Quantum states are the key mathematical objects in quantum mechanics, and entanglement lies at the heart of the nascent fields of quantum information processing and computation. However, there has not been a general, necessary and sufficient, and operational separability condition to determine whether an arbitrary quantum state is entangled or separable. In this paper, we show that whether a quantum state is entangled or not is determined by a threshold within the quantum state. We first introduce the concept of \emph{finer} and \emph{optimal} separable states based on the properties of separable states in the role of higher-level witnesses. Then we show that any bipartite quantum state can be decomposed into a convex mixture of its optimal entangled state and its optimal separable state. Furthermore, we show that whether an arbitrary quantum state is entangled or separable, as well as positive partial transposition (PPT) or not, is determined by the robustness of its optimal entangled state to its optimal separable state with reference to a crucial threshold. Moreover, for an arbitrary quantum state, we provide operational algorithms to obtain its optimal entangled state, its optimal separable state, its best separable approximation (BSA) decomposition, and its best PPT approximation decomposition while it was an open question that how to calculate the BSA in high-dimension systems.
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.
We provide a constructive algorithm to find the best separable approximation to an arbitrary density matrix of a composite quantum system of finite dimensions. The method leads to a condition of separability and to a measure of entanglement.
We investigate the separability of arbitrary $n$-dimensional multipartite identical bosonic systems. An explicit relation between the dimension and the separability is presented. In particular, for $n=3$, it is shown that the property of PPT (positive partial transpose) and the separability are equivalent for tripartite systems.
We investigate the separability of quantum states based on covariance matrices. Separability criteria are presented for multipartite states. The lower bound of concurrence proposed in Phys. Rev. A. 75, 052320 (2007) is improved by optimizing the local orthonormal observables.
The separability and entanglement of quantum mixed states in $\Cb^2 \otimes \Cb^3 \otimes \Cb^N$ composite quantum systems are investigated. It is shown that all quantum states $ρ$ with positive partial transposes and rank $r(ρ)\leq N$ are separable.
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.
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 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.
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 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 separability problem is formulated in terms of a characterization of a single entanglement witness. More specifically, we show that any (in general multipartite) state \varrho is separable if and only if a specially constructed entanglement witness W_{\varrho} is weakly optimal, i.e., its expectation value vanishes on at least one product vector. Interestingly, the witness can always be chosen to be decomposable. Our result changes the conceptual aspect of the separability problem and rises some questions about properties of positive maps.
We found three qubit Greenberger-Horne-Zeilinger diagonal states which tells us that the partial separability of three qubit states violates the distributive rules with respect to the two operations of convex sum and intersection. The gaps between the convex sets involving the distributive rules are of nonzero volume.
We construct seven kinds of three-qubit biseparable states to show that every class of biseparable states in the partial separability classification proposed by Szalay and Kökényesi [S. Szalay and Z. Kökényesi, Phys. Rev. A 86, 032341 (2012)] is nonempty.
We review the problem of discriminating entangled states from separable states for bipartite systems. We formally define what entangled states are, present some important criteria to detect entanglement, and show how they can be classified according to their capability to perform some precisely defined tasks.
We show that all density operators of 2$\times N$--dimensional quantum systems that remain invariant after partial transposition with respect to the first system are separable. Based on this criterion, we derive a sufficient separability condition for general density operators in such quantum systems. We also give a simple proof of the separability criterion in $2\times 2$--dimensional systems [A. Peres, Phys. Rev. Lett {\bf 77}, 1413 (1996)]
We reduce the question whether a given quantum mixed state is separable or entangled to the problem of existence of a certain full family of commuting normal matrices whose matrix elements are partially determined by components of the pure states constituting a decomposition of the considered mixture. The method reproduces many known entanglement and/or separability criteria, and provides yet another geometrical characterization of mixed separable states.
Quantum entanglement plays crucial roles in quantum information processing. Quantum entangled states have become the key ingredient in the rapidly expanding field of quantum information science. Although the nonclassical nature of entanglement has been recognized for many years, considerable efforts have been taken to understand and characterize its properties recently. In this review, we introduce some recent results in the theory of quantum entanglement. In particular separability criteria based on the Bloch representation, covariance matrix, normal form and entanglement witness; lower bound…
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.
