In this paper, we mainly discuss the separability of $n$-partite quantum states from elements of density matrices. Practical separability criteria for different classes of $n$-qubit and $n$-qudit quantum states are obtained. Some of them are also sufficient conditions for genuine entanglement of $n$-partite quantum states. Moreover, one of the resulting criteria is also necessary and sufficient for a class of $n$-partite states.
For an N-partite quantum system we show that separability implies inequalities on Bell correlations which are stronger than the local reality inequalities by a factor 2^{(N-1)/2}.
We develop separability criteria to identify non-$k$-separability $(k = 2,3,\ldots,n)$ and genuine multipartite entanglement in different classes of mixed $n$-partite quantum states using elements of density matrices. With the help of these criteria, we detect non-$k$-separability in $n$-qudit GHZ and W states respectively added with white noise. We also discuss the experimental implementation of our criteria by means of local observables.
We give a new separability criterion, a necessary condition for separability of $N$-partite quantum states. The criterion is based on the Bloch representation of a $N$-partite quantum state and makes use of multilinear algebra, in particular, the matrization of tensors. Our criterion applies to {\it arbitrary} $N$-partite quantum states in $\mathcal{H}=\mathcal{H}^{d_1}\otimes \mathcal{H}^{d_2} \otimes ... \otimes \mathcal{H}^{d_N}.$ The criterion can test whether a $N$-partite state is entangled and can be applied to different partitions of the $N$-partite system. We provide examples that sho…
We study the norms of the Bloch vectors for arbitrary $n$-partite quantum states. A tight upper bound of the norms is derived for $n$-partite systems with different individual dimensions. These upper bounds are used to deal with the separability problems. Necessary conditions are presented for $\mathbf m$-separable states in $n$-partite quantum systems. Based on the upper bounds, classification of multipartite entanglement is illustrated with detailed examples.
The partial separability of multipartite qubit density matrixes is strictly defined. We give a reduction way from N-partite qubit density matrixes to bipartite qubit density matrixes, and prove a necessary condition that a N-partite qubit density matrix to be partially separable is its reduced density matrix to satisfy PPT condition.
We investigate the detection of entanglement in $n$-partite quantum states. We obtain practical separability criteria to identify genuinely entangled and non-separable mixed quantum states. No numerical optimization or eigenvalue evaluation is needed, and our criteria can be evaluated by simple computations involving components of the density matrix. We provide examples in which our criteria perform better than all known separability criteria. Specifically, we are able to detect genuine $n$-partite entanglement which has previously not been identified. In addition, our criteria can be used in …
In this paper, we discuss the partial separability and its criteria problems of multipartite qubit mixed-states. First we strictly define what is the partial separability of a multipartite qubit system. Next we give a reduction way from N-partite qubit density matrixes to bipartite qubit density matrixes, and prove a necessary condition that a N-partite qubit mixed-state to be partially separable is its reduction to satisfy the PPT condition. PACC numbers: 03.67.Mn; 03.65.Ud; 03.67.Hk
We derive hierarchies of separability criteria that identify the different degrees of entanglement ranging from bipartite to genuine multi-partite in mixed quantum states of arbitrary size.
We give a direct tensor decomposition for any density matrix into Hermitian operators. Based upon the decomposition we study when the mixed states are separable and generalize the separability indicators to multi-partite states and show that a density operator is separable if and only if the separable indicator is non-negative. We then derive two bounds for the separable indicator in terms of the spectrum of the factor operators in the tensor summands.
The separability detecting problem of mixed states is one of the fundamental problems in quantum information theory. In the last 20 years, almost all methods are based on the sufficient or necessary conditions for entanglement. However, in this paper, we only need one algorithm to solve the problem. We propose a tensor optimization method to check whether an $m$-partite quantum mixed state is separable or not and give a decomposition for it if it is. We first convert the separability discrimination problem of mixed states to the positive Hermitian decomposition problem of Hermitian tensors. Th…
We derive a combinatorial criterion for detecting k-separability of N-partite Dicke states. The criterion is efficiently computable and implementable without full state tomography. We give examples in which the criterion succeeds, where known criteria fail.
The detection of multipartite entanglement in arbitrary dimensional systems is investigated. We derive useful $k$-separability criteria of mixed $n$-partite ($n\geq 3$) quantum states to detect $k$-nonseparable $n$-partite quantum states. Our criteria can be expressed by the elements of the density matrix, which allows a simple and practical evaluation and computation. They are experimentally accessible without quantum state tomography.
