Complementarity relations for wave-particle duality are saturated only for pure, single-quanton, quantum states. For a completely incoherent state, it is known that wave and particle quantifiers can reach zero, and hence no information about the wave and particle aspects of the system can be obtained. This means that the information is being shared with another systems, and quantum correlations can be seen as responsible for the loss of purity of the quanton, provided that the quanton is part of a multipartite pure quantum system. In this paper, by exploring the purity of bi- and tri-partite pure quantum states, we show that it is possible to obtain complete complementarity relations. This procedure allows us to create a general framework for obtaining complete complementarity relations for a subsystem that belongs to an arbitrary multi-partite quantum system in a pure state. Besides, by some simple examples, we show that if the predictability measure is changed then the correlation measure must also be changed in order to obtain complete complementarity relations for pure cases.
We derive complementarity relations for arbitrary quantum states of multiparty systems, of arbitrary number of parties and dimensions, between the purity of a part of the system and several correlation quantities, including entanglement and other quantum correlations as well as classical and total correlations, of that part with the remainder of the system. We subsequently use such a complementarity relation, between purity and quantum mutual information in the tripartite scenario, to provide a bound on the secret key rate for individual attacks on a quantum key distribution protocol.
We introduce a new feature of no-signaling (Bell) non-local theories, namely, when a system of multiple parties manifests genuine non-local correlation, then there cannot be arbitrarily high non-local correlation among any subset of the parties. We call this feature, \textit{complementarity of genuine multipartite non-locality}. We use Svetlichny's criterion for genuine multipartite non-locality and non-local games to derive the complementarity relations under no-signaling constraints. We find that the complementarity relations are tightened for the much stricter quantum constraints. We compar…
Based on quantitative complementarity relations (QCRs), we analyze the multipartite correlations in four-qubit cluster-class states. It is proven analytically that the average multipartite correlation $E_{ms}$ is entanglement monotone. Moreover, it is also shown that the mixed three-tangle is a correlation measure compatible with the QCRs in this kind of quantum states. More arrestingly, with the aid of the QCRs, a set of hierarchy entanglement measures is obtained rigorously in the present system.
Based on the quantitative complementarity relations, we analyze thoroughly the properties of multipartite quantum correlations and entanglement in four-qubit pure states. We find that, unlike the three-qubit case, the single residual correlation, the genuine three- and four-qubit correlations are not suited to quantify entanglement. More interestingly, from our qualitative and numerical analysis, it is conjectured that the sum of all the residual correlations may constitute a good measure for the total multipartite entanglement in the system.
We present a method to derive Bell monogamy relations by connecting the complementarity principle with quantum non-locality. The resulting monogamy relations are stronger than those obtained from the no-signaling principle alone. In many cases, they yield tight quantum bounds on violation of single and multiple qubit correlation Bell inequalities. In contrast with the two-qubit case, a rich structure of possible violation patterns is shown to exist in the multipartite scenario.
A simple entanglement measure for multipartite pure states is formulated based on the partial entropy of a series of reduced density matrices. Use of the proposed new measure to distinguish disentangled, partially entangled, and maximally entangled multipartite pure states is illustrated.
In this paper, we give out some effective criterions which can be used to judge the separability of multipartite pure states. We obtain the relationship between separability and Schmidt decomposable of multipartite pure states in Theorem1. The first criterion derived from Theorem2 dose not need the Schmidt decomposition which is hard to find for multipartite states. Theorem3 is more profound which can be used to deduce Corollary1 which is one of the main results in [1]. Finally, we give out an algorithm which can be used to judge the separability of multipartite pure states effectively.
