Motivated by the Kronecker product approximation technique, we have developed a very simple method to assess the inseparability of bipartite quantum systems, which is based on a realigned matrix constructed from the density matrix. For any separable state, the sum of the singular values of the matrix should be less than or equal to 1. This condition provides a very simple, computable necessary criterion for separability, and shows powerful ability to identify most bound entangled states discussed in the literature. As a byproduct of the criterion, we give an estimate for the degree of entanglement of the quantum state.
In quant-ph/0205017 v1 Chen, Wu and Yang formulated a necessary separability criterion based on a realignment method for matrices. This note is to point out that this criterion is identical to the necessary cross norm criterion previously put forward by this author.
It is shown that the enhanced (nonlinear) realignment criterion is equivalent to the family of linear criteria based on correlation tensor. These criteria generalize the original (linear) realignment criterium and give rise to the family of entanglement witnesses. An appropriate limiting procedure is proposed which leads to a novel class of witnesses which are as powerful as the enhanced realignment criterion.
It is challenging task to detect and measure genuine multipartite entanglement. We investigate the problem by considering the average based positive partial transposition(PPT) criterion and the realignment criterion. Sufficient conditions for detecting genuine tripartite entanglement are presented. We also derive lower bounds for the genuine tripartite entanglement concurrence with respect to the conditions. While the PPT criterion and the realignment criterion are powerful for detecting bipartite entanglement and for providing lower bounds of bipartite concurrences, our results give an effect…
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…
The separability of bipartite non-Gaussian states is studied by applying the realignment criterion with the technique of functional analysis. The realignment criterion is given as one inequality in contrast to the infinitive number of inequalities based on the moments. We give the necessary and sufficient condition of inseparability for non-Gaussian states prepared by photon subtraction or addition from symmetric Gaussian states. The entanglement criterion of non-Gaussian states evolved in thermal noise and amplitude damping environment is also obtained.
Separability problem, to decide whether a given state is entangled or not, is a fundamental problem in quantum information theory. We propose a powerful and computationally simple separability criterion, which allows us to detect the entanglement of many bound entangled states. The criterion is strictly stronger than the dV criterion, the computable cross-norm or realignment criterion and its optimal nonlinear entanglement witnesses. Furthermore, this criterion can be generalized to an analogue of permutation separability criteria in the even-partite systems.
Quantum entanglement is the core resource in quantum information processing and quantum computing. It is an significant challenge to effectively characterize the entanglement of quantum states. Recently, elegant separability criterion is presented in [Phys. Rev. Lett. 125, 200501 (2020)] by Elben et al. based on the first three partially transposed (PT) moments of density matrices. Then in [Phys. Rev. Lett. 127, 060504 (2021)] Yu \emph{et al.} proposed two general powerful criteria based on the PT moments. In this paper, based on the realignment operations of matrices we propose entanglement d…
It is known that realignment crierion is necessary but not a sufficient criterion for lower as well as higher dimensional system. In this work, we first consider a two-qubit system and derived the necessary and sufficient condition based on realignment operation for a particular class of two-qubit system. Thus we solved the problem of if and only if condition partially for a particular class of two-qubit state. We have shown that the derived necessary and sufficient condition detects two-qubit entangled states, which are not detected by the realignment criterion. Next, we discuss the higher di…
Based on the matrix realignment and partial transpose, we develop an approach to entangling power and operator entanglement of quantum unitary operators. We demonstrate efficiency of the approach by studying several unitary operators on qudits, and indicate that these two matrix rearrangements are not only powerful for studying separability problem of quantum states, but also useful in studying entangling capabilities of quantum operators.
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…
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…
In this paper, in terms of the relation between the state and the reduced states of it, we obtain two inequalities which are valid for all separable states in infinite-dimensional bipartite quantum systems. One of them provides an entanglement criterion which is strictly stronger than the computable cross-norm or realignment (CCNR) criterion.
