子系统的纠缠分析提供了通用且可扩展的途径。
The general class of Gaussian Schmidt-number witness operators for bipartite systems is studied. It is shown that any member of this class is reducible to a convex combination of two types of Gaussian operators using local operations and classical communications. This gives rise to a simple operational method, which is solely based on measurable covariance matrices of quantum states. Our method bridges the gap between theory and experiment of entanglement quantification. In particular, we certify lower bounds of the Schmidt number of squeezed thermal and phase-randomized squeezed vacuum states…
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。
In this short note we show two completely opposite methods of constructing entangled states. Given a bipartite state $γ\in M_k\otimes M_k$, define $γ_S=(Id+F)γ(Id+F)$, $γ_A=(Id-F)γ(Id-F)$, where $F\in M_k\otimes M_k$ is the flip operator. In the first method, entanglement is a consequence of the inequality $\text{rank}(γ_S)<\sqrt{\text{rank}(γ_A)}$. In the second method, there is no correlation between $γ_S$ and $γ_A$. These two methods show how diverse is quantum entanglement. We prove that any bipartite state $γ\in M_k\otimes M_k$ satisfies $\displaystyle SN(γ)\geq\max \left\{ \frac{\text{ra…
Bipartite representations and many-body entanglement of pure states of $N$ indistinguishable particles
本研究提出适用于玻色子与费米子的N个全同粒子纯态类二分表示，推导精确类Schmidt展开，分析纠缠单调性与近似可靠性，为全同粒子多体纠缠研究提供新形式体系。
Nonlocal unitary operations can create quantum entanglement between distributed particles, and the quantification of created entanglement is a hard problem. It corresponds to the concepts of entangling and assisted entangling power when the input states are, respectively, product and arbitrary pure states. We analytically derive them for Schmidt-rank-two bipartite unitary and some complex bipartite permutation unitaries. In particular, the entangling power of permutation unitary of Schmidt rank three can take only one of two values: $\log_2 9 - 16/9$ or $\log_2 3$ ebits. The entangling power, …
We introduce the notion of a Schmidt number of a bipartite density matrix, characterizing the minimum Schmidt rank of the pure states that are needed to construct the density matrix. We prove that Schmidt number is nonincreasing under local quantum operations and classical communication. We show that $k$-positive maps witness Schmidt number, in the same way that positive maps witness entanglement. We show that the family of states which is made from mixing the completely mixed state and a maximally entangled state have increasing Schmidt number depending on the amount of maximally entangled st…
Bipartite maximally entangled states have the property that the largest Schmidt coefficient reaches its lower bound. However, for multipartite states the standard Schmidt decomposition generally does not exist. We use a generalized Schmidt decomposition and the geometric measure of entanglement to characterize three-qubit pure states and derive a single-parameter family of maximally entangled three-qubit states. The paradigmatic Greenberger-Horne-Zeilinger (GHZ) and W states emerge as extreme members in this family of maximally entangled states. This family of states possess different trends o…
Schmidt rank of bipartite pure state serves as a testimony of entanglement. It is a monotone under local operation + classical communications (LOCC) and puts restrictions in LOCC convertibility of quantum states. Identifying the Schmidt rank of an unknown quantum state therefore seek importance from information theoretic perspective. In this work it is shown that a modified version of Hardy's argument, which reveals the contradiction of quantum theory with local realism, turns out to be useful for inspecting the minimal Schmidt rank of the unknown state and hence also the minimal dimension of …
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。
Bound entanglement with a nonpositive partial transposition (NPT) does not exist. For any NPT entangled state a distillation procedure can be based on a certain number of copies. This number is the minimal Schmidt rank of a pure state needed to witness the NPT entanglement under study.
The Schmidt decomposition is an important tool in the study of quantum systems especially for the quantification of the entanglement of pure states. However, the Schmidt decomposition is only unique for bipartite pure states, and some multipartite pure states. Here a generalized Schmidt decomposition is given for states which are equivalent to depolarized pure states. Experimental methods for the identification of this class of mixed states are provided and some examples are discussed which show the utility of this description. A particularly interesting example provides, for the first time, a…
Implementing nonlocal unitary operators is an important and hard question in quantum computing and cryptography. We show that any bipartite nonlocal unitary operator of Schmidt rank three on the $(d_A \times d_B)$-dimensional system is locally equivalent to a controlled unitary when $d_A$ is at most three. This operator can be locally implemented assisted by a maximally entangled state of Schmidt rank $r=\min\{d_A^2,d_B\}$. We further show that stochastic-equivalent nonlocal unitary operators are indeed locally equivalent, and propose a sufficient condition on which nonlocal and controlled uni…
This paper has been withdrawn.
There has been spectacular progress in the field of quantum information in recent decades. The development of this field highlights the importance of the role of entanglement in quantum computing, quantum teleportation and quantum cryptography. These notes serve to provide a gentle introduction to the entanglement of bipartite states. In these notes, we introduce Bell's theorem in the form derived by Clauser, Horne, Shimony and Holt. We discuss the Schmidt decomposition and the Peres-Horodecki criterion in the entanglement of pure and mixed bipartite states. Finally, we describe a teleportatio…
We discuss disentanglement of pure bipartite quantum states within the framework of the schemes developed for entanglement splitting and broadcasting of entanglement.
A complete analysis of entangled bipartite qutrit pure states is carried out based on a simple entanglement measure. An analysis of all possible extremally entangled pure bipartite qutrit states is shown to reduce, with the help of SLOCC transformations, to three distinct types. The analysis and the results should be helpful for finding different entanglement types in multipartite pure state systems.
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