Barbara M. Terhal
We review the criteria for separability and quantum entanglement, both in a bipartite as well as a multipartite setting. We discuss Bell inequalities, entanglement witnesses, entropic inequalities, bound entanglement and several features of multipartite entanglement. We indicate how these criteria bear on the experimental detection of quantum entanglement.
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Based on the generators of $SU(n)$ we present inequalities for detecting quantum entanglement for $2 \otimes d$ and $M \otimes N$ systems. These inequalities provide a sufficient condition of entanglement for bipartite mixed states and give rise to an experimental way of entanglement detection.
We present the general scheme for construction of noiseless networks detecting entanglement with the help of linear, hermiticity-preserving maps. We show how to apply the method to detect entanglement of unknown state without its prior reconstruction. In particular, we prove there always exists noiseless network detecting entanglement with the help of positive, but not completely positive maps. Then the generalization of the method to the case of entanglement detection with arbitrary, not necessarily hermiticity-preserving, linear contractions on product states is presented.
Monogamy of bipartite correlations leads, for arbitrary pure multi-qubit states, to simple conditions able to indicate various types of multipartite entanglement by being capable to exclude the possibility of k-separability.
Detecting multilevel entanglement from light-based entanglement witnesses
本文提出了一种基于电场的不等式，用于检测多级纠缠，适用于多体量子系统。
We present entanglement witness operators for detecting genuine multipartite entanglement. These witnesses are robust against noise and require only two local measurement settings when used in an experiment, independent from the number of qubits. This allows detection of entanglement for an increasing number of parties without a corresponding increase in effort. The witnesses presented detect states close to GHZ, cluster and graph states. Connections to Bell inequalities are also discussed.
We derive necessary conditions in terms of the variances of position and momentum linear combinations for all kinds of separability of a multi-party multi-mode continuous-variable state. Their violations can be sufficient for genuine multipartite entanglement, provided the combinations contain both conjugate variables of all modes. Hence a complete state determination, for example by detecting the entire correlation matrix of a Gaussian state, is not needed.
We explore the possibility of entanglement detection in continuous variable systems by entanglement witnesses based on covariance matrices, constructible from random homodyne measurements. We propose new linear constraints characterizing the entanglement witnesses based on second moments, and use them in a semidefinite program providing the optimal entanglement test for given random measurements. We test the method on the class of squeezed vacuum states and study the efficiency of entanglement detection in general unknown covariance matrices.
We show a general approach for detecting genuine multipartite entanglement (GME) and partial inseparability in many-body-systems by means of macroscopic observables (such as the energy) only. We show that the obtained criteria, the "GME gap" and "the k-entanglement gap", detect large areas of genuine multipartite entanglement and partial entanglement in typical many body states, which are not detected by other criteria. As genuine multipartite entanglement is a necessary property for several quantum information theoretic applications such as e.g. secret sharing or certain kinds of quantum comp…
Detecting $k$-nonseparability and $k$-partite Entanglement with Generalized Skew Information and Mutually Unbiased Measurements
本文提出基于广义Wigner-Yanase斜信息和互无偏测量的充分判据，用于检测多体系统中的$k$-不可分性与$k$体纠缠，并通过实例验证其检测能力，且可应用于量子网络深度识别。
It is possible to achieve an arbitrary amount of entanglement between two atoms using only spontaneously emitted photons, linear optics, single photon sources and projective measurements. This is in contrast to all current experimental proposals for entangling two atoms, which are fundamentally restricted to one entanglement bit or ebit.
