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 are weaker than the parent criterion, can be strengthened in such a way to obtain a new family of criteria that are independent of the original one. This conjecture is supported by numerical examples for the low dimensional cases. These ideas can be applied to the study of quantum channels, leading to a relation between the rate of contraction of a map and its ability to preserve entanglement.
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.
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.
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…
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…
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.
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.
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…
For a quantum state in a bipartite system represented as a density matrix, researchers used the realignment matrix and functions on its singular values to study the separability of the quantum state. We obtain bounds for elementary symmetric functions of singular values of realignment matrices. This answers some open problems proposed by Lupo, Aniello, and Scardicchio. As a consequence, we show that the proposed scheme by these authors for testing separability would not work if the two subsystems of the bipartite system have the same dimension.
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…
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.
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…
We present a necessary and sufficient product criterion for bipartite quantum states based on the rank of realignment matrix of density matrix. Then, this approach is generalized to multipartite systems. We first introduce the concept of semiproduct in a similar manner to the semiseparable and prove that semiproduct is equivalent to fully product. Therefore, a quantum state is bipartite product with respect to all possible partitions implies fully product which is different from the case of separability. For pure states, it can easily be seen that several necessary and sufficient separability …
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 present a framework for deciding whether a quantum state is separable or entangled using covariance matrices of locally measurable observables. This leads to the covariance matrix criterion as a general separability criterion. We demonstrate that this criterion allows to detect many states where the familiar criterion of the positivity of the partial transpose fails. It turns out that a large number of criteria which have been proposed to complement the positive partial transpose criterion - the computable cross norm or realignment criterion, the criterion based on local uncertainty relatio…
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 entangle…
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 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 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 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.
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.
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…
We quantify the usefulness of a bipartite quantum state in the ancilla-assisted channel discrimination of arbitrary quantum channels, formally defining a worst-case-scenario channel discrimination power for bipartite quantum states. We show that such a quantifier is deeply connected with the operator Schmidt decomposition of the state. We compute the channel discrimination power exactly for pure states, and provide upper and lower bounds for general mixed states. We show that highly entangled states can outperform any state that passes the realignment criterion for separability. Furthermore, w…
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.
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 study the stochastic local operation and classical communication (SLOCC) equivalence for arbitrary dimensional multipartite quantum states. For multipartite pure states, we present a necessary and sufficient criterion in terms of their coefficient matrices. This condition can be used to classify some SLOCC equivalent quantum states with coefficient matrices having the same rank. For multipartite mixed state, we provide a necessary and sufficient condition by means of the realignment of matrix. Some detailed examples are given to identify the SLOCC equivalence of multipartite quantum states.
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.
We derive an analytical lower bound for the concurrence of a bipartite quantum state in arbitrary dimension. A functional relation is established relating concurrence, the Peres-Horodecki criterion and the realignment criterion. We demonstrate that our bound is exact for some mixed quantum states. The significance of our method is illustrated by giving a quantitative evaluation of entanglement for many bound entangled states, some of which fail to be identified by the usual concurrence estimation method.
In this paper, the realignment criterion and the RCCN criterion of separability for states in infinite-dimensional bipartite quantum systems are established. Let $H_A$ and $H_B$ be complex Hilbert spaces with $\dim H_A\otimes H_B=+\infty$. Let $ρ$ be a state on $H_A\otimes H_B$ and $\{δ_k\}$ be the Schmidt coefficients of $ρ$ as a vector in the Hilbert space ${\mathcal C}_2(H_A)\otimes{\mathcal C}_2(H_B)$. We introduce the realignment operation $ρ^R$ and the computable cross norm $\|ρ\|_{\rm CCN}$ of $ρ$ and show that, if $ρ$ is separable, then $\|ρ^{R}\|_{\rm Tr}=\|ρ\|_{\rm CCN}=\sum\limits_k…
A group of symmetric operators are introduced to carry out the separability criterion for bipartite and multipartite quantum states. Every symmetric operator, represented by a symmetric matrix with only two nonzero elements, and their arbitrary linear combinations are found to be entanglement witnesses. By using these symmetric operators, Wootters' separability criterion for two-qubit states can be generalized to bipartite and multipartite systems in arbitrary dimensions.
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…
We introduce algebraic sets in the complex projective spaces for the mixed states in bipartite quantum systems as their invariants under local unitary operations. The algebraic sets of the mixed state have to be the union of the linear subspaces if the mixed state is separable. Some examples are given and studied based on our criterion
We provide a necessary and sufficient condition for separability of Gaussian states of bipartite systems of arbitrarily many modes. The condition provides an operational criterion since it can be checked by simple computation. Moreover, it allows us to find a pure product-state decomposition of any given separable Gaussian state. Our criterion is independent of the one based on partial transposition, and is strictly stronger.
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.
A decomposition form is introduced in this report to establish a criterion for the bi-partite separability of Bell diagonal states. A such criterion takes a quadratic form of the coefficients of a given Bell diagonal states and can be derived via a simple algorithmic calculation of its invariants. In addition, the criterion can be extended to a quantum system of higher dimension.