Ichikawa et al. [Phys. Rev. A 78, 052105 (2008)] showed that exchange symmetry gives rise to simple characterization of whether multipartite pure quantum states being either globally entangled or fully separable. In this Brief Report, we provide a simple alternative approach and some extension to their conclusions.
The qudit state for j = 3=2 with density matrix of the form corresponding to X-state of two-qubits is studied from the point of view of entanglement and separability properties. The method of qubit portrait of qudit states is used to get the entropic inequalities for the entangled state of the single qudit. The tomographic probability representation of the qudit X-state under consideration and its Shannon and q-entropic characteristics are presented in explicit form.
We show that any multi-qudit entanglement witness leads to a non-separability indicator for quantum optical fields, which involves intensity correlations. We get, e.g., necessary and sufficient conditions for intensity or intensity-rate correlations to reveal polarization entanglement. We also derive separability conditions for experiments involving multiport interferometers, now feasible with integrated optics. We show advantages of using intensity rates rather than intensities, e.g., a mapping of Bell inequalities to ones for optical fields. The results have implication for studies of non-cl…
The known Peres-Horodecki criterion and scaling criterion of separability are considered on examples of three-mode and four-mode Gaussian states of electromagnetic field. It is shown that the principal minors of the photon quadrature dispersion matrix are sensitive to the change of scaling parameters. An empirical observation has shown that the bigger the modulus of negative principal minors, the more entangled the state.
The general expression with the physical significance and positive definite condition of the eigenvalues of $4\times 4$ Hermitian and trace-one matrix are obtained. This implies that the eigenvalue problem of the $4\times 4$ density matrix is generally solved. The obvious expression of Peres' separability condition for an arbitrary state of two qubits is then given out and it is very easy to use. Furthermore, we discuss some applications to the calculation of the entanglement, the upper bound of the entanglement, and a model of the transfer of entanglement in a qubit chain through a noisy chan…
We present a geometric approach to the characterization of separability and entanglement in pure Gaussian states of an arbitrary number of modes. The analysis is performed adapting to continuous variables a formalism based on single subsystem unitary transformations that has been recently introduced to characterize separability and entanglement in pure states of qubits and qutrits [arXiv:0706.1561]. In analogy with the finite-dimensional case, we demonstrate that the $1 \times M$ bipartite entanglement of a multimode pure Gaussian state can be quantified by the minimum squared Euclidean distan…
We examine the entanglement of general mixed states of a two qubit Heisenberg XYZ chain in the presence of a magnetic field, and its detection by means of different criteria. Both the exact separability conditions and the weaker conditions implied by the disorder and the von Neumann entropic criteria are analyzed. The ensuing limit temperatures for entanglement in thermal states of different XYZ models are then examined and compared with the limit temperature of the symmetry-breaking solution in a mean field type approximation. The latter, though generally lower, can also be higher than the ex…
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 investigate separability and entanglement of mixed states in ${\cal C}^2\otimes{\cal C}^2\otimes{\cal C}^N$ three party quantum systems. We show that all states with positive partial transposes that have rank $\le N$ are separable. For the 3 qubit case (N=2) we prove that all states $ρ$ that have positive partial transposes and rank 3 are separable. We provide also constructive separability checks for the states $ρ$ that have the sum of the rank of $ρ$ and the ranks of partial transposes with respect to all subsystems smaller than 15N-1.
We present a review of the problem of finding out whether a quantum state of two or more parties is entangled or separable. After a formal definition of entangled states, we present a few criteria for identifying entangled states and introduce some entanglement measures. We also provide a classification of entangled states with respect to their usefulness in quantum dense coding, and present some aspects of multipartite entanglement.
Quantum correlations in the state of four-level atom are investigated by using generic unitary transforms of the classical (diagonal) density matrix. Partial cases of pure state, $X$-state, Werner state are studied in details. The geometrical meaning of unitary Hilbert reference-frame rotations generating entanglement in the initially separable state is discussed. Characteristics of the entanglement in terms of concurrence, entropy and negativity are obtained as functions of the unitary matrix rotating the reference frame.
