We investigate the detection of entanglement in $n$-partite quantum states. We obtain practical separability criteria to identify genuinely entangled and non-separable mixed quantum states. No numerical optimization or eigenvalue evaluation is needed, and our criteria can be evaluated by simple computations involving components of the density matrix. We provide examples in which our criteria perform better than all known separability criteria. Specifically, we are able to detect genuine $n$-partite entanglement which has previously not been identified. In addition, our criteria can be used in today's experiment.
We investigate genuinely entangled $N$-qubit states with no $N$-partite correlations in the case of symmetric states. Using a tensor representation for mixed symmetric states, we obtain a simple characterization of the absence of $N$-partite correlations. We show that symmetric states with no $N$-partite correlations cannot exist for an even number of qubits. We fully identify the set of genuinely entangled symmetric states with no $N$-partite correlations in the case of three qubits, and in the case of rank-2 states. We present a general procedure to construct families for an arbitrary odd nu…
We find an algebraic formula for the N-partite concurrence of N qubits in an X-matrix. X- matricies are density matrices whose only non-zero elements are diagonal or anti-diagonal when written in an orthonormal basis. We use our formula to study the dynamics of the N-partite entanglement of N remote qubits in generalized N-party Greenberger-Horne-Zeilinger (GHZ) states. We study the case when each qubit interacts with a partner harmonic oscillator. It is shown that only one type of GHZ state is prone to entanglement sudden death; for the rest, N-partite entanglement dies out momentarily. Algeb…
A genuinely $N$-partite entangled state may display vanishing $N$-partite correlations measured for arbitrary local observables. In such states the genuine entanglement is noticeable solely in correlations between subsets of particles. A straightforward way to obtain such states for odd $N$ is to design an `anti-state' in which all correlations between an odd number of observers are exactly opposite. Evenly mixing a state with its anti-state then produces a mixed state with no $N$-partite correlations, with many of them genuinely multiparty entangled. Intriguingly, all known examples of `entan…
We construct tri-qubit genuinely entangled states which have positive partial transposes with respect to bi-partition of systems. These examples disprove a conjecture [L. Novo, T. Moroder and O. G\" uhne, Phys.Rev.A {88}, 012305 (2013)] which claims that PPT mixtures are necessary and sufficient for biseparability of three qubits.
We derive a general framework to identify genuinely multipartite entangled mixed quantum states in arbitrary-dimensional systems and show in exemplary cases that the constructed criteria are stronger than those previously known. Our criteria are simple functions of the given quantum state and detect genuine multipartite entanglement that had not been identified so far. They are experimentally accessible without quantum state tomography and are easily computable as no optimization or eigenvalue evaluation is needed.
We give a direct tensor decomposition for any density matrix into Hermitian operators. Based upon the decomposition we study when the mixed states are separable and generalize the separability indicators to multi-partite states and show that a density operator is separable if and only if the separable indicator is non-negative. We then derive two bounds for the separable indicator in terms of the spectrum of the factor operators in the tensor summands.
We prove, in a multipartite setting, that it's always feasible to exactly transform a genuinely $m$-partite entangled state with sufficient many copies to any other $m$-partite state via local quantum operation and classical communication. This result affirms the comparability of multipartite pure entangled states.