Inspired by the realignment or computable cross norm criterion, we present a new result about the characterization of quantum entanglement. Precisely, an interesting class of inequalities satisfied by all separable states of a bipartite quantum system is derived. These inequalities induce new separability criteria that generalize the realignment criterion.
We give an explicit tight lower bound for the entanglement of formation for arbitrary bipartite mixed states by using the convex hull construction of a certain function. This is achieved by revealing a novel connection among the entanglement of formation, the well-known Peres-Horodecki and realignment criteria. The bound gives a quite simple and efficiently computable way to evaluate quantitatively the degree of entanglement for any bipartite quantum state.
We show that any state which violates the computable cross norm (or realignment) criterion for separability also violates the separability criterion of the local uncertainty relations. The converse is not true. The local uncertainty relations provide a straightforward construction of nonlinear entanglement witnesses for the cross norm criterion.
Quantifying mixed-state entanglement via partial transpose and realignment moments
Technical University of Munich（慕尼黑工业大学）；Munich Center for Quantum Science and Technology (MCQST)（慕尼黑量子科学与技术中心）；Technology Innovation Institute（技术创新研究所）
Negativity is regarded as an important measure of entanglement in quantum information theory. In contrast to other measures of entanglement, it is easily computable for bipartite states in arbitrary dimensions. In this paper, based on the negativity and realignment, we provide a set of entanglement-sharing constraints for multipartite states, where the entanglement is not necessarily limited to bipartite and pure states, thus aiding in the quantification of constraints for entanglement-sharing. These may find applications in studying many-body systems.
The problem on detecting the entanglement of a bipartite state is significant in quantum information theory. In this article, we apply the Ky Fan norm to the revised realignment matrix of a bipartite state. Specifially, we consider a family of separable criteria for bipartite states, and present when the density matrix corresponds to a state is real, the criteria is equivalent to the enhanced realignment criterion. Moreover, we present analytical lower bounds of concurrence and the convex-roof extended negativity for arbitrary dimensional systems.
We show that a special type of measurements, called symmetric informationally complete positive operator-valued measures (SIC POVMs), provide a stronger entanglement detection criterion than the computable cross-norm or realignment criterion based on local orthogonal observables. As an illustration, we demonstrate the enhanced entanglement detection power in simple systems of qubit and qutrit pairs. This observation highlights the significance of SIC POVMs for entanglement detection.
Employing a recently proposed separability criterion we develop analytical lower bounds for the concurrence and for the entanglement of formation of bipartite quantum systems. The separability criterion is based on a nondecomposable positive map which operates on state spaces with even dimension N >= 4, and leads to a class of nondecomposable optimal entanglement witnesses. It is shown that the bounds derived here complement and improve the existing bounds obtained from the criterion of positive partial transposition and from the realignment criterion.
We study the concurrence of arbitrary dimensional bipartite quantum systems. An explicit analytical lower bound of concurrence is obtained, which detects entanglement for some quantum states better than some well-known separability criteria, and improves the lower bounds such as from the PPT, realignment criteria and the Breuer's entanglement witness.
Entanglement detection problem is one of the important problem in quantum information theory. Gurvit showed that this problem is NP complete and thus this may be the possible reason that only one criterion is not sufficient to detect all entangled states. There are some powerful entanglement detection criterion such as partial transposition criterion, realignment criterion but it may not be possible to implement them successfully in the experiment. This situation can be avoided if the entanglement is detected through the construction of witness operator method. In this work, we take an analyti…
Motivated by the Peres-Horodecki criterion and the realignment criterion we develop a more powerful method to identify entangled states for any bipartite system through a universal construction of the witness operator. The method also gives a new family of positive but non-completely positive maps of arbitrary high dimensions which provide a much better test than the witness operators themselves. Moreover, we find there are two types of positive maps that can detect 2xN and 4xN bound entangled states. Since entanglement witnesses are physical observables and may be measured locally our constru…
By using the partial transpose and realignment method,we study the time evolution of the bound entanglement under the bilinear-biquadratic Hamiltonian. For the initial Horodecki's bound entangled state, it keeps bound entangled for some time, while for the initial bound entangled states constructed from the unextendable product basis, they become free once the time evolution begins. The time evolution provides a new way to construct bound entangled states, and also gives a method to free bound entanglement.