A new necessary separability criterion that relates the structures of the total density matrix and its reductions is given. The method used is based on the realignment method [K. Chen and L.A. Wu, Quant. Inf. Comput. 3, 193 (2003)]. The new separability criterion naturally generalizes the reduction separability criterion introduced independently in previous work of [M. Horodecki and P. Horodecki, Phys. Rev. A 59, 4206 (1999)] and [N.J. Cerf, C. Adami and R.M. Gingrich, Phys. Rev. A 60, 898 (1999)]. In special cases, it recovers the previous reduction criterion and the recent generalized partia…
We prove that all inseparable Gaussian states of two modes can be distilled into maximally entangled pure states by local operations. Using this result we show that a bipartite Gaussian state of arbitrarily many modes can be distilled if and only if its partial transpose is not positive.
For a given density matrix $ρ$ of a bipartite quantum system an asymptotical separability criterion is suggested. Using the continuous ensemble method, a sequence of separable density matrices is built which converges to $ρ$ if and only if $ρ$ is separable. The convergence speed is evaluated and for any given tolerance parameter $κ$ an iterative procedure is suggested which decides in finite number of steps if there exists a separable density matrix $ρ_κ$ which differs from the matrix $ρ$ by at most $κ$.
Bipartite quantum states are classified into three categories: separable states, bound entangled states, and free entangled states. It is of great importance to characterize these families of states for the development of quantum information science. In this paper, I show that the separable states and the bound entangled states have a common spectral property. More precisely, I prove that for undistillable -- separable and bound entangled -- states, the eigenvalue vector of the global system is majorized by that of the local system. This result constitutes a new sufficient condition for distil…
In this paper we present a necessary and sufficient condition of distinguishability of bipartite quantum states. It is shown that the operators to reliably distinguish states need only rounds of projective measurements and classical comunication. We also present a necessary condition of distinguishability of bipartite quantum states which is simple and general. With this condition one can get many cases of indistinguishability. The conclusions may be useful in understanding the essence of nonlocality and calculating the distillable entanglement and the bound of distillable entanglement.
We construct a density matrix whose elements are written in terms of expectation values of non-Hermitian operators and their products for arbitrary dimensional bipartite states. We then show that any expression which involves matrix elements can be reformulated by the expectation values of these non-Hermitian operators and vice versa. We consider the condition of pure states and pure product states and rewrite them in terms of expectation values and density matrix elements respectively. We utilize expectation values of these operators to present the condition for separability of ${C}^d \otimes…
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 construct a family of bipartite states of arbitrary dimension whose eigenvalues of the partially transposed matrix can be inferred directly from the block structure of the global density matrix. We identify from this several subfamilies in which the PPT criterion is both necessary and sufficient. A sufficient criterion of separability is obtained, which is fundamental for the discussion. We show how several examples of states known to be classifiable by the PPT criterion indeed belong to this general set. Possible uses of these states in numerical analysis of entanglement and in the search …
Photon subtraction and addition are experimental means of generating non-Gaussian states from Gaussian states. Coherent subtraction or addition is a combination of photon subtractions or additions. The resultant states are quite general non-Gaussian states. The states can be photon number entangled states with arbitrary coefficients. We derive the entanglement conditions for several classes of coherent subtraction or coherent addition bipartite continuous variable states. One of the entanglement conditions is necessary and sufficient.
Quantum steering describes the ability of one observer to nonlocally affect the other observer's state through local measurements, which represents a new form of quantum nonlocal correlation and has potential applications in quantum information and quantum communication. In this paper, we propose a computable steering criterion that is applicable to bipartite quantum systems of arbitrary dimensions. The criterion can be used to verify a wide range of steerable states directly from a given density matrix without constructing measurement settings. Compared with the existing steering criteria, it…
A new criterion necessary and sufficient for the separability of pure bipartite systems for arbitrary finite dimensions is demonstrated; and the corresponding finer quantitative measures or characterizations of entanglement (beyond mere separability or non-separability determination) are discussed. Based on this criterion, we proved that the well-known Peres-Horodecki positivity-of-partial-transform criterion is also necessary and sufficient for separability in the case of pure bipartite systems. The maximum value of entanglement, and the corresponding maximally-entangled states are also worke…
We describe a criterion for the detection of entanglement between two multi-boson systems. The criterion is based on calculating correlations of Gell-Mann matrices with a fixed boson number on each subsystem. This applies naturally to systems such as two entangled spinor Bose-Einstein condensates. We apply our criterion to several experimentally motivated examples, such as an $ S^z S^z $ entangled BECs, ac Stark shift induced two-mode squeezed BECs, and photons under parametric down conversion. We find that entanglement can be detected for all parameter regions for the most general criterion. …
We propose a entanglement measure for pure $M \otimes N$ bipartite quantum states. We obtain the measure by generalizing the equivalent measure for a $2 \otimes 2$ system, via a $2 \otimes 3$ system, to the general bipartite case. The measure emphasizes the role Bell states have, both for forming the measure, and for experimentally measuring the entanglement. The form of the measure is similar to generalized concurrence. In the case of $2 \otimes 3$ systems, we prove that our measure, that is directly measurable, equals the concurrence. It is also shown that in order to measure the entanglemen…
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…