A set of multipartite orthogonal quantum states is strongly nonlocal if it is locally irreducible for every bipartition of the subsystems [Phys. Rev. Lett. 122, 040403 (2019)]. Although this property has been shown in three-, four- and five-partite systems, the existence of strongly nonlocal sets in $N$-partite systems remains unknown when $N\geq 6$. In this paper, we successfully show that a strongly nonlocal set of orthogonal entangled states exists in $(\mathbb{C}^d)^{\otimes N}$ for all $N\geq 3$ and $d\geq 2$, which for the first time reveals the strong quantum nonlocality in general $N$-…
After introducing the partially separable concept, we proved the equivalence between the partial separability of a given $m$-partite subsystem with $m$ qubits and the purity of states of this $m$-partite subsystem for a pure state in multipartite systems with arbitrary finite $n(>m)$ qubits. Furthermore, we give out the operational realizations (corollaries) of our theorem, which are the sufficient and necessary criterions of partial separability of states and can be used to classification of states. Our results are helpful to understand and describe quantum entanglement in multipartite system…
We show how to quantify tri-partite entanglement using entropies derived from experimental correlations. We use a multi-partite generalization of the entanglement of formation that is greater than zero if and only if the state is genuinely multi-partite entangled. We develop an entropic witness for tripartite entanglement, and show that the degree of violation of this witness places a lower limit on the tripartite entanglement of formation. We test our results in the three-qubit regime using the GHZ-Werner state and the W-Werner state, and in the high-dimensional pure-state regime using the tr…
We investigate conditions on a finite set of multi-partite product vectors for which separable states with corresponding product states have unique decomposition, and show that this is true in most cases if the number of product vectors is sufficiently small. In the three qubit case, generic five dimensional spaces give rise to faces of the convex set consisting of all separable states, which are affinely isomorphic to the five dimensional simplex with six vertices. As a byproduct, we construct three qubit entangled PPT edge states of rank four with explicit formulae. This covers those entangl…
We study how multi-partite entanglement evolves under the paradigm of separable operations, which include the local operations and classical communication (LOCC) as a special case. We prove that the average "decay" of entanglement induced by a separable operation is measure independent (among SL-invariant ones) and state independent: the ratio between the average output entanglement and the initial entanglement is solely a function of the separable operation, regardless of the input state and of the SL-invariant entanglement measure being used. We discuss the "disentangling power" of a quantum…
Non-classical correlations between measurement results make entanglement the essence of quantum physics and the main resource for quantum information applications. Surprisingly, there are $n$-particle states which do not exhibit $n$-partite correlations at all but still are genuinely $n$-partite entangled. We introduce a general construction principle for such states, implement them in a multiphoton experiment and analyze their properties in detail. Remarkably, even without $n$-partite correlations, these states do violate Bell inequalities showing that there is no classical, i.e., local reali…
We study the quantum coherence and its distribution of N-partite GHZ and W states of bosonic fields in the noninertial frames with arbitrary number of acceleration observers. We find that the coherence of both GHZ and W state reduces with accelerations and freezes in the limit of infinite accelerations. The freezing value of coherence depends on the number of accelerated observers. The coherence of N-partite GHZ state is genuinely global and no coherence exists in any subsystems. For the N-partite W state, however, the coherence is essentially bipartite types, and the total coherence is equal …
Entanglement is the resource to overcome the restriction of operations to Local Operations assisted by Classical Communication (LOCC). The Maximally Entangled Set (MES) of states is the minimal set of n-partite pure states with the property that any truly n-partite entangled pure state can be obtained deterministically via LOCC from some state in this set. Hence, this set contains the most useful states for applications. In this work we characterize the MES for generic three qutrit states. Moreover, we analyze which generic three qutrit states are reachable (and convertible) under LOCC transfo…
We study the norms of the Bloch vectors for arbitrary $n$-partite quantum states. A tight upper bound of the norms is derived for $n$-partite systems with different individual dimensions. These upper bounds are used to deal with the separability problems. Necessary conditions are presented for $\mathbf m$-separable states in $n$-partite quantum systems. Based on the upper bounds, classification of multipartite entanglement is illustrated with detailed examples.
The separability detecting problem of mixed states is one of the fundamental problems in quantum information theory. In the last 20 years, almost all methods are based on the sufficient or necessary conditions for entanglement. However, in this paper, we only need one algorithm to solve the problem. We propose a tensor optimization method to check whether an $m$-partite quantum mixed state is separable or not and give a decomposition for it if it is. We first convert the separability discrimination problem of mixed states to the positive Hermitian decomposition problem of Hermitian tensors. Th…
We generalize Greenberger-Horne-Zeilinger (GHZ) theorem to an arbitrary number of D-dimensional systems. Contrary to conventional approaches using compatible composite observables, we employ incompatible and concurrent observables, whose common eigenstate is still a generalized GHZ state. It is these concurrent observables which enable to prove a genuinely N-partite and D-dimensional GHZ theorem. Our principal idea is illustrated for a four-partite system with D which is an arbitrary multiple of 3. By extending to N qudits, we show that GHZ theorem holds as long as N is not divisible by all no…
The partial separability of multipartite qubit density matrixes is strictly defined. We give a reduction way from N-partite qubit density matrixes to bipartite qubit density matrixes, and prove a necessary condition that a N-partite qubit density matrix to be partially separable is its reduced density matrix to satisfy PPT condition.
