We propose a range criterion which is a sufficient and necessary condition satisfied by two pure states transformable with each other under reversible stochastic local operations assisted with classical communication. We also provide a systematic method for seeking all kinds of true entangled states in the $2\times{M}\times{N}$ system, and can effectively distinguish them by means of the range criterion. The efficiency of the criterion and the method is exhibited by the classification of true entanglement in some types of the tripartite systems.
We present an entanglement criterion for multiqubits by using the quantum correlation tensors which rely on the expectation values of the Pauli operators for a multiqubit state. Our criterion explains not only the total entanglement of the system but also the partial entanglement in subsystems. It shows that we have to consider the subsystem entanglements in order to obtain the full description for multiqubit entanglements. Furthermore, we offer an extension of the entanglement to multiqudits.
We study the quantum separability problem by using general symmetric informationally complete measurements and present a separability criterion for arbitrary dimensional bipartite systems. We show by detailed examples that our criterion is more powerful than the existing ones in entanglement detection.
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.
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.
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.
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.
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.
An entanglement monotone, which is invariant under the determinant 1 SLOCC operations and measures the true quadripartite entanglement, is explicitly constructed.
We derive criterion in the form of inequality based on quantum Fisher information and quantum variance to detect multipartite entanglement. It can be regarded as complementary of the well-established PPT criterion in the sense that it can also detect bound entangled states. The inequality is motivated by Y.Akbari-Kourbolagh $et\ al.$[Phys. Rev A. 99, 012304 (2019)] which introduced a multipartite entanglement criterion based on quantum Fisher information. Our criterion is experimentally measurable for detecting any $N$-qudit pure state mixed with white noisy. We take several examples to illust…
We propose a directly measurable criterion for the entanglement of two qubits. We compare the criterion with other criteria, and we find that for pure states, and some mixed states, it coincides with the state's concurrency. The measure can be obtained with a Bell state analyser and the ability to make general local unitary transformations. However, the procedure fails to measure the entanglement of a general mixed two-qubit state.
The local uncertainty relation (LUR) criteria for quantum entanglement, which is dependent on chosen observables, is developed recent. In the paper, applying the uncertainty principle, an entanglement criteria for multipartite Gaussian states is given, which is implemented by a minimum optimization computer program and independent on observalbes.
We propose entanglement criteria for multipartite systems via symmetric informationally complete (SIC) measurement and general symmetric informationally complete (GSIC) measurement. We apply these criteria to detect entanglement of multipartite states, such as the convex of Bell states, entangled states mixed with white noise. It is shown that these criteria are stronger than some existing ones.
Great advances have been achieved in studying characteristics of entanglement for fundamentals of quantum mechanics and quantum information processing. However, even for N-qubit systems, the problem of entanglement criterion has not been well solved. In this Letter, using the method of state decomposition and high order singular value decomposition (HOSVD), we propose a necessary and sufficient entanglement criterion for general N-qubit systems. As an example, we apply our method to study the multi-qubit W state with white noise. We not only obtain the separability critical point, which is tig…
A general and computable criterion for k-(in)separability in continuous multipartite quantum systems is presented. The criterion can be experimentally implemented with a finite and comparatively low number of local observables. We discuss in detail how the detection quality can be optimised.
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.
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 present local invariants of multi-partite pure or mixed states, which can be easily calculated and have a straight-forward physical meaning. As an application, we derive a new entanglement criterion for arbitrary mixed states of $n$ parties. The new criterion is weaker than the partial transposition criterion but offers advantages for the study of multipartite systems. A straightforward generalization of these invariants allows for the construction of a complete set of observable polynomial invariants.
We prove a necessary and sufficient condition for the occurrence of entanglement in two two-level systems, simple enough to be of experimental interest. Our results are illustrated in the context of a spin star system analyzing the exact entanglement evolution of the central couple of spins.
We develop a strong and computationally simple entanglement criterion. The criterion is based on an elementary positive map Phi which operates on state spaces with even dimension N >= 4. It is shown that Phi detects many entangled states with positive partial transposition (PPT) and that it leads to a class of optimal entanglement witnesses. This implies that there are no other witnesses which can detect more entangled PPT states. The map Phi yields a systematic method for the explicit construction of high-dimensional manifolds of bound entangled states.