Mutually unbiased bases, mutually unbiased measurements and general symmetric informationally complete measurements are three related concepts in quantum information theory. We investigate multipartite systems using these notions and present some criteria detecting entanglement of arbitrary high dimensional multi-qudit systems and multipartite systems of subsystems with different dimensions. It is proved that these criteria can detect the $k$-nonseparability ($k$ is even) of multipartite qudit systems and arbitrary high dimensional multipartite systems of $m$ subsystems with different dimensio…
We construct a set of criteria detecting genuine multipartite entanglement in arbitrary dimensional multipartite systems. These criteria are optimally suited for detecting multipartite entanglement in n-qubit Dicke states with m-excitations. In a detailed analysis we show that the criteria are also more robust to noise than any other criterion known so far, especially with increasing system size. Furthermore it is shown that the number of required local observables scales only polynomially with size, thus making the criteria experimentally feasible.
The problem of entanglement detection is a long standing problem in quantum information theory. One of the primary procedures of detecting entanglement is to find the suitable positive but non-completely positive maps. Here we try to give a generic prescription to construct a positive map that can be useful for such scenarios. We study a class of positive maps arising from Lindblad structures. We show that two famous positive maps viz. transposition and Choi map can be obtained as a special case of a class of positive maps having Lindblad structure. Generalizing the transposition map to a one …
Detecting genuine multipartite entanglement using moments of positive maps
本文提出利用正映射的矩检测真正的多体纠缠，通过示例展示其在三体和四体纠缠检测中的有效性，并提出实验实现方案。
In quantum information theory, the reliable and effective detection of entanglement is of paramount importance. However, given an unknown state, assessing its entanglement is a challenging task. To attack this problem, we investigate the use of random local measurements, from which entanglement witnesses are then constructed via semidefinite programming methods. We propose a scheme of successively increasing the number of measurements until the presence of entanglement can be unambiguously concluded, and investigate its performance in various examples.
We present a set of inequalities for detecting quantum entanglement of $2\otimes d$ quantum states. For $2\otimes 2$ and $2\otimes 3$ systems, the inequalities give rise to sufficient and necessary separability conditions for both pure and mixed states. For the case of $d>3$, these inequalities are necessary conditions for separability, which detect all entangled states that are not positive under partial transposition and even some entangled states with positive partial transposition. These inequalities are given by mean values of local observables and present an experimental way of detecting…
We demonstrate the principle of one-sided device-independent continuous variable (CV) quantum information. In situations of no trust, we show by enactment how the use of standard CV entanglement criteria can mislead Charlie into thinking Alice and Bob share entanglement, when the data is actually generated classically using a Local Hidden Variable theory based on the Wigner function. We distinguish between criteria that demonstrate CV entanglement, and criteria that demonstrate the CV Einstein-Podolsky-Rosen (EPR) steering paradox. We show that the latter, but not the former, are necessarily o…
Entanglement witness is an effective method to detect entanglement in unknown states without doing full tomography. One of the most widespread schemes of witnessing entanglement is measuring its fidelity with respect to a pure entangled state. Recently, a large class of states whose entanglement can not be detected with the fidelity witness has been discovered in Phys.Rev.Lett \textbf{124},200502(2020). They are called unfaithful states. In this paper we propose a new way to detect entanglement by calculating the lower bound of entanglement using measurement results. Numerical simulation shows…
Entanglement detection and estimation are fundamental in quantum information science. Compared with discrete-variable states, for which lots of efficient entanglement detection criteria and lower bounds of entanglement measures have been proposed, the continuous-variable entanglement is much less understood. Here we shall present a family of entanglement witnesses based on continuous-variable local orthogonal observables (CVLOOs) to detect and estimate entanglement of Gaussian and non-Gaussian states, especially for bound entangled states. By choosing an optimal set of CVLOOs our entanglement …
We derive several entanglement conditions employing non-hermitian operators. We start with two conditions that were derived previously for field mode operators, and use them to derive conditions that can be used to show the existence of field-atom entanglement and entanglement between groups of atoms. The original conditions can be strengthened by making them invariant under certain sets of local unitary transformations, such as Gaussian operations. We then apply these conditions to several examples, such as the Dicke model. We conclude with a short discussion of how local uncertainty relation…