We investigate how much amount of Greenberger-Horne-Zeilinger (GHZ) entanglement is required in order to prepare a given multipartite state by local operations and classical communication (LOCC).We present a LOCC procedure that asymptotically converts GHZ states into an arbitrary multipartite pure state, whose conversion rate is given by the multipartite discord of the state. This reveals that the GHZ-entanglement cost of preparing a pure state is not higher than the multipartite discord of the state. It also provides an operational interpretation of multipartite discord for pure states, namel…
We propose an explicit formula for an entanglement measure of pure multipartite quantum states, then study a general pure tripartite state in detail, and at end we give some simple but illustrative examples on four-qubits and m-qubits states.
Monogamy of bipartite correlations leads, for arbitrary pure multi-qubit states, to simple conditions able to indicate various types of multipartite entanglement by being capable to exclude the possibility of k-separability.
In this note we generalize Nielsen's marjoization criterion for the convertibility of bipartite pure states [Phys. Rev. Lett \textbf{83}, 436(1999)] to a special class of multipartite pure states which have generalized Schmidt decompositions.
In this paper, we construct a measure of entanglement by generalizing the quadric polynomial of the Segre variety for general multipartite states. We give explicit expressions for general pure three-partite and four-partite states. Moreover, we will discuss and compare this measure of entanglement with the generalized concurrence.
We construct a measure of entanglement for general pure multipartite states based on Segre variety. We also construct a class of entanglement monotones based on the Plücker coordinate equations of the Grassmann variety. Moreover, we discuss and compare these measures of entanglement.
We propose concurrence classes for general pure multipartite states based on an orthogonal complement of a positive operator valued measure on quantum phase. In particular, we construct $W^{m}$ class, $GHZ^{m}$, and $GHZ^{m-1}$ class concurrences for general pure $m$-partite states. We give explicit expressions for $W^{3}$ and $GHZ^{3}$ class concurrences for general pure three-partite states and for $W^{4}$, $GHZ^{4}$, and $GHZ^{3}$ class concurrences for general pure four-partite states.
We propose a sufficient and necessary separability criterion for pure states in multipartite and high dimensional systems. Its main advantage is operational and computable. The obvious expressions of this criterion can be given out by the coefficients of components of the pure state. In the end, we simply mention a principle method how to define and obtain the measures of entanglement in multipartite and high dimensional systems.
In this Letter we suggest a method of convex rigid frames in the studies of the multipartite quNit pure-states. We illustrate what are the convex rigid frames and what is the method of convex rigid frames. As the applications we use this method to solve some basic problems and give some new results (three theorems): The problem of the partial separability of the multipartite quNit pure-states and its geometric explanation; The problem of the classification of the multipartite quNit pure-states, and give a perfect explanation of the local unitary transformations; Thirdly, we discuss the invaria…
This paper will address the question of the distillation of entanglement from a finite number of multi-partite mixed states. It is shown that if one can distill a pure entangled state from n copies of a mixed state $σ_{ABC...}$ there must be at least a subspace in whole Hilbert space of the all copies such that the projection of $σ_{ABC...}^{\otimes n}$ onto the subspace is a pure entangled state. We also show that the purification of entanglement or distillation of entanglement can be carried out by local joint projective measurements with the help of classical communication and local general…
We consider one copy of a quantum system prepared in one of two non-orthogonal pure product states of multipartite distributed among separated parties. We show that there exist protocols which obtain optimal probability in the sense of conclusive discrimination by means of local operations and classical communications(LOCC) as good as by global operations. Also, we show a protocol which minimezes the average number of local operations. Our result implies that two product pure multipartite states might not have the non-local property though more than two can have.
We study the dynamical entanglement distribution in a multipartite system. The initial state is a maximally entangled two level atom with a single photon field. Next a sequence of atoms are sent, one at the time, and interact with the field. We show that the which way information initially stored only in the field is now distributed among the parties of the global system. We obtain the corresponding complementarity relations in analytical form. We show that this dynamics may lead to a quantum eraser phenomenon provided that measurements of the probe atoms are performed in a basis which maximiz…
The spectral decomposition of all (n,2,2) Bell operators ($2^{2^n}$ in number, $n \ge 2$), as introduced by Werner and Wolf, is done. Its implications on the characterization of Bell operators as probes of entanglement are considered in detail.