Determining whether a quantum state is separable or entangled is a problem of fundamental importance in quantum information science. It has recently been shown that this problem is NP-hard. There is a highly inefficient `basic algorithm' for solving the quantum separability problem which follows from the definition of a separable state. By exploiting specific properties of the set of separable states, we introduce a new classical algorithm that solves the problem significantly faster than the `basic algorithm', allowing a feasible separability test where none previously existed e.g. in 3-by-3-…
We determine the conditions under which general dimer-type spin chains with $XYZ$ couplings of arbitrary range in a general transverse field will exhibit an exactly separable parity-breaking eigenstate. We also provide sufficient conditions which ensure that it will be a ground state. We then examine the exact side limits at separability of the entanglement between any two spins in a finite chain, showing that in the vicinity of separability, the system will loose all signatures of dimerization, with pairwise entanglement approaching infinite range and becoming independent of separation and in…
We focus on determining the separability of an unknown bipartite quantum state $ρ$ by invoking a sufficiently large subset of all possible entanglement witnesses given the expected value of each element of a set of mutually orthogonal observables. We review the concept of an entanglement witness from the geometrical point of view and use this geometry to show that the set of separable states is not a polytope and to characterize the class of entanglement witnesses (observables) that detect entangled states on opposite sides of the set of separable states. All this serves to motivate a classica…
To determine whether a given multipartite quantum state is separable with respect to some partition we construct a family of entanglement measures R_m. This is done utilizing generalized concurrences as building blocks which are defined by flipping of M constituents and indicate states that are separable with regard to bipartitions when vanishing. Further, we provide an analytically computable lower bound for R_m via a simple ordering relation of the convex roof extension. Using the derived lower bound, we illustrate the effect of the isotropic noise on a family of four-qubit mixed states for …
Equilibrium states of infinite extended lattice systems at high temperature are studied with respect to their entanglement. Two notions of separability are offered. They coincide for finite systems but differ for infinitely extended ones. It is shown that for lattice systems with localized interaction for high enough temperature there exists no local entanglement. Even more quasifree states at high temperature are also not distillably entangled for all local regions of arbitrary size. For continuous systems entanglement survives for all temperatures. In mean field theories it is possible, that…
We define the separability and entanglement notion for particle with spin $s=1$. We consider two cases. In the first the particle is composed of two fermions with $s_1=1/2$ and $s_2=1/2$. In the second case the state is the qutrit state which is not composed system. The notion of negativity and concurrence is defined for the qutrit state. The concurrence and negativity of entangled and separable qutrit states determined by the parameters of the density matrix are explicitly calculated. The maximum entanglement of the qutrit state is observed for maximum values of non diagonal matrix elements o…
Separability and entanglement for n-qubits systems are quantified by using Hilbert-Schmidt (HS) decompositions in which the density matrices are decomposed into various terms representing certain one qubit, two-qubits,and larger qubits measurements. The present method is more general than previous methods for bipartite systems as it can be used for quantification of entanglement for large n-qubits systems.We demonstrate the use of such method by analyzing three qubits GHZ states, and three qubits general Bell states produced by a certain multiplications of Braid operators, operating on the com…
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…
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…
In this paper we give a general integral representation for separable states in the tensor product of infinite dimensional Hilbert spaces and provide the first example of separable states that are not countably decomposable. We also prove the structure theorem for the quantum communication channels that are entanglement-breaking, generalizing the finite-dimensional result of M. Horodecki, Ruskai and Shor. In the finite dimensional case such channels can be characterized as having the Kraus representation with operators of rank 1. The above example implies existence of infinite-dimensional enta…
In the discussion about the quantumness of NMR computation a conclusion is done that computational states are separable and therefore can not be entangled. This conclusion is based on the assumption that the initial density matrix of an individual molecule coincides with whole sample molecules distribution over single molecule energy levels. This means that quantum stochasticity is replaced by classical stochasticity. In the present paper it is shown, that quantum NMR computation can create genuine entangled states if initial system states are thermodynamical equilibrium ones. A separability a…
