Ming Li, Shao-Ming Fei, Zhi-Xi Wang
We investigate the separability of arbitrary dimensional tripartite sys- tems. By introducing a new operator related to transformations on the subsystems a necessary condition for the separability of tripartite systems is presented.
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In this Letter we find the new criteria of separability of multipartite qubit density matrixes. Especially, we discuss in detail the criteria of separability for tripartite qubit density matrixes. We find the sufficient and necessary conditions of separability for tripartite qubit density matrixes, and give two corollaries. The second corollary can be taken as the criterion of existence of entanglement for tripartite qubit density matrixes. In concrete application, its steps are quite simple and easy to operate. Some examples, discussions and the generalization to more high dimensional multipa…
In this paper, we present a method to construct full separability criteria for tripartite systems of qubits. The spirit of our approach is that a tripartite pure state can be regarded as a three-order tensor that provides an intuitionistic mathematical formulation for the full separability of pure states. We extend the definition to mixed states and given out the corresponding full separability criterion. As applications, we discuss the separability of several bound entangled states, which shows that our criterion is feasible.
Tripartite three mode Gaussian state undergoes parametric amplification and amplitude damping as well as thermal noise is studied. In the case of a state totally symmetrically interacting with the environment, the time dependent correlation matrix of the state in evolution is given. The conditions for fully separability and fully entanglement of the final tripartite three mode Gaussian state are worked out.
Starting with a set of conditions for bipartite separability of arbitrary quantum states in any dimension and expressed in terms of arbitrary operators whose commutator is a $c$-number, we derive a hierarchy of conditions for tripartite separability of continuous-variable three-mode quantum states. These conditions have the form of inequalities for higher-order moments of linear combinations of the mode operators. They enable one to distinguish between all possible kinds of tripartite separability, while the strongest violation of these inequalities is a sufficient condition for genuine tripar…
In this paper, an intuitive approach is employed to generalize the full separability criterion of tripartite quantum states of qubits to the higher-dimensional systems (Phys. Rev. A \textbf{72}, 022333 (2005)). A distinct characteristic of the present generalization is that less restrictive conditions are needed to characterize the properties of full separability. Furthermore, the formulation for pure states can be conveniently extended to the case of mixed states by utilizing the kronecker product approximate technique. As applications, we give the analytic approximation of the criterion for …
The density matrix of a graph is the combinatorial laplacian matrix of a graph normalized to have unit trace. In this paper we generalize the entanglement properties of mixed density matrices from combinatorial laplacian matrices of graphs discussed in Braunstein {\it et al.} Annals of Combinatorics, {\bf 10}(2006)291 to tripartite states. Then we proved that the degree condition defined in Braunstein {\it et al.} Phys. Rev. A {\bf 73}, (2006)012320 is sufficient and necessary for the tripartite separability of the density matrix of a nearest point graph.
We study separability properties in a 5-dimensional set of states of quantum systems composed of three subsystems of equal but arbitrary finite Hilbert space dimension d. These are the states, which can be written as linear combinations of permutation operators, or, equivalently, commute with unitaries of the form UxUxU. We compute explicitly the following subsets and their extreme points: (1) triseparable states, which are convex combinations of triple tensor products, (2) biseparable states, which are separable for a twofold partition of the system, and (3) states with positive partial trans…
We study separability properties in a 5-dimensional set of states of quantum systems composed of three subsystems of equal but arbitrary finite Hilbert space dimension. These are the states, which can be written as linear combinations of permutation operators, or, equivalently, commute with unitaries of the form UxUxU. We compute explicitly the following subsets: (1) triseparable states, which are convex combinations of triple tensor products, (2) biseparable states, which are separable for a twofold partition of the system, and (3) states with positive partial transpose with respect to such a…
It is possible to prepare classical optical beams which cannot be characterized by a tensor product of vectors describing each of their degrees of freedom. Here we report the experimental creation of such a non-separable, tripartite GHZ-like state of path, polarization and transverse modes of a classical laser beam. We use a Mach-Zehnder interferometer with an additional mirror and other optical elements to perform measurements that violate Mermin's inequality. This demonstration of a classical optical analogue of tripartite entanglement paves the path to novel optical applications inspired by…
Extracting work from quantum system is one of the important areas in quantum thermodynamics. As a significant thermodynamic quantity, the ergotropy gap characterizes the difference between the global and local maximum extractable works. We derive an analytical upper bound of the ergotropic gap with respect to $d\times d\times d$ tripartite separable states. This bound also provides a necessary criterion for the separability of tripartite states. Detailed examples are presented to illustrate the efficiency of this separability criterion.