Following previous work, we distinguish between genuine $N$-partite entanglement and full $N$-partite inseparability. Accordingly, we derive criteria to detect genuine multipartite entanglement using continuous variable (position and momentum) measurements. Our criteria are similar but different to those based on the van Loock-Furusawa inequalities, which detect full $N$-partite inseparability. We explain how the criteria can be used to detect the genuine $N$-partite entanglement of continuous variable states generated from squeezed and vacuum state inputs, including the continuous variable Gr…
We derive tight quadratic inequalities for all kinds of hybrid separable-inseparable $n$-particle density operators on an arbitrary dimensional space. This methodology enables us to truly derive a tight quadratic inequality as tests for full $n$-partite entanglement in various Bell-type correlation experiments on the systems that may not be identified as a collection of qubits, e.g., those involving photons measured by incomplete detectors. It is also proved that when the two measured observables are assumed to precisely anti-commute, a stronger quadratic inequality can be used as a witness of…
Identifying the $k$-partite entanglement and $k$-nonseparability of general $N$-partite quantum states are fundamental issues in quantum information theory. By use of computable inequalities of nonlinear operators, we present some simple and powerful $k$-partite entanglement and $k$-nonseparability criteria that works very well and allow for a simple and inexpensive test for the whole hierarchy of $k$-partite entanglement and $k$-separability of $N$-partite systems with $k$ running from $N$ down to 2. We illustrate their strengths by considering several examples in which our criteria perform b…
We explain several separability criteria which rely on uncertainty relations. For the derivation of these criteria uncertainty relations in terms of variances or entropies can be used. We investigate the strength of the separability conditions for the case of two qubits and show how they can improve entanglement witnesses.
The entanglement detection via local measurements can be experimentally implemented. Based on mutually unbiased measurements and general symmetric informationally complete positive-operator-valued measures, we present separability criteria for bipartite quantum states, which, by theoretical analysis, are stronger than the related existing criteria via these measurements. Two detailed examples are supplemented to show the efficiency of the presented separability criteria.
We present separability criteria for both bipartite and multipartite quantum states. These criteria include the criteria based on the correlation matrix and its generalized form as special cases. We show by detailed examples that our criteria are more powerful than the positive partial transposition criterion, the realignment criterion and the criteria based on the correlation matrices.
The recent proposed realignment separability criterion for mixed is analyzed. We identify the essential part of this criterion is a swap operator followed by a partial transposition. Then we analyze the separability criterion of permutation indices of the density matrix for multipartite state which is a generalization of the realignment criterion. We give a method to solve the equivalence problem of the separability criterion of permutation indices of the density matrix. For tripartite state, we show the non-trivial separability criteria are either partial transposition criterion or realignmen…
The correlation matrices or tensors in the Bloch representation of density matrices are encoded with entanglement properties. In this paper, based on the Bloch representation of density matrices, we give some new separability criteria for bipartite and multipartite quantum states. Theoretical analysis and some examples show that the proposed criteria can be more efficient than the previous related criteria.
We study separability criteria in multipartite quantum systems of arbitrary dimensions by using the Bloch representation of density matrices. We first derive the norms of the correlation tensors and obtain the necessary conditions for separability under partition of tripartite and four-partite quantum states. Moreover, based on the norms of the correlation tensors,we obtain the separability criteria by matrix method. Using detailed examples, our results are seen to be able to detect more entangled states than previous studies. Finally, necessary conditions of separability for multipartite syst…
In this paper, we show that an arbitrary separable state can be the output of a certain entanglement-breaking channel corresponding exactly to the input of a maximally entangled state. A necessary and sufficient separability criterion and some sufficient separability criteria from entanglement-breaking channels are given.
In this Letter we find the new criteria of separability of multipartite qubit density matrixes. Especially, we discuss in detail the criteria of separability for tripartite qubit density matrixes. We find the sufficient and necessary conditions of separability for tripartite qubit density matrixes, and give two corollaries. The second corollary can be taken as the criterion of existence of entanglement for tripartite qubit density matrixes. In concrete application, its steps are quite simple and easy to operate. Some examples, discussions and the generalization to more high dimensional multipa…
We present a method to derive separability criteria for the different classes of multiparticle entanglement, especially genuine multiparticle entanglement. The resulting criteria are necessary and sufficient for certain families of states. Further, the criteria are superior to all known entanglement criteria for many other families; also they allow the detection of bound entanglement. We next demonstrate that they are easily implementable in experiments and discuss applications to the decoherence of multiparticle entangled states.
Detection power of separability criteria based on a correlation tensor is tested within a family of generalized isotropic state in $d_1 \otimes d_2$. For $d_1 \neq d_2$ all these criteria are weaker than positive partial transposition (PPT) criterion. Interestingly, our analysis supports the recent conjecture that a criterion based on symmetrically informationaly complete positive operator-valued measure (SIC-POVMs) is stronger than realignment criterion.