An entanglement bound based on local measurements is introduced for multipartite pure states. It is the upper bound of the geometric measure and the relative entropy of entanglement. It is the lower bound of minimal measurement entropy. For pure bipartite states, the bound is equal to the entanglement entropy. The bound is applied to pure tripartite qubit states and the exact tripartite relative entropy of entanglement is obtained for a wide class of states.
Necessary and sufficient conditions for the equivalence of arbitrary n-qubit pure quantum states under Local Unitary (LU) operations are derived. First, an easily computable standard form for multipartite states is introduced. Two generic states are shown to be LU-equivalent iff their standard forms coincide. The LU-equivalence problem for non--generic states is solved by presenting a systematic method to determine the LU operators (if they exist) which interconvert the two states.
In this paper we present the necessary and sufficient conditions of separability for multipartite pure states. These conditions are very simple, and they don't require Schmidt decomposition or tracing out operations. We also give a necessary condition for a local unitary equivalence class for a bipartite system in terms of the determinant of the matrix of amplitudes and explore a variance as a measure of entanglement for multipartite pure states.
We show that pure states of multipartite quantum systems are multiseparable (i.e. give separable density matrices on tracing any party) if and only if they have a generalized Schmidt decomposition. Implications of this result for the quantification of multipartite pure-state entanglement are discussed. Further, as an application of the techniques used here, we show that any purification of a bipartite PPT bound entangled state is tri-inseparable, i.e. has none of its three bipartite partial traces separable.
We show that genuine multipartite entanglement of all multipartite pure states in arbitrary finite dimension can be detected in a device-independent way by employing bipartite Bell inequalities on states that are deterministically generated from the initial state via local operations. This leads to an efficient scheme for large classes of multipartite states that are relevant in quantum computation or condensed-matter physics, including cluster states and the ground state of the Affleck-Kennedy-Lieb-Tasaki (AKLT) model. For cluster states the detection of genuine multipartite entanglement invo…
The quantification and classification of quantum entanglement is a very important and still open question of quantum information theory. In this paper, we describe an entanglement measure for multipartite pure states (the minimum of Shannon's entropy of orthogonal measurements). This measure is additive, monotone under LOCC, and coincides with the reduced von Neumann entropy on bipartite states. A method for numerical calculation of this measure by genetic algorithms is also presented. Moreover, the minimization of entropy technique is extended to fermionic states.
The restrictions that nature places on the distribution of correlations in a multipartite quantum system play fundamental roles in the evolution of such systems, and yield vital insights into the design of protocols for the quantum control of ensembles with potential applications in the field of quantum computing. We show how this entanglement sharing behavior may be studied in increasingly complex systems of both theoretical and experimental significance and demonstrate that entanglement sharing, as well as other unique features of entanglement, e.g. the fact that maximal information about a …
Local operations assisted by classical communication (LOCC) constitute the free operations in entanglement theory. Hence, the determination of LOCC transformations is crucial for the understanding of entanglement. We characterize here almost all LOCC transformations among pure multipartite multilevel states. Combined with the analogous results for qubit states shown by Gour \emph{et al.} [J. Math. Phys. 58, 092204 (2017)], this gives a characterization of almost all local transformations among multipartite pure states. We show that nontrivial LOCC transformations among generic, fully entangled…
We propose an entanglement tensor to compute the entanglement of a general pure multipartite quantum state. We compare the ensuing tensor with the concurrence for bipartite state and apply the tensor measure to some interesting examples of entangled three-qubit and four-qubit states. It is shown that in defining the degree of entanglement of a multi-partite state, one needs to make assumptions about the willingness of the parties to cooperate. We also discuss the degree of entanglement of the multi-qubit $\ket{W_{M}}$-states.