We first propose a new separability criterion based on algebraic-geometric invariants of bipartite mixed states introduced in [1], then prove that for all low ranks r <m+n-2, generic rank r mixed states in mxn systems have relatively high Schmidt numbers (thus entangled) by this separability criterion. This also means that the algebraic-geometric separability criterion proposed here can be used to dectect all low rank entangled mixed states outside a measure zero set.
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 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…
We apply the Walborn [Walborn S P, Taketani B G, Salles A, Toscano F and de Matos Filho R L 2009 Entropic Entanglement Criteria for Continuous Variables Phys. Rev. Lett. 103 160505] Shannon Entropic Entanglement Criterion (SEEC), to simple harmonic oscillators, a system typically discussed in undergraduate quantum mechanics courses. In particular, we investigate the entanglement of a system of coupled harmonic oscillators. A simple form of entanglement criterion in terms of their interaction is found. It is shown that a pair of interacting ground state coupled oscillators are more likely to be…
We use matched quantum entanglement witnesses to study the separable criteria of continuous variable states. The witness can be written as an identity operator minus a Gaussian operator. The optimization of the witness then is transformed to an eigenvalue problem of a Gaussian kernel integral equation. It follows a separable criterion not only for symmetric Gaussian quantum states, but also for non-Gaussian states prepared by photon adding to or/and subtracting from symmetric Gaussian states. Based on Fock space numeric calculation, we obtain an entanglement witness for more general two-mode s…
We study the entanglement detection by using mutually unbiased measurements and provide a quantum separability criterion that can be experimentally implemented for arbitrary $d$-dimensional bipartite systems. We show that this criterion is more effective than the criterion based on mutually unbiased bases. For isotropic states our criterion becomes both necessary and sufficient.
Experimental procedures are presented for the rapid detection of entanglement of unknown arbitrary quantum states. The methods are based on the entanglement criterion using accessible correlations and the principle of correlation complementarity. Our first scheme essentially establishes the Schmidt decomposition for pure states, with few measurements only and without the need for shared reference frames. The second scheme employs a decision tree to speed up entanglement detection. We analyze the performance of the methods using numerical simulations and verify them experimentally for various s…
We formulate an entanglement criterion using Peres-Horodecki positive partial transpose operations combined with the Schrödinger-Robertson uncertainty relation. We show that any pure entangled bipartite and tripartite state can be detected by experimentally measuring mean values and variances of specific observables. Those observables must satisfy a specific condition in order to be used, and we show their general form in the $2\times 2$ (two qubits) dimension case. The criterion is applied on a variety of physical systems including bipartite and multipartite mixed states and reveals itself to…
We revisit the criterion of multi-particle entanglement based on the overlaps of a given quantum state $ρ$ with maximally entangled states. For a system of $m$ particles, each with $N$ distinct states, we prove that $ρ$ is $m$-particle negative partial transpose (NPT) entangled, if there exists a maximally entangled state $|{\rm MES}>$, such that $<{\rm MES}|ρ|{\rm MES}>>{1}/{N}$. While this sufficiency condition is weaker than the Peres-Horodecki criterion in all cases, it applies to multi-particle systems, and becomes especially useful when the number of particles ($m$) is large. We also con…
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…
This paper gives a criterion for detecting the entanglement of a quantum state, and uses it to study the relationship between topological and quantum entanglement. It is fundamental to view topological entanglements such as braids as entanglement operators and to associate to them unitary operators that are capable of creating quantum entanglement. The entanglement criterion is used to explore this connection. The paper discusses non-locality in the light of this criterion.
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.
We find that a class of entanglement measures for bipartite pure state can be expressed by the average values of quantum operators, which are related to any complete basis of one partite operator space. Two specific examples are given based on two different ways to generalize Pauli matrices to $d$ dimensional Hilbert space and the case for identical particle system is also considered. In addition, applying our measure to mixed state case will give a sufficient condition for entanglement.