Mutually unbiased bases (MUBs) and symmetric informationally complete (SIC) positive operator-valued measurements (POVMs) are two related topics in quantum information theory. They are generalized to mutually unbiased measurements (MUMs) and general symmetric informationally complete (GSIC) measurements, respectively, that are both not necessarily rank 1. We study the quantum separability problem by using these measurements and present separability criteria for bipartite systems with arbitrary dimensions and multipartite systems of multi-level subsystems. These criteria are proved to be more e…
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…
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.
By combining a parameterized Hermitian matrix, the realignment matrix of the bipartite density matrix $ρ$ and the vectorization of its reduced density matrices, we present a family of separability criteria, which are stronger than the computable cross norm or realignment (CCNR) criterion. With linear contraction methods, the proposed criteria can be used to detect the multipartite entangled states that are biseparable under any bipartite partitions. Moreover, we show by examples that the presented multipartite separability criteria can be more efficient than the corresponding multipartite real…
A general separability condition on the second moment (covariance matrix) for continuous variable two-party systems is derived by an analysis analogous to the derivation of the Kennard's uncertainty relation without referring to the non-negativity of the partially transposed density matrix. This separability criterion is generally more stringent than that used by Simon which is based on the non-negativity of partially transposed density matrix, and thus this criterion may be useful in the analysis of general continuous two-party systems. Another separability criterion used by Duan et al. is sh…
In tri-partite systems, there are three basic biseparability, $A$-$BC$, $B$-$CA$ and $C$-$AB$ biseparability according to bipartitions of local systems. We begin with three convex sets consisting of these basic biseparable states in the three qubit system, and consider arbitrary iterations of intersections and/or convex hulls of them to get convex cones. One natural way to classify tri-partite states is to consider those convex sets to which they belong or do not belong. This is especially useful to classify partial entanglement of mixed states. We show that the lattice generated by those thre…
Separability is an important problem in theory of quantum entanglement. By using the Bloch representation of quantum states in terms of the Heisenberg-Weyl observable basis, we present a new separability criterion for bipartite quantum systems. It is shown that this criterion can be better than the previous ones in detecting entanglement. The results are generalized to multipartite quantum states.
Recently P. Wocjan and M. Horodecki [quant-ph/0503129] gave a characterization of combinatorially independent permutation separability criteria. Combinatorial independence is a necessary condition for permutations to yield truly independent criteria meaning that that no criterion is strictly stronger that any other. In this paper we observe that some of these criteria are still dependent and analyze why these dependencies occur. To remove them we introduce an improved necessary condition and give a complete classification of the remaining permutations. We conjecture that the remaining class of…
We introduce an entanglement criterion to exclude full separability of quantum states. We present numerical evidence that the criterion is necessary and sufficient for the class of GHZ diagonal three-qubit states and estimate the volume of bound entangled states within this class. Finally, we extend our approach to bound entangled states which are not GHZ diagonal.
The purpose of this paper is to obtain a sufficient and necessary condition as a criteria to test whether an arbitrary multipartite state is entangled or not. Based on the tensor expression of a multipartite pure state, the paper shows that a state is separable iff $| \mathbf{C}(ρ)| $ =0 for pure states and iff $C(ρ)$ vanishes for mixed states.
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.
Mutually unbiased measurements (MUMs) are generalized from the concept of mutually unbiased bases (MUBs) and include the complete set of MUBs as a special case, but they are superior to MUBs as they do not need to be rank one projectors. We investigate entanglement detection using sets of MUMs and derived separability criteria for $d$-dimensional multipartite systems, and arbitrary high-dimensional bipartitie and multipartite systems. These criteria provide experimental implementation in detecting entanglement of unknown quantum states.
Using a recently introduced framework, we derive criteria for quantum k-separability, which are very easily computed. In the case k = 2, our criteria are equally strong to the best methods known so far, while in all other cases there are currently no comparable criteria known. We also show how the criteria can be implemented experimentally.