Unextendible product bases (UPBs) are interesting mathematical objects arising in composite Hilbert spaces that have found various applications in quantum information theory, for instance in a construction of bound entangled states or Bell inequalities without quantum violation. They are closely related to another important notion, completely entangled subspaces (CESs), which are those that do not contain any fully separable pure state. Among CESs one finds a class of subspaces in which all vectors are not only entangled, but are genuinely entangled. Here we explore the connection between UPBs…
This paper focuses on the geometric phase of entangled states of bi-partite systems under bi-local unitary evolution. We investigate the relation between the geometric phase of the system and those of the subsystems. It is shown that (1) the geometric phase of cyclic entangled states with non-degenerate eigenvalues can always be decomposed into a sum of weighted non-modular pure state phases pertaining to the separable components of the Schmidt decomposition, though the same cannot be said in the non-cyclic case, and (2) the geometric phase of the mixed state of one subsystem is generally diff…
We present a general technique for hiding a classical bit in multipartite quantum states. The hidden bit, encoded in the choice of one of two possible density operators, cannot be recovered by local operations and classical communication without quantum communication. The scheme remains secure if quantum communication is allowed between certain partners, and can be designed for any choice of quantum communication patterns to be secure, but to allow near perfect recovery for all other patterns. The maximal probability of unwanted recovery of the hidden bit, as well as the maximal error for allo…
We propose two classes of the generalized concurrence vectors of the multipartite systems consisting of qubits. Making use of them, we are able to, respectively, describe and quantify GHZ-class and W-class entanglement both in total and between arbitrary two partite in the multipartite system consisting of qubits. In the case of pure state of three qubits that one partite is separable, it is shown to exactly back to the usual Wootters' concurrence after introduce a whole concurrence vector. In principle, our method is applicable to any $N$-partite systems consisting of $N$ qubits.
We show that for an m-partite quantum system, there is a ball of radius 2^{-(m/2-1)} in Frobenius norm, centered at the identity matrix, of separable (unentangled) positive semidefinite matrices. This can be used to derive an epsilon below which mixtures of epsilon of any density matrix with 1 - epsilon of the maximally mixed state will be separable. The epsilon thus obtained is exponentially better (in the number of systems) than existing results. This gives a number of qubits below which NMR with standard pseudopure-state preparation techniques can access only unentangled states; with parame…
An orthogonal product basis of a composite Hilbert space is genuinely nonlocal if the basis states are locally indistinguishable across every bipartition. From an operational point of view such a basis corresponds to a separable measurement that cannot be implemented by local operations and classical communication (LOCC) unless all the parties come together in a single location. In this work we classify genuinely nonlocal product bases into different categories. Our classification is based on state elimination property of the set via orthogonality-preserving measurements when all the parties a…
We introduce a new measure for the genuinely N-partite (all-party) entanglement of N-qubit states using the trace distance metric, and find an algebraic formula for the GHZ-diagonal states. We then use this formula to show how the all-party entanglement of experimentally produced GHZ states of an arbitrary number of qubits may be bounded with only four measurements.