Einstein-Podolsky-Rosen steering is a kind of powerful nonlocal quantum resource in quantum information processing such as quantum cryptography and quantum communication. Many criteria have been proposed in the past few years to detect steerability, both analytically and numerically, for bipartite quantum systems. We propose effective criteria for tripartite steerability and genuine tripartite steerability of three-qubit quantum states by establishing connections between the tripartite steerability (resp. genuine tripartite steerability) and the tripartite entanglement (resp. genuine tripartit…
We construct a single observable measurement of which mean value on four copies of an {\it unknown} two-qubit state is sufficient for unambiguous decision whether the state is separable or entangled. In other words, there exists a universal collective entanglement witness detecting all two-qubit entanglement. The test is directly linked to a function which characterizes to some extent the entanglement quantitatively. This function is an entanglement monotone under so--called local pure operations and classical communication (pLOCC) which preserve local dimensions. Moreover it provides tight up…
An uncertainty relation is introduced for a symmetric arrangement of three mutually unbiased bases in continuous variable phase space, and then used to derive a bipartite entanglement criterion based on the variance of global operators composed of these three phase space variables. We test this criterion using spatial variables of photon pairs, and show that the entangled photons are correlated in three pairs of bases.
We present a criterion for multiparticle entanglement based on covariance matrices. On the one hand, the criterion allows to detect bound entangled states which are not detected by other criteria; on the other hand, some strongly entangled pure states such as the GHZ states are not detected. We show, however, that this is a general phenomenon: No separability criterion based on two-particle correlations can recognize the entanglement in the family of graph states, to which the GHZ states belong.
Detecting genuine multipartite entanglement via machine learning
本文研究用监督与半监督机器学习检测三量子比特真正多体纠缠，随机生成密度矩阵训练SVM，并改进S4VM的分组与迭代预测方法，数值模拟表明可显著提升预测精度。
We devise a novel protocol to detect genuinely multipartite entangled states by harnessing quantum non-Markovian operations. We utilize a particular type of non-Markovianity known as the eternal non-Markovianity to construct a non-complete positive map to filter out the bi-separable states and detect genuine multipartite entanglement. We further propose a witness operator to detect genuinely multipartite entangled states experimentally based on this theory. Our study sheds light on a hitherto unexplored connection between entanglement theory and quantum non-Markovianity.
We derive a set of Bell-type inequalities for arbitrarily high-dimensional systems, based on the assumption of partial separability in the hybrid local-nonlocal hidden variable model. Partially entangled states would not violate the inequalities, and thus upon violation, these Bell-type inequalities are sufficient conditions to detect the full $N$-particle entanglement and validity of the hybrid local-nonlocal hidden variable description.
Quadrature squeezed cylindrically polarized modes contain entanglement not only in the polarization and spatial electric field variables but also between these two degrees of freedom [1]. In this paper we present tools to generate and detect this entanglement. Experimentally we demonstrate the generation of quadrature squeezing in cylindrically polarized modes by mode transforming a squeezed Gaussian mode. Specifically, -1.2 dB of amplitude squeezing are achieved in the radially and azimuthally polarized mode. Furthermore, theoretically it is shown how the entanglement contained within these m…
Detecting and protecting entanglement through nonlocality, variational entanglement witness, and nonlocal measurements
针对CHSH不等式无法完全覆盖所有纠缠态的问题，提出变分纠缠见证与非定域测量框架，在保留量子纠缠的同时提升检测可靠性，推动量子技术发展。
Detecting unfaithful entanglement by multiple fidelities
本文通过组合多个保真度改进纠缠见证，证明可认证不忠实量子态的纠缠，并分析其数学结构及提出最优设计搜索算法。
We present an entanglement criterion for two mode squeezed states which relies on particle counting only. The proposed inequality is optimal for the state under consideration and robust against particle losses up to 2/3. As it does not involve measurements of quadratures - which is typically very challenging for atomic modes - it renders the detection of atomic many-particle entanglement feasible in many different settings. Moreover it bridges the gap between entanglement verification for a qubit and criteria for continuous variables measured by homodyne detection. We illustrate its applicatio…
We present a general argument showing that the temperature as well as other thermodynamical state variables can qualify as entanglement witnesses for spatial entanglement. This holds for a variety of systems and we exemplify our ideas using a simple free non-interacting Bosonic gas. We find that entanglement can exist at arbitrarily high temperatures, provided that we can probe smaller and smaller regions of space. We then discuss the relationship between the occurrence of Bose-Einstein condensation and our conditions for the presence of entanglement and compare the respective critical tempera…