After introducing the partially separable concept, we proved the equivalence between the partial separability of a given $m$-partite subsystem with $m$ qubits and the purity of states of this $m$-partite subsystem for a pure state in multipartite systems with arbitrary finite $n(>m)$ qubits. Furthermore, we give out the operational realizations (corollaries) of our theorem, which are the sufficient and necessary criterions of partial separability of states and can be used to classification of states. Our results are helpful to understand and describe quantum entanglement in multipartite system…
The so-called permutation separability criteria are simple operational conditions that are necessary for separability of mixed states of multipartite systems: (1) permute the indices of the density matrix and (2) check if the trace norm of at least one of the resulting operators is greater than one. If it is greater than one then the state is necessarily entangled. A shortcoming of the permutation separability criteria is that many permutations give rise to dependent separability criteria. Therefore, we introduce a necessary condition for two permutations to yield independent criteria called c…
We revisit the application of different separability criteria by recourse to an exhaustive Monte Carlo exploration involving the pertinent state-space of pure and mixed states. The corresponding chain of implications of different criteria is in such a way numerically elucidated. We also quantify, for a bipartite system of arbitrary dimension, the proportion of states $ρ$ that can be distilled according to a definite criterion. Our work can be regarded as a complement to the recent review paper by B. Terhal [Theor. Comp. Sci. {\bf 287} (2002) 313]. Some questions posed there receive an answer h…
In this paper, we present the separability criteria to identify non-$k$-separability and genuine multipartite entanglement in mixed multipartite states using elements of density matrices. Our criteria can detect the non-$k$-separability of Dicke class of states, anti W states and mixtures thereof and higher dimensional W class of states. We then investigate the performance of our criteria by considering $N$-qubit Dicke states with arbitrary excitations added with white noise and mixture of $N$-qudit W state with white noise. We also study the robustness of our criteria against white noise. Fur…
We study the separability of symmetric bipartite quantum states and show that a single correlation measurement is sufficient to detect the entanglement of any bipartite symmetric state with a non-positive partial transpose. We also discuss entanglement conditions and entanglement witnesses for states with a positive partial transpose.
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.
It is pointed out that separability problem for arbitrary multi-partite states can be fully solved by a finite size, elementary recursive algorithm. In the worse case scenario, the underlying numerical procedure, may grow doubly exponentially with the state's rank. Nevertheless, we argue that for generic states, analysis of concurrence matrices essentially reduces the task of solving separability problem in $m \times n$ dimensions to solving a set of linear equations in about $\binom{mn+D-1}{D}$ variables, where $D$ decreases with $mn$ and for large $mn$ it should not exceed 4. Moreover, the s…
Entanglement as a vital resource for information processing can be described by special properties of the quantum state. Using the well-known Weyl basis we propose a new Bloch decomposition of the quantum state and study its separability problem. This decomposition enables us to find an alternative characterization of the separability based on the correlation matrix. We shaw that the criterion is effective in detecting entanglement for the isotropic states, Bell-diagonal states and some PPT entangled states. We also use the Weyl operators to construct an detecting operator for quantum teleport…
The study of the entanglement properties of systems of N fermions has attracted considerable interest during the last few years. Various separability criteria for pure states of N identical fermions have been recently discussed but, excepting the case of two-fermions systems, these criteria are difficult to implement and of limited value from the practical point of view. Here we advance simple necessary and sufficient separability criteria for pure states of N identical fermions. We found that to be identified as separable a state has to comply with one single identity involving either the pur…
Usual separability criteria applicable to distinguishable particles are not applicable to identical particles. Here we show that Partial transposition and symmetrization (or anti symmetrization) of density matrix of bipartite boson systems (fermion systems) can be used to check whether a state is separable or not.By using determinant based separability criteria it has been found that bipartite states of identical particles are always separable. It is found that partial transposition and symmetrization is equivalent to realignment criteria proposed by Wu.
We present a general scheme that allows for construction of scalar separability criteria from positive but not completely positive maps. The concept is based on a decomposition of every positive map $Λ$ into a difference of two completely positive maps $Λ_1$, $Λ_2$, i.e., $Λ=Λ_1-Λ_2$. The scheme may be also treated as a generalization of the known entropic inequalities, which are obtained from the reduction map. An analysis performed on few classes of states shows that the new scalar criteria are stronger than the entropic inequalities and furthermore, when derived from indecomposable maps all…
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…
We investigate classification and detection of entanglement of multipartite quantum states in a very general setting, and obtain efficient $k$-separability criteria for mixed multipartite states in arbitrary dimensional quantum systems. These criteria can be used to distinguish $n-1$ different classes of multipartite inseparable states and can detect many important multipartite entangled states such as GHZ states, W states, anti W states, and mixtures thereof. They detect $k$-nonseparable $n$-partite quantum states which have previously not been identified. Here $k=2,3,\cdots,n$. No optimizati…
We explore the subtle relationships between partial separability and entanglement of subsystems in multiqubit quantum states and give experimentally accessible conditions that distinguish between various classes and levels of partial separability in a hierarchical order. These conditions take the form of bounds on the correlations of locally orthogonal observables. Violations of such inequalities give strong sufficient criteria for various forms of partial inseparability and multiqubit entanglement. The strength of these criteria is illustrated by showing that they are stronger than several ot…