A general $n$-partite state $| Ψ>$ of a composite quantum system can be regarded as an element in a Hilbert tensor product space $\HH = \otimes_{k=1}^n \HH_k$, where the dimension of $\HH_k$ is $d_k$ for $k = 1,..., n$. Without loss of generality we may assume that $d_1 \le...\le d_n$. A separable (Hartree) $n$-partite state $| ϕ>$ can be described by $| ϕ> = \otimes_{k=1}^n | ϕ^{(k)}>$ with $| ϕ^{(k)}> \in \HH_k$. We show that $σ:= \min \{< Ψ| ϕ_Ψ> : | Ψ> \in \HH,.$ $. < Ψ| Ψ> = 1\}$ is a positive number, where $| ϕ_Ψ>$ is the nearest separable state to $| Ψ>$. We call $σ$ the minimum Hartree…
We show that spin squeezing criteria commonly used for entanglement detection can be erroneous, if the probe is not symmetric. We then derive a lower bound on squeezing for separable states in spin systems probed asymmetrically. Using this we further develop a procedure that allows us to verify the degree of entanglement of a quantum state in the spin system. Finally, we apply our method for entanglement verification to existing experimental data, and use it to prove the existence of tri-partite entanglement in a spin squeezed atomic ensemble.
We address the question of whether or not global entanglement of a quantum state can be inferred from local properties. Specifically, we are interested in genuinely multiparticle entangled states whose two-body marginals are all separable, but where the entanglement can be proven using knowledge of a subset of the marginals only. Using an iteration of semidefinite programs we prove that for any possible marginal configuration up to six particles multiqubit states with the desired properties can be found. We then present a method to construct states with the same properties for more particles i…
Three distant labs A, B and C, having no prior entanglement can establish a shared GHZ state, when one of them say A sends two particles to B and C for their local actions. The mediating particles remain separable from each other and from the particles of A, B and C. We prove that in this way, GHZ states are shared with a probability $\frac{1}{7}$. We also show how separable particles can be mediated to establish arbitrary $d-$ dimensional Bell states between distant labs. Our method is constructive and allows generaization of GHZ sharing between any number of parties and in any dimension. The…
In this paper, we discuss the partial separability and its criteria problems of multipartite qubit mixed-states. First we strictly define what is the partial separability of a multipartite qubit system. Next we give a reduction way from N-partite qubit density matrixes to bipartite qubit density matrixes, and prove a necessary condition that a N-partite qubit mixed-state to be partially separable is its reduction to satisfy the PPT condition. PACC numbers: 03.67.Mn; 03.65.Ud; 03.67.Hk
We consider the concept of "the permutationally invariant (PI) part of a density matrix," which has proven very useful for both efficient quantum state estimation and entanglement characterization of $N$-qubit systems. We show here that the concept is, in fact, basis-dependent, but that this basis dependence makes it an even more powerful concept than has been appreciated so far. By considering the PI part $ρ^{\rm PI}$ of a general (mixed) $N$-qubit state $ρ$, we obtain: (i) strong bounds on quantitative nonseparability measures, (ii) a whole hierarchy of multi-partite separability criteria (o…
In this paper, we will show that a vanishing generalized concurrence of a separable state can be seen as an algebraic variety called the Segre variety. This variety define a quadric space which gives a geometric picture of separable states. For pure, bi- and three-partite states the variety equals the generalized concurrence. Moreover, we generalize the Segre variety to a general multipartite state by relating to a quadric space defined by two-by-two subdeterminants.
We construct three-qubit entanglement witnesses with the properties that all the partial transposes have the spanning properties. These witnesses determine faces for separable states whose interior lies in the interior of PPT states, and so they detect sets of PPT entangled states with nonempty interiors. These witnesses also detect all kinds of bi-separable pure product states. Our construction depends on the observation that multi-partite entanglement witnesses correspond to multi-linear maps which are positive.