Detecting entanglement from macroscopic measurements of the electric field and its fluctuations
本文提出基于电场正交分量和总荧光的纠缠见证族，无需态层析即可从几乎任意方向探测大型开放量子系统中的纠缠，适用于冷原子、囚禁离子、色心和超导量子比特等泡利群发射体。
We propose a Partial Lorentz Transformation (PLT) test for detecting entanglement in a two qubit system. One can expand the density matrix of a two qubit system in terms of a tensor product of $(\mathbb{I}, \vecσ)$. The matrix $A$ of the coefficients that appears in such an expansion can be "squared" to form a $4\times4$ matrix $B$. It can be shown that the eigenvalues $λ_0, λ_1, λ_2, λ_3$ of $B$ are positive. With the choice of $λ_0$ as the dominant eigenvalue, the separable states satisfy $\sqrt{λ_1}+\sqrt{λ_2}+\sqrt{λ_3}\leq \sqrt{λ_0}$. Violation of this inequality is a test of entanglemen…
We propose a more general method for detecting a set of entanglement measures, i.e. negativities, in an \emph{arbitrary} tripartite quantum state by local operations and classical communication. To accomplish the detection task using this method, three observers, Alice, Bob and Charlie, do not need to perform the partial transposition maps by the structural physical approximation; instead, they are only required to collectively measure some functions via three local networks supplemented by a classical communication. With these functions, they are able to determine the set of negativities rela…
We discuss an experimentally amenable class of two-particle states of motion giving rise to nonlocal spatial interference under position measurements. Using the concept of modular variables, we derive a separability criterion which is violated by these non-Gaussian states. While we focus on the free motion of material particles, the presented results are valid for any pair of canonically conjugate continuous variable observables and should apply to a variety of bipartite interference phenomena.
One of the important approaches to detect quantum entanglement is using linear entanglement witnesses EWs. In this paper, by determining the envelope of the boundary hyper-planes defined by a family of linear EWs, a set of powerful nonlinear optimal EWs is manipulated. These EWs enable us to detect some three qubits bound MUB (mutually unbiased bases) diagonal entangled states, i.e., the PPT (positive partial transpose) entangled states. Also, in some particular cases, the introduced nonlinear optimal EWs are powerful enough to separate the bound entangled regions from the separable ones. Fina…
We implement a protocol to determine the degree of entanglement between a qubit and the rest of the system on a quantum computer. The protocol is based on results obtained in paper [Frydryszak et al. (2017)]. This protocol is tested on a 5-qubit superconducting quantum processor called ibmq-ourense provided by the IBM company. We determine the values of entanglement of the Schrödinger cat and the Werner states prepared on this device and compare them with the theoretical ones. In addition, a protocol for determining the entanglement of rank-2 mixed states is proposed. We apply this protocol to…
We show how the presence of entanglement in a bipartite Gaussian state can be detected by the amount of work extracted by a continuos variable Szilard-like device, where the bipartite state serves as the working medium of the engine. We provide an expression for the work extracted in such a process and specialize it to the case of Gaussian states. The extractable work provides a sufficient condition to witness entanglement in generic two-mode states, becoming also necessary for squeezed thermal states. We extend the protocol to tripartite Gaussian states, and show that the full structure of in…
Improved criteria of detecting multipartite entanglement structure
本文利用张量积空间划分构造纠缠见证，并结合半定规划与梯度下降算法，提出改进的多体纠缠结构检测判据，实验表明其性能优于现有方法。
We propose a scheme for the observation of micro-macro entanglement in photon number based on amplifying and de-amplifying a single-photon entangled state in combination with homodyne quantum state tomography. The created micro-macro entangled state, which exists between the amplification and de-amplification steps, is a superposition of two components with mean photon numbers that differ by approximately a factor of three. We show that for reasonable values of photon loss it should be possible to detect micro-macro photon-number entanglement where the macro system has a mean number of one hun…
We consider a subgroup of unitary transformations on a mode of light induced by a Mach-Zehnder Interferometer and an algebra of observables describing a photon-number detector proceeded by an interferometer. We explore the uncertainty principles between such observables and their usefulness in performing a Bell-like experiment to show a violation of the CHSH inequality, under physical assumption that the detector distinguishes only zero from non-zero number of photons. We show which local settings of the interferometers lead to a maximal violation of the CHSH inequality.