In this paper, an intuitive mathematical formulation is provided to generalize the residual entanglement for tripartite systems of qubits [Phys. Rev. A 61, 052306 (2000)] to the tripartite systems in higher dimension. The spirit lies in the tensor treatment of tripartite pure states [Phys. Rev. A 72, 022333 (2005)]. A distinct characteristic of the present generalization is that the formulation for higher dimensional systems is invariant under permutation of the subsystems, hence is employed as a criterion to test the existence of genuine tripartite entanglement. Furthermore, the formulation f…
In this paper, an intuitive mathematical formulation is provided to generalize the residual entanglement for tripartite systems of qubits (Phys. Rev. A \textbf{61}, 052306 (2000)) to the tripartite systems in higher dimension. The spirit lies in the tensor treatment of tripartite pure states (Phys. Rev. A \textbf{72}, 022333 (2005)). A distinct characteristic of the present generalization is that the formulation for higher dimensional systems is invariant under permutation of the subsystems, hence is employed as a criterion to test the existence of genuine tripartite entanglement. Furthermore,…
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.
In recent years weak values have been used to explore interesting quantum features in novel ways. In particular, the real part of the weak value of the momentum operator has been widely studied, mainly in connection with (nonlocal) Bohmian trajectories. Here we focus on the imaginary part and its role in relation with the entanglement of a bipartite system. We establish an entanglement criterion based on weak momentum correlations, that allows to discern whether the entanglement is encoded in the amplitude and/or in the phase of the wave function. Our results throw light on the physical role o…
Genuine multipartite entanglement (GME) offers more significant advantages in quantum information compared with entanglement. We propose a sufficient criterion for the detection of GME based on local sum uncertainty relations for chosen observables of subsystems. We apply the criterion to detect the GME properties of noisy $n$-partite W state when $n = 3, 4, 5$ and $6$, and find that the criterion can detect more noisy W states when $n$ ranges from 4 to 6. Moreover, the criterion is also used to detect the genuine entanglement of $3$-qutrit state. The result is stronger than that based on GME …
We show that quantum entanglement and the Heisenberg uncertainty principle are inextricably connected. Toward this end, a complete set of commuting observables (CSCO) criterion for the entanglement is developed. Assuming (A1,A2,...) and (B1,B2,...) being two CSCO's for a given system, and C being the matrix, Cij = i [Bi,Aj], for each given row i (i=1,2,...) if at least one matrix element Cij (j=1,2,...) is nonzero, then for the simultaneous eigenstates |ψ)=|A1',A2',...) of (A1,A2,...), the simultaneous measurements of (B1,B2,...) are, in general,entangled. The only exception is when all the si…
We generalize entanglement detection with covariance matrices for an arbitrary set of observables. A generalized uncertainty relation is constructed using the covariance and commutation matrices, then a criterion is established by performing a partial transposition on the operators. The method is highly efficient and versatile in the sense that the set of measurement operators can be freely chosen, do not need to be complete, and there is no constraint on the commutation relations. The method is particularly suited for systems with higher dimensionality since the computations do not scale with…
A necessary and sufficient condition for characterization and quantification of entanglement of any bipartite Gaussian state belonging to a special symmetry class is given in terms of classicality measures of one-party states. For Gaussian states whose local covariance matrices have equal determinants it is shown that separability of a two-party state and classicality of one party state are completely equivalent to each other under a nonlocal operation, allowing entanglement features to be understood in terms of any available classicality measure.
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.
We transform the way of finding entanglement criterion into two steps: to obtain necessary criterion of separability by maximizing an algebra function for a set of characteristic variables of the witness operator and the given number of partitions, then to obtain the sufficient criterion by minimizing an algebra function with respect to the characteristic variables for a given quantum state. Our method avoids the semi-definite program calculation in the witness operator entanglement detection. The necessary and sufficient criterion of separability for the three qubit X shaped state is given to…
We propose an entanglement criterion, specially designed for mixed states, based on uncertainty relation and the Wigner-Yanase skew information. The variances in this uncertainty relation does not involve any classical mixing uncertainty, and thus turns out to be purely of quantum mechanical nature. We show that any mixed entangled state can be characterized by our criterion. We demonstrate its utility for several generalized mixed entangled state including Werner states and it turns out to be stronger than any other known criterion in identifying the correct domain of relevant parameters for …
We propose an entanglement criterion based on local uncertainty relations (LURs) in a stronger form than the original LUR criterion introduced in [H. F. Hofmann and S. Takeuchi, Phys. Rev. A \textbf{68}, 032103 (2003)]. Using arbitrarily chosen operators $\{\hat{A}_{k}\}$ and $\{\hat{B}_{k}\}$ of subsystems A and B, the tighter LUR criterion, which may be used not only for discrete variables but also for continuous variables, can detect more entangled states than the original criterion.