We show that for all $n\ge3$, an example of an $n$-partite quantum correlation that is not genuinely multipartite nonlocal but rather exhibiting anonymous nonlocality, that is, nonlocal but biseparable with respect to all bipartitions, can be obtained by locally measuring the $n$-partite Greenberger-Horne-Zeilinger (GHZ) state. This anonymity is a manifestation of the impossibility to attribute unambiguously the underlying multipartite nonlocality to any definite subset(s) of the parties, even though the correlation can indeed be produced by nonlocal collaboration involving only such subsets. …
Chen (quant-ph/0611126) has recently claimed ``exponential violation of local realism by separable states", in the sense that multi-partite separable quantum states are supposed to give rise to correlations and fluctuations that violate a Bell-type inequality that Chen takes to be satisfied by local realism. However, this can not be true since all predictions (including all correlations and fluctuations) that separable quantum states give rise to have a local realistic description and thus satisfy all Bell-type inequalities, and this holds for all number of parties. Since Chen claims otherwise…
In [\href{http://www.sciencedirect.com/science/article/pii/037596019290711T}{Phys. Lett. A {\bf 166}, 293 (1992)}] Popescu characterized nonlocality of pure $n$-partite entangled systems by studying bipartite violation of local realism when $n-2$ number of parties perform projective measurements on their particles. A pertinent question in this scenario is whether similar characterization is possible for $n$-partite mixed entangled states also. In the present work we have followed an analogous approach so as to explore whether given a tripartite mixed entangled state the conditional bipartite s…
We present a simple family of Bell inequalities applicable to a scenario involving arbitrarily many parties, each of which performs two binary-outcome measurements. We show that these inequalities are members of the complete set of full-correlation Bell inequalities discovered by Werner-Wolf-Zukowski-Brukner. For scenarios involving a small number of parties, we further verify that these inequalities are facet-defining for the convex set of Bell-local correlations. Moreover, we show that the amount of quantum violation of these inequalities naturally manifests the extent to which the underlyin…
We introduce the concept of "absolutely classical" spin states, in analogy to absolutely separable states of bi-partite quantum systems. Absolutely classical states are states that remain classical under any unitary transformation applied to them. We investigate the maximum ball of absolutely classical states centered on the fully mixed state that can be inscribed into the set of classical states, and derive a lower bound for its radius as function of the total spin quantum number. The result is compared to the case of absolutely separable states.
Bipartite and global entanglement are analyzed for the ground state of a system of $N$ spin 1/2 particles interacting via a collective spin-spin coupling described by the Lipkin-Meshkov-Glick (LMG) Hamiltonian. Under certain conditions which includes the special case of a super-symmetry, the ground state can be constructed analytically. In the case of an anti-ferromagnetic coupling and for an even number of particles this state undergoes a smooth crossover as a function of the continuous anisotropy parameter $γ$ from a separable ($γ=\infty $) to a maximally entangled many-particle state ($γ=0$…
We review and generalize the recently introduced framework of entropy vectors for detecting and quantifying genuine multipartite entanglement in high dimensional multicomponent quantum systems. We show that these ideas can be extended to discriminate among other forms of multipartite entanglement. In particular, we develop methods to test whether density matrices are: decomposable, i.\ e.\ separable with respect to certain given partitions of the subsystems; $k$-separable, i.\ e.\ separable with respect to $k$ partitions of the subsystems; $k$-partite entangled, i.e. there is entanglement amon…
We derive an upper bound on the action of a direct product of two quantum maps (channels) acting on multi-partite quantum states. We assume that the individual channels $Λ_j$ affect single-particle states so, that for an arbitrary input $ρ_j$, the distance $D_j (Λ_j [ ρ_j ], ρ_j)$ between the input $ρ_j$ and the output $Λ_j [ ρ_j ]$ of the channel is less than $ε$. Given this assumption we show that for an arbitrary {\em separable} two-partite state $ρ_{12}$ the distance between the input $ρ_{12}$ and the output $Λ_1\otimesΛ_2[ρ_{12} ]$ fulfills the bound $D_{12} (Λ_1 \otimes Λ_2 [ ρ_{12} ], ρ…
We devise a novel protocol to detect genuinely multipartite entangled states by harnessing quantum non-Markovian operations. We utilize a particular type of non-Markovianity known as the eternal non-Markovianity to construct a non-complete positive map to filter out the bi-separable states and detect genuine multipartite entanglement. We further propose a witness operator to detect genuinely multipartite entangled states experimentally based on this theory. Our study sheds light on a hitherto unexplored connection between entanglement theory and quantum non-Markovianity.
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…