We propose the Wigner separability entropy as a measure of complexity of a quantum state. This quantity measures the number of terms that effectively contribute to the Schmidt decomposition of the Wigner function with respect to a chosen phase space decomposition. We prove that the Wigner separability entropy is equal to the operator space entanglement entropy, measuring entanglement in the space of operators, and, for pure states, to twice the entropy of entanglement. The quantum to classical correspondence between the Wigner separability entropy and the separability entropy of the classical …
We present a family of entanglement measures R_m which act as indicators for separability of n-qubit quantum states into m subsystems for arbitrary 2 \leq m \leq n. The measure R_m vanishes if the state is separable into m subsystems, and for m = n it gives the Meyer-Wallach measure while for m = 2 it reduces, in effect, to the one introduced recently by Love et al. The measures R_m are evaluated explicitly for the GHZ state and the W state (and its modifications, the W_k states) to show that these globally entangled states exhibit rather distinct behaviors under the measures, indicating the u…
In this paper we develop a mathematical framework for the characterisation of separability and entanglement of formation (EoF) of general bipartite states. These characterisations are of the variational kind, meaning that separability and EoF are given in terms of a function which is to be minimized over the manifold of unitary matrices. A major benefit of such a characterisation is that it directly leads to a numerical procedure for calculating EoF. We present an efficient minimisation algorithm and an apply it to the bound entangled 3X3 Horodecki states; we show that their EoF is very low an…
Hilbert-Schmidt (HS) decompositions are employed for analyzing systems of n-qubits, and a qubit with a qudit. Negative eigenvalues, obtained by partial-transpose (PT) plus local unitary transformations (PTU) for one qubit from the whole system, are used for indicating inseparability. A sufficient criterion for full separability of the n-qubits and qubit-qudit systems is given. We use the singular value decomposition (SVD) for improving the criterion for full separability. General properties of entanglement and separability are analyzed for a system of a qubit and a qudit and n-qubits systems, …
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…
We study the concepts of compatibility and separability and their implications for quantum and classical systems. These concepts are illustrated on a macroscopic model for the singlet state of a quantum system of two entangled spin 1/2 with a parameter reflecting indeterminism in the measurement procedure. By varying this parameter we describe situations from quantum, intermediate to classical and study which tests are compatible or separated. We prove that for classical deterministic systems the concepts of separability and compatibility coincide, but for quantum systems and intermediate syst…
It was shown in [Augusiak et al.,\;Phys. Rev. A \textbf{77}, 030301(R) (2008)] that discrimination between entanglement and separability in a two qubit state can be achieved by a measurement of a single observable on four copies of it. Moreover, a pseudo entanglement monotone $π$ was proposed to quantify entanglement in such states. The main goal of the present paper is to show that close relationship between $π$ and concurrence reported there is a result of sharing the same underlying construction of a spin flipped matrix. We also show that monogamy of entanglement can be rephrased in terms o…
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…
The problem of determining whether a given quantum state is entangled lies at the heart of quantum information processing, which is known to be an NP-hard problem in general. Despite the proposed many methods such as the positive partial transpose (PPT) criterion and the k-symmetric extendibility criterion to tackle this problem in practice, none of them enables a general, effective solution to the problem even for small dimensions. Explicitly, separable states form a high-dimensional convex set, which exhibits a vastly complicated structure. In this work, we build a new separability-entanglem…
We construct a single observable measurement of which mean value on four copies of an {\it unknown} two-qubit state is sufficient for unambiguous decision whether the state is separable or entangled. In other words, there exists a universal collective entanglement witness detecting all two-qubit entanglement. The test is directly linked to a function which characterizes to some extent the entanglement quantitatively. This function is an entanglement monotone under so--called local pure operations and classical communication (pLOCC) which preserve local dimensions. Moreover it provides tight up…
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.
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…
Quantum mechanics, in principle, allows for processes with indefinite causal order. However, most of these causal anomalies have not yet been detected experimentally. We show that every such process can be simulated experimentally by means of non-Markovian dynamics with a measurement on additional degrees of freedom. Explicitly, we provide a constructive scheme to implement arbitrary acausal processes. Furthermore, we give necessary and sufficient conditions for open system dynamics with measurement to yield processes that respect causality locally, and find that tripartite entanglement and no…