We consider four norms on tensor product spaces that have appeared in quantum information theory and demonstrate duality relationships between them. We show that the product numerical radius is dual to the robustness of entanglement, and we similarly show that the S(k)-norm is dual to the projective tensor norm. We show that, analogous to how the product numerical radius and the S(k)-norm characterize k-block positivity of operators, there is a natural version of the projective tensor norm that characterizes Schmidt number. In this way we obtain an elementary new proof of the cross norm criter…
We study how the realignment criterion (also called computable cross-norm criterion) succeeds asymptotically in detecting whether random states are separable or entangled. We consider random states on $\C^d \otimes \C^d$ obtained by partial tracing a Haar-distributed random pure state on $\C^d \otimes \C^d \otimes \C^s$ over an ancilla space $\C^s$. We show that, for large $d$, the realignment criterion typically detects entanglement if and only if $s \leq (8/3π)^2 d^2$. In this sense, the realignment criterion is asymptotically weaker than the partial transposition criterion.
Entanglement concurrence has been widely used for featuring entanglement in quantum experiments. As an entanglement monotone it is related to specific quantum Tsallis entropy. Our goal in this paper is to propose a new parameterized bipartite entanglement monotone which is named as $q$-concurrence inspired by general Tsallis entropy. We derive an analytical lower bound for the $q$-concurrence of any bipartite quantum entanglement state by employing positive partial transposition criterion and realignment criterion, which shows an interesting relationship to the strong separability criteria. Th…
The correlation matrix (CM) criterion is a recently derived powerful sufficient condition for the presence of entanglement in bipartite quantum states of arbitrary dimensions. It has been shown that it can be stronger than the positive partial transpose (PPT) criterion, as well as the computable cross norm or realignment (CCNR) criterion in different situations. However, it remained as an open question whether there existed sets of states for which the CM criterion could be stronger than both criteria simultaneously. Here, we give an affirmative answer to this question by providing examples of…
It is shown that, every entangled state in an infinite-dimensional composite system has a simple entanglement witness of the form $αI+T$ with $α$ a nonnegative number and $T$ a finite rank self-adjoint operator. We also provide two methods of constructing entanglement witness and apply them to obtain some entangled states that cannot be detected by the PPT criterion and the realignment criterion.
In this paper, a class of indecomposable positive finite rank elementary operators of order $(n,n)$ are constructed. This allows us to give a simple necessary and sufficient criterion for separability of pure states in bipartite systems of any dimension in terms of positive elementary operators of order $(2,2)$ and get some new mixed entangled states that can not be detected by the positive partial transpose (PPT) criterion and the realignment criterion.