For any bipartite systems, a universal entanglement witness of rank-4 for pure states is obtained and a class of finite rank entanglement witnesses is constructed. In addition, a method of detecting entanglement of a state only by entries of its density matrix with respect to some product basis is obtained.
The problem of bound entanglement detection is a challenging aspect of quantum information theory for higher dimensional systems. Here, we propose an indecomposable positive map for two-qutrit systems, which is shown to generate a class of positive partial transposed (PPT) states. A corresponding witness operator is constructed and shown to be weakly optimal and locally implementable. Further, we perform a structural physical approximation of the indecomposable map to make it a completely positive one, and find a new PPT entangled state which is not detectable by certain other well-known entan…
Detecting entanglement in any measurement using quantum networks
针对复合量子测量中纠缠检测缺乏严谨方法的问题，本文定义了测量纠缠见证者概念，证明任意纠缠测量在无输入网络量子导引中具优势，且星型网络中秩一投影纠缠测量可设备无关检测，为测量纠缠检测提供新途径。
We show how a straightforward Bayesian updating procedure allows one to detect and quantify entanglement from any finite set of measurement results. The measurements do not have to be tomographically complete, and may consist of POVMs rather than von Neumann measurements. One obtains a probability that one's state is entangled and an estimate of any desired entanglement measure, including their error bars. As an example we consider (tomographically incomplete) spin correlation measurements on both 2-qubit and 3-qubit states. As byproducts we obtain estimates of the volume of entangled states v…
Detecting multipartite entanglement via complete orthogonal basis
本文基于完全正交基（COB）研究任意$n$体量子系统中的多体纠缠，利用统一的生成元简化计算，推导出可操作的判据，能比现有方法检测出更多真正三方与多体纠缠态。
Detecting entanglement and nonlocality with minimum observable length
本文针对量子纠缠与非定域性的检测需求，扩展“检测长度”框架以覆盖多类纠缠与非定域性现象，构建分析模型确定检测长度并利用半定规划技术提供上界，且验证短检测长度见证者在特定条件下噪声鲁棒性更优。
While the detection of entanglement has been proved already to be quite a difficult task, experimental quantification of entanglement is even more challenging. In this work, we derive an analytical lower bound for the concurrence of a bipartite mix quantum state in arbitrary dimension. The lower bound is experimentally implementable in a feasible way, which enabling quantification of entanglement in a broad variety of cases
We introduce a weak form of the realignment separability criterion which is particularly suited to detect continuous-variable entanglement and is physically implementable (it requires linear optics transformations and homodyne detection). Moreover, we define a family of states, called Schmidt-symmetric states, for which the weak realignment criterion reduces to the original formulation of the realignment criterion, making it even more valuable as it is easily computable especially in higher dimensions. Then, we focus in particular on Gaussian states and introduce a filtration procedure based o…
Multipartite entanglement plays an essential role in both quantum information science and many-body physics. Due to the exponentially large dimension and complex geometric structure of the state space, the detection of entanglement in many-body systems is extremely challenging in reality. Conventional means, like entanglement witness and entropy criterion, either highly depend on the prior knowledge of the studied systems or the detection capability is relatively weak. In this work, we propose a framework for designing multipartite entanglement criteria based on permutation moments, which have…
We present a method for detecting the entanglement of a state using non-equilibrium processes. A comparison of relative entropies allows us to construct an entanglement witness. The relative entropy can further be related to the quantum Jarzynski equality, allowing non-equilibrium work to be used in entanglement detection. To exemplify our results, we consider two different spin chains.