We propose a method of constructing the separability criteria for multipartite quantum states on the basis of entanglement witnesses. The entanglement witnesses are obtained by finding the maximal expectation values of Hermitian operators and then optimizing over all possible Hermitian operators. We derive a set of tripartite separability criteria for the four-qubit Greenberger--Horne--Zeilinger (GHZ) diagonal states. The derived criterion set contains four criteria that are necessary and sufficient for the tripartite separability of the highly symmetric four-qubit GHZ diagonal states; the cri…
We derive a necessary and sufficient condition for the separability of tripartite three mode Gaussian states, that is easy to check for any such state. We give a classification of the separability properties of those systems and show how to determine for any state to which class it belongs. We show that there exist genuinely tripartite bound entangled states and point out how to construct and prepare such states.
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.
Separability and lower bounds of quantum entanglement based on realignment
该研究基于密度矩阵重排和约化密度矩阵向量化构造矩阵，提出两体及多体系统可分性判据，推导纠缠度量新下界，在纠缠检测与估计上优于现有结果。
We generalize the definition of strong positive partial transpose (SPPT) to the multipartite system. The tripartite case was first considered by X.-Y. Yu and H. Zhao [ Int. J. Theor. Phys.,54, 292, (2015)]. In this extension, unfortunately, desired properties such as the PPT of SPPT states the separability of super and pure SPPT states are not preserved. In contrast, this paper provides an alternative generalization to multipartite cases with these properties preserved. We also provide sufficient conditions for the separability of SPPT states.
We study separability criteria in multipartite quantum systems of arbitrary dimensions by using the Bloch representation of density matrices. We first derive the norms of the correlation tensors and obtain the necessary conditions for separability under partition of tripartite and four-partite quantum states. Moreover, based on the norms of the correlation tensors,we obtain the separability criteria by matrix method. Using detailed examples, our results are seen to be able to detect more entangled states than previous studies. Finally, necessary conditions of separability for multipartite syst…
Separability from Multipartite Measures
研究三体纯态的完全可分离性，提出三阶负性作为必要充分条件，并扩展至混合态，应用于共形场论。
We study the concurrence of arbitrary multipartite mixed quantum states. An explicit lower bound of the concurrence is derived, which detects quantum entanglement of some states better than some separability criteria, and gives sufficient conditions for distilling GHZ states from tripartite states. An interesting relations between the lower bound of the concurrence for bipartite states and for tripartite states has been presented.
Separability criteria and lower bounds of entanglement via realignment
该研究针对量子信息科学中纠缠检测与量化的核心问题，基于密度矩阵重排和约化密度矩阵向量化构造增强型可分性判据，推导并发度等的更紧下界，通过示例验证了结果优势。
Einstein-Podolsky-Rosen steering is a kind of powerful nonlocal quantum resource in quantum information processing such as quantum cryptography and quantum communication. Many criteria have been proposed in the past few years to detect steerability, both analytically and numerically, for bipartite quantum systems. We propose effective criteria for tripartite steerability and genuine tripartite steerability of three-qubit quantum states by establishing connections between the tripartite steerability (resp. genuine tripartite steerability) and the tripartite entanglement (resp. genuine tripartit…
Quantum coherence and quantum entanglement are two strong pillars in quantum information theory. We study here for the possibility of any connection between these two important aspects of quantum mechanics while studying the entanglement detection problem for the detection of bipartite higher dimensional entangled states and multipartite entangled states. To achieve our goal, we derive the lower bound $L$ of $l_{1}$ norm of coherence of bipartite qubit-qudit system using the criterion that detect entanglement. Furthermore, we deduce the upper bound $U$ of $l_{1}$ norm of coherence of separable…
In the present work, we study tripartite quantum steering of quantum correlations arising from two local dichotomic measurements on each side in the two types of partially device-independent scenarios: $1$-sided device-independent scenario where one of the parties performs untrusted measurements while the other two parties perform trusted measurements and $2$-sided device-independent scenario where one of the parties performs trusted measurements while the other two parties perform untrusted measurements. We demonstrate that tripartite steering in the $2$-sided device-independent scenario is w…
We derive an analytical lower bound for the concurrence of tripartite quantum mixed states. A functional relation is established relating concurrence and the generalized partial transpositions.
Genuine multipartite entanglement plays important roles in quantum information processing. The detection of genuine multipartite entanglement has been long time a challenging problem in the theory of quantum entanglement. We propose a criterion for detecting genuine tripartite entanglement of arbitrary dimensional tripartite states based on quantum Fisher information. We show that this criterion is more effective for some states in detecting genuine tripartite entanglement by detailed example.
Entanglement of multi-qubit quantum graph states and studies structural properties of tripartite graphs with quantum computing
本文提出一种构建多粒子纠缠量子态的方法，用于表示加权三元图。通过计算多粒子状态的纠缠距离，研究了三元图结构的特性，并利用量子编程探讨了其应用。
The quantum discord of bipartite systems is one of the best-known measures of non-classical correlations and an important quantum resource. In the recent work appeared in [Phys. Rev. Lett 2020, 124:110401], the quantum discord has been generalized to multipartite systems. In this paper, we give analytic solutions of the quantum discord for tripartite states with fourteen parameters.