We study multipartite entanglement under stochastic local operations and classical communication (SLOCC) and propose the entanglement classification under SLOCC for arbitrary-dimensional multipartite ($n$-qudit) pure states via the rank of coefficient matrix, together with the permutation of qudits. The ranks of the coefficient matrices have been proved to be entanglement monotones. The entanglement classification of the $2 \otimes 2 \otimes 2 \otimes 4$ system is discussed in terms of the generalized method, and 22 different SLOCC families are found.
We first define a quantity exhibiting the usefulness of bipartite quantum states for teleportation, called the quantum teleportation capability, and then investigate its restricted shareability in multi-party quantum systems. In this work, we verify that the quantum teleportation capability has a monogamous property in its shareability for arbitrary three-qutrit pure states by employing the monogamy inequality in terms of the negativity.
Consider a system consisting of $n$ $d$-dimensional quantum particles and arbitrary pure state $Ψ$ of the whole system. Suppose we simultaneously perform complete von Neumann measurements on each particle. One can ask: what is the minimal possible value $S[Ψ]$ of the entropy of outcomes joint probability distribution? We show that $S[Ψ]$ coincides with entanglement entropy for bipartite states. We compute $S[Ψ]$ for two sample multipartite states: the hexacode state ($n=6, d=2$) and determinant states ($n=d$). The generalization of determinant states to the case $d<n$ is considered.
We show that not all 4-party pure states are GHZ reducible (i.e., can be generated reversibly from a combination of 2-, 3- and 4-party maximally entangled states by local quantum operations and classical communication asymptotically) through an example, we also present some properties of the relative entropy of entanglement for those 3-party pure states that are GHZ reducible, and then we relate these properties to the additivity of the relative entropy of entanglement.
The entanglement quantification and classification of multipartite quantum states are two important research fields in quantum information. In this work, we study the entanglement of arbitrary-dimensional multipartite pure states by looking at the averaged partial entropies of various bipartite partitions of the system, namely, the so-called Manhattan distance ($l_1$ norm) of averaged partial entropies (MAPE), and it is proved to be an entanglement measure for pure states. We connected the MAPE with the coefficient matrices, which are important tools in entanglement classification and reexpres…
We propose a method to detect genuine quantum correlation for multi-qubit pure states. We then derive a complementarity relations for pure quantum states of N qubits. We prove that in all many-qubit systems there exist strict monogamy laws for quantum correlations. On the other hand, it is known that the entanglement monogamy equality proposed by Coffman, Kundu, and Wootters is in general not true for multiqubit states. Inducing from the CKW equality, we find a proper form of entanglement monogamy equality for arbitrary quantum states. The total quantum correlation of qubit k with the remainin…
The necessary and sufficient conditions for the equivalence of arbitrary n-qubit pure quantum states under Local Unitary (LU) operations derived in [B. Kraus Phys. Rev. Lett. 104, 020504 (2010)] are used to determine the different LU-equivalence classes of up to five-qubit states. Due to this classification new parameters characterizing multipartite entanglement are found and their physical interpretation is given. Moreover, the method is used to derive examples of two n-qubit states (with n>2 arbitrary) which have the properties that all the entropies of any subsystem coincide, however, the s…
If a pure state of a multipartite quantum system is Borromean, that is, its density matrix becomes product after tracing out any its component then the initial state is product itself. This shows the essentially classical nature of Borromean correlations which can not be achieved by entangled pure states.