Inseparability criteria for continuous and discrete bipartite quantum states based on moments of annihilation and creation operators are studied by developing the idea of Shchukin-Vogel criterion [Phys. Rev. Lett. {\bf 95}, 230502 (2005)]. If a state is separable, then the corresponding matrix of moments is separable too. Thus, we derive generalized criteria, based on the separability properties of the matrix of moments, are thus derived. In particular, a new criterion based on realignment of moments in the matrix is proposed as an analogue of the standard realignment criterion for density mat…
In this paper we develop an approach for detecting entanglement, which is based on measuring quantum correlations and constructing a correlation matrix. The correlation matrix is then used for defining a family of parameters, named Correlation Minor Norms, which allow one to detect entanglement. This approach generalizes the computable cross-norm or realignment (CCNR) criterion, and moreover requires measuring a state-independent set of operators. Furthermore, we illustrate a scheme which yields for each Correlation Minor Norm a separable state that maximizes it. The proposed entanglement dete…
Let $n\geq 2$ and $Φ_{n,t,π}: M_n({\mathbb C}) \rightarrow M_n({\mathbb C})$ be a linear map defined by $Φ_{n,t,π}(A)=(n-t)\sum_{i=1}^nE_{ii}AE_{ii}+t\sum_{i=1}^nE_{i,π(i)}AE_{i,π(i)}^†-A$, where $0\leq t\leq n$, $E_{ij}$s are the matrix units and $π$ is a non-identity permutation of $(1,2,\cdots,n)$. Denote by $\{{ F}_s: s=1,2\ldots, k\}$ the set of all minimal cycles of $π$ and $l(π)=\max\{\# { F}_s: s=1,2,\ldots,k\}$ the length of $π$. It is shown that the Hermitian matrix $W_{n,t,π}$ induced by $Φ_{n,t,π}$ is an indecomposable entanglement witness if and only if $π^2\not={\rm id}$ (the ide…
Optimal entanglement criteria from trace invariants
S. N. Bose National Centre for Basic Sciences（S.N. 玻色基础科学国家中心）；Indian Statistical Institute（印度统计研究所）；Université de Toulouse（图卢兹大学）；Laboratoire de Physique Théorique, Université de Toulouse, CNRS, UPS（理论物理实验室，图卢兹大学，法国国家科学研究中心）
We present a generalized partial transposition separability criterion for the density matrix of a multipartite quantum system. This criterion comprises as special cases the famous Peres-Horodecki criterion and the recent realignment criterion in [O. Rudolph, quant-ph/0202121] and [K. Chen, L.A. Wu, quant-ph/0205017]. It involves only straightforward matrix manipulations and is easy to apply. A quantitative measure of entanglement based on this criterion is also obtained.
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…
The bound entangled state carries noisy entanglement and it is very hard to distill but the usefulness of bound entangled states has been depicted in different applications. This article represents a comparative dynamical study of an open quantum system for one of the bound entangled states proposed by Bennett et al. The study is conducted under the influence of Heisenberg, bi-linear bi-quadratic and Dzyaloshinskii-Moriya (DM) interaction. During the study, an auxiliary qutrit interacts with one of the qutrits of the selected two qutrit bound entangled state through different interactions. The…
A family of separability criteria based on correlation matrix (tensor) is provided. Interestingly, it unifies several criteria known before like e.g. CCNR or realignment criterion, de Vicente criterion and derived recently separability criterion based on SIC POVMs. It should be stressed that, unlike the well known Correlation Matrix Criterion or criterion based on Local Uncertainty Relations, the new criteria are linear in the density operator and hence one may find new classes of entanglement witnesses and positive maps. Interestingly, there is a natural generalization to multipartite scenari…
We introduce and study a class of entanglement criteria based on the idea of applying local contractions to an input multipartite state, and then computing the projective tensor norm of the output. More precisely, we apply to a mixed quantum state a tensor product of contractions from the Schatten class $S_1$ to the Euclidean space $\ell_2$, which we call entanglement testers. We analyze the performance of this type of criteria on bipartite and multipartite systems, for general pure and mixed quantum states, as well as on some important classes of symmetric quantum states. We also show that pr…
Iterative optimization in quantum metrology and entanglement theory using semidefinite programming
Durham University（杜伦大学）；University of the Basque Country UPV/EHU（巴斯克自治大学）；HUN-REN Wigner Research Centre for Physics（匈牙利研究与教育网络维格纳物理研究中心）；Donostia International Physics Center DIPC（圣塞巴斯蒂安国际物理中心）；Department of Theoretical Physics, University of Szeged（塞格德大学理论物理系）；HUN-REN Institute for Nuclear Research（匈牙利研究与教育网络核物理研究所）；IKERBASQUE, Basque Foundation for Science（伊克尔巴斯科，巴斯克科学基金会）