We discuss the problem of determining whether the state of several quantum mechanical subsystems is entangled. As in previous work on two subsystems we introduce a procedure for checking separability that is based on finding state extensions with appropriate properties and may be implemented as a semidefinite program. The main result of this work is to show that there is a series of tests of this kind such that if a multiparty state is entangled this will eventually be detected by one of the tests. The procedure also provides a means of constructing entanglement witnesses that could in princip…
We introduce a sequence of numerical tests that can determine the entanglement or separability of a state even when there is not enough information to completely determine its density matrix. Given partial information about the state in the form of linear constraints on the density matrix, the sequence of tests can prove that either all states satisfying the constraints are entangled, or there is at least one separable state that satisfies them. The algorithm works even if the values of the constraints are only known to fall in a certain range. If the states are entangled, an entanglement witn…
Even-parity precession protocol for detecting nonclassicality and entanglement
本研究提出偶宇称进动协议，仅通过不同时间点的均匀进动变量测量，即可检测多种量子系统的非经典性与纠缠，还可检测被原有类似协议遗漏的态，填补了偶数比特GHZ纠缠仅用集体自旋测量检测的长期空白。
We discuss the possibility of observing quantum nonlocality using the so-called mode entanglement, analyzing the differences between different types of particles in this context. We first discuss the role of coherent states in such experiments, and we comment on the existence of coherent states in nature. The discussion of coherent states naturally raises questions about the role of particle statistics in this problem. Although the Pauli exclusion principle precludes coherent states with a large number of fermionic particles, we find that a large number of fermionic coherent states, each conta…
Einstein-Podolsky-Rosen (EPR) steering is a form of quantum nonlocality which is intermediate between entanglement and Bell nonlocality. EPR steering is a resource for quantum key distribution that is device independent on only one side in that it certifies bipartite entanglement when one party's device is not characterized while the other party's device is fully characterized. In this work, we introduce two types of genuine tripartite EPR-steering, and derive two steering inequalities to detect them. In a semi-device-independent scenario where only the dimensions of two parties are assumed, t…
We present methods for detecting entanglement around symmetric Dicke states. In particular, we consider N-qubit symmetric Dicke states with N/2 excitations. In the first part of the paper we show that for large N these states have the smallest overlap possible with states without genuine multi-partite entanglement. Thus these states are particulary well suited for the experimental examination of multi-partite entanglement. We present fidelity-based entanglement witness operators for detecting multipartite entanglement around these states. In the second part of the paper we consider entanglemen…
Genuine multipartite entanglement plays important roles in quantum information processing. The detection of genuine multipartite entanglement has been long time a challenging problem in the theory of quantum entanglement. We propose a criterion for detecting genuine tripartite entanglement of arbitrary dimensional tripartite states based on quantum Fisher information. We show that this criterion is more effective for some states in detecting genuine tripartite entanglement by detailed example.
Detecting strongly non-Gaussian entanglement
本文提出基于部分转置态矩的连续变量纠缠判据及多副本读出方案，可检测包括NOON态在内的强非高斯纠缠，并对实验损耗和噪声具有鲁棒性。
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