Resource theory of quantum coherence originated like entanglement in quantum information theory. However, still now proper classification of quantum states is missing under coherence. In this work, we have provided a classification of states under local incoherent operations. We have succeeded in deriving the necessary and sufficient condition for which two pure multipartite states are equivalent under stochastic local incoherent operations and classical communications (SLICC) and local incoherent operations and classical communications (LICC). In particular, we have succeeded in characterizin…
The tensor rank (also known as generalized Schmidt rank) of multipartite pure states plays an important role in the study of entanglement classifications and transformations. We employ powerful tools from the theory of homogeneous polynomials to investigate the tensor rank of symmetric states such as the tripartite state $\ket{W_3}=\tfrac{1}{\sqrt{3}}(\ket{100}+\ket{010}+\ket{001})$ and its $N$-partite generalization $\ket{W_N}$. Previous tensor rank estimates are dramatically improved and we show that (i) three copies of $\ket{W_3}$ has rank either 15 or 16, (ii) two copies of $\ket{W_N}$ has…
In a recent paper, Walgate et. al. demonstrated that any two orthogonal multipartite pure states can be optimally distinguished using only local operations. We utilise their result to show that this is true for any two multiparty pure states, in the sense of inconclusive discrimination. There are also certain regimes of conclusive discrimination for which the same also applies, although we can only conjecture that the result is true for all conclusive regimes. We also discuss a class of states that can be distinguished locally according to any discrimination measure, as they can be locally rec…
In this paper, we consider the problem of how to quantify entanglement for any multipartite quantum states. For bipartite pure states partial entropy is a good entanglement measure. By using partial entropy, we firstly introduce the Combinatorial Entropy of Fully entangled states (CEF) which can be used to quantify entanglement for any fully entangled pure states. In order to quantify entanglement for any multipartite states we also need another concept the Entanglement Combination (EC) which can be used to completely describe the entanglement between any parties of the given quantum states. C…
We introduce algebriac sets in the products of complex projective spaces for multipartite mixed states, which are independent of their eigenvalues and only measure the "position" of their eigenvectors, as their non-local invariants (ie. remaining invariant after local untary transformations). These invariants are naturally arised from the physical consideration of checking multipartite mixed states by measuring them with multipartite separable pure states. The algebraic sets have to be the sum of linear subspaces if the multipartite mixed state is separable, and thus we give a new separability…
For any finite number of parts, measurements and outcomes in a Bell scenario we estimate the probability of random $N$-qu$d$it pure states to substantially violate any Bell inequality with uniformly bounded coefficients. We prove that under some conditions on the local dimension the probability to find any significant amount of violation goes to zero exponentially fast as the number of parts goes to infinity. In addition, we also prove that if the number of parts is at least 3, this probability also goes to zero as the the local Hilbert space dimension goes to infinity.
The Schmidt number is of crucial importance in characterizing the bipartite pure states. We explore and propose here a generalization of Schmidt number for states in multipartite systems. It is shown to be entanglement monotonic and valid for both pure and mixed states. In addition, the corresponding generalization of multipartite Schmidt coefficients is introduced. Our approach is applicable for systems with arbitrary number of parties and for arbitrary dimensions.
We provide methods for computing the geometric measure of entanglement for two families of pure states with both experimental and theoretical interests: symmetric multiqubit states with non-negative amplitudes in the Dicke basis and symmetric three-qubit states. In addition, we study the geometric measure of pure three-qubit states systematically in virtue of a canonical form of their two-qubit reduced states, and derive analytical formulae for a three-parameter family of three-qubit states. Based on this result, we further show that the W state is the maximally entangled three-qubit state wit…
Characterizing the transformation and classification of multipartite entangled states is a basic problem in quantum information. We study the problem under two most common environments, local operations and classical communications (LOCC), stochastic LOCC and two more general environments, multi-copy LOCC (MCLOCC) and multi-copy SLOCC (MCSLOCC). We show that two transformable multipartite states under LOCC or SLOCC are also transformable under MCLOCC and MCSLOCC. What's more, these two environments are equivalent in the sense that two transformable states under MCLOCC are also transformable un…
The geometric measure of entanglement of a pure state, defined by its distance to the set of pure separable states, is extended to multipartite mixed states. We characterize the nearest disentangled mixed state to a given mixed state with respect to this measure by a system of equations. The entanglement eigenvalue for a mixed state is introduced. For a given mixed state, we show that its nearest disentangled mixed state is associated with its entanglement eigenvalue.