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…
We first present a generalized criterion for maximally entangled states of 2, 3, 4, 5, 6, 8 and in theory to arbitrary-number qubits. By this criterion, some known highly entangled multi-qubit states are examined and a new genuine eight-qubit maximally entangle state is obtained. For the 4, 7 and 8 qubits system in which no maximally multi-qubit entangled states (MMES) is thought to exist before, we find that the proved and most suspected MMESes, are not completely mixed in subsystem with a critical-number qubits, below which the subsystems are all completely mixed. We believe that the new cri…
In this paper, we show a direct method of deriving the Peres-Horodecki criterion for the two-qubit states from the Hill-Wootters formula for the entanglement of formation. Although the Peres-Horodecki criterion and the Hill-Wootters formula are established results in the field of quantum information theory, they are proved independently and connections between them are not discussed precisely. In this paper, we clarify these connections. First, we replace the original Peres-Horodecki criterion with its another equivalent statement found by Augusiak et al. Second, we obtain an analytical form o…
We analyse the problem of distillation of entanglement of mixed states in higher dimensional compound systems. Employing the positive maps method [M. Horodecki et al., Phys. Lett. A 223 1 (1996)] we introduce and analyse a criterion of separability which relates the structures of the total density matrix and its reductions. We show that any state violating the criterion can be distilled by suitable generalization of the two-qubit protocol which distills any inseparable two-qubit state. Conversely, all the states which can be distilled by such a protocol must violate the criterion. The proof in…
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 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…
We derive a general criterion to detect entangled states in multi-partite systems based on the symmetric logarithmic derivative quantum Fisher information. This criterion is a direct consequence of the fact that separable states do not improve the accuracy upon estimating one-parameter family of quantum channels. Our result is a generalization of the previously known criterion for one-parameter unitary channel to any one-parameter quantum channel. We discuss several examples to illustrate our criterion. The proposed criterion is extended to the case of open quantum systems and we briefly discu…
Based on the Pauli spin operators we develop the notion of the spin-correlation matrix for the two-qubit system. If this matrix is non-zero, the measure of the correlation between the qubits is the average of the non-zero elements. Trivially, for zero matrix the bipartite is uncorrelated. This criterion turns out to be a necessary and sufficient condition for the full correlation, where it includes information on both entanglement and correlation other than entanglement. Moreover, we discuss to what extent this criterion can give information on the entanglement of the system. The criterion is …
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.
We provide a novel criterion for identifying quantum correlation, which allows us to find connections between Bell type inequalities, entanglement detection, and correlation. We utilize the criterion to construct witness operators that can detect genuine multi-qubit entanglement with fewer local measurements. The connection between identifications of quantum correlation and Mermin's inequality is discussed. Detection of genuine four-level tripartite entanglement with two local measurement settings is shown in the same manner. Further, through the criterion of quantum correlation, we derive a n…
A Multipartite entangled state has many different kinds of entanglement specified by the number of partitions. The most essential example of multipartite entanglement is the entanglement of multi-qubit Greenberger-Horne-Zeilinger (GHZ) state in white noise. We explicitly construct the entanglement witnesses for these states with stabilizer generators of the GHZ states. For a $N$ qubit GHZ state in white noise, we demonstrate the necessary and sufficient criterion of separability when it is divided into $k$ parties with $N\leq 2k-1$ for arbitrary $N$ and $k$. The criterion covers more than a ha…
We relate the notion of entanglement for quantum systems composed of two identical constituents to the impossibility of attributing a complete set of properties to both particles. This implies definite constraints on the mathematical form of the state vector associated with the whole system. We then analyze separately the cases of fermion and boson systems, and we show how the consideration of both the Slater-Schmidt number of the fermionic and bosonic analog of the Schmidt decomposition of the global state vector and the von Neumann entropy of the one-particle reduced density operators can su…