We explore the quantum correlation distribution in multipartite quantum states based on the square of quantum discord (SQD). For tripartite quantum systems, we derive the necessary and sufficient condition for the SQD to satisfy the monogamy relation. Particularly, we prove that the SQD is monogamous for three-qubit pure states, based on which a genuine tripartite quantum correlation measure is introduced. In addition, we also address the quantum correlation distributions in four-qubit pure states. As an example, we investigate multipartite quantum correlations in the dynamical evolution of mu…
We introduce two operational entanglement measures which are applicable for arbitrary multipartite (pure or mixed) states. One of them characterizes the potentiality of a state to generate other states via local operations assisted by classical communication (LOCC) and the other the simplicity of generating the state at hand. We show how these measures can be generalized to two classes of entanglement measures. Moreover, we compute the new measures for pure few-partite systems and use them to characterize the entanglement contained in a three-qubit state. We identify the GHZ- and the W-state a…
For a given pure state of multipartite system, the concurrence vector is defined by employing the defining representation of generators of the corresponding rotation groups. The norm of concurrence vector is considered as a measure of entanglement. For multipartite pure state, the concurrence vector is regarded as the direct sum of concurrence subvectors in the sense that each subvector is associated with a pair of particles. It is proposed to use the norm of each subvector as the contribution of the corresponding pair in entanglement of the system.
We estimate the probability of random $N$-qudit pure states violating full-correlation Bell inequalities with two dichotomic observables per site. These inequalities can show violations that grow exponentially with $N$, but we prove this is not the typical case. For many-qubit states the probability to violate any of these inequalities by an amount that grows linearly with $N$ is vanishingly small. If each system's Hilbert space dimension is larger than two, on the other hand, the probability of seeing \emph{any} violation is already small. For the qubits case we discuss furthermore the conseq…
Reversible state transformations under entanglement non-increasing operations give rise to entanglement measures. It is well known that asymptotic local operations and classical communication (LOCC) are required to get a simple operational measure of bipartite pure state entanglement. For bipartite mixed states and multipartite pure states it is likely that a more powerful class of operations will be needed. To this end \cite{BPRST01} have defined more powerful versions of state transformations (or reducibilities), namely LOCCq (asymptotic LOCC with a sublinear amount of quantum communication)…
We investigate the system-environment information flow from the point of view ofcomplete complementarity relations. We consider some commonly used noisy quantum channels:Amplitude damping, phase damping, bit flip, bit-phase flip, phase flip, depolarizing, and correlatedamplitude damping. By starting with an entangled bipartite pure quantum state, with the linearentropy being the quantifier of entanglement, we study how entanglement is redistributed and turnedinto general correlations between the degrees of freedom of the whole system. For instance, it ispossible to express the entanglement ent…
Quantum entanglement and nonlocality are inextricably linked. However, while entanglement is necessary for nonlocality, it is not always sufficient in the standard Bell scenario. We derive sufficient conditions for entanglement to give rise to genuine multipartite nonlocality in networks. We find that any network where the parties are connected by bipartite pure entangled states is genuine multipartite nonlocal, independently of the amount of entanglement in the shared states and of the topology of the network. As an application of this result, we also show that all pure genuine multipartite e…
The degree to which a pure quantum state is entangled can be characterized by the distance or angle to the nearest unentangled state. This geometric measure of entanglement, already present in a number of settings [A. Shimony, Ann. NY. Acad. Sci. 755, p.675 (1995) and H. Barnum and N. Linden, J. Phys. A: Math. Gen. 34, p.6787 (2001)], is explored for bipartite and multipartite pure and mixed states. It is determined for arbitrary two-qubit mixed states and for generalized Werner and isotropic states, and is also applied to certain multipartite mixed states.