We propose a direct measure of tripartite quantum correlation in an arbitrary symmetric pure state of N correlated two-level atoms (qubits). We compute the third order moments of the collective pseudo-spin operators in terms of the individual atomic operators of the atoms in the assembly and find that all the bipartite quantum correlation terms among the atoms cancel out leaving only the all possible tripartite quantum correlation terms among them. We observe that the third order moments are made up of solely the tripartite quantum correlations among the atoms. This helps to extract out only t…
The role of entanglement is discussed in the Harrow-Hassidim-Lloyd (HHL) algorithm. We compute all tripartite entanglement at every steps of the HHL algorithm. The tripartite entanglement is generated in the first quantum phase estimation (QPE) step. However, it turns out that amount of the generated entanglement is not maximal except very rare cases. In the second rotation step some tripartite entanglement is annihilated. Thus, the net tripartite entanglement is diminished. At the final inverse-QPE step the matrix inversion task is completed at the price of complete annihilation of the entang…
An interesting monogamy equation with the form of Pythagorean theorem is found for $2\otimes 2\otimes n$-dimensional pure states, which reveals the relation among bipartite concurrence, concurrence of assistance, and genuine tripartite entanglement. At the same time, a genuine tripartite entanglement monotone as a generalization of 3-tangle is naturally obtained for $(2\otimes 2\otimes n)$- dimensional pure states in terms of a distinct idea. For mixed states, the monogamy equation is reduced to a monogamy inequality. Both results for tripartite quantum states can be employed to multipartite q…
We define genuine total, classical and quantum correlations in tripartite systems. The measure we propose is based on the idea that genuine tripartite correlation exists if and only if the correlation between any bipartition does not vanish. We find in a symmetrical tripartite state, for total correlation and classical correlation, the genuine tripartite correlations are no less than pair-wise correlations. However, the genuine quantum tripartite correlation can be surpassed by the pair-wise quantum correlations. Analytical expressions for genuine tripartite correlations are obtained for pure …
We investigate economic protocol to securely distribute and reconstruct a single-qubit quantum state between two users via a tripartite entangled state in cavity QED. Our scheme is insensitive to both the cavity decay and the thermal field.
Quantum discord and quantum uncertainty are two important features of the quantum world. In this work, the relation between entropic uncertainty relation and the shareability of quantum discord is studied. First, by using tripartite quantum-memory-assisted entropic uncertainty relation, an upper bound for the shareability of quantum discord among different partites of a composite system is obtained. It is also shown that, for a specific class of tripartite states, the obtained relation could be expressed as monogamy of quantum discord. Moreover, it is illustrated that the relation could be gen…
From an operational criterion of physical reality, a quantifier of realism-based nonlocality was recently introduced for two-part quantum states. This measure has shown to capture aspects that are rather different from Bell nonlocality. Here we take a step further and introduce a tripartite realism-based nonlocality quantifier. We show that this measure reduces to genuine tripartite entanglement for a certain class of pure tripartite states and manifests itself in correlated mixed states even in the absence of quantum correlations. A case study for noisy GHZ and W states points out the existen…
Quantum correlations of tripartite mixed states in the black hole quantum atmosphere
研究黑洞量子大气中三方混合态的量子关联，通过引入哈特尔 - 霍金局部温度刻画局部霍金效应影响，发现相关量极值在近视界区域，且三方非定域性更脆弱，为三方混合态相关特性提供统一刻画。
We construct the tripartite Bell-type inequalities of product states for l1-norm of coherence, relative entropy of coherence and skew information. Some three-qubit entangled states violate these inequalities. Particulary, the tripartite Bell-type inequality for relative entropy of coherence is always violated by the W class pure or mixed states as well as the GHZ class pure or mixed states, being used as entanglement witness.
We present an interesting monogamy equation for $(2 \otimes 2 \otimes n)$-dimensional pure states, by which a quantity is found to characterize the tripartite entanglement with the GHZ type and W typeentanglements as a whole. In particular, we, for the first time, reveals that for any quantum state of a pair of qubits, the difference between the two remarkable entanglement measures, concurrence and negativity, characterizes the W type entanglement of tripartite pure states with the two-qubit state as reduced density.
Polygamy relations for tripartite and multipartite quantum systems
本文研究了三方与多方量子系统的一妻多夫性质，在三方系统中构建了解集并找到其下界作为辅助量子纠缠具有一妻多夫性质的充要条件，在多方系统中给出了n-qubit系统两种划分的广义定义并构建了相关不等式，同时利用几何图形解释了这些关系。
The equivalence of tripartite pure states under local unitary transformations is investigated. The nonlocal properties for a class of tripartite quantum states in $\Cb^K \otimes \Cb^M \otimes \Cb^N$ composite systems are investigated and a complete set of invariants under local unitary transformations for these states is presented. It is shown that two of these states are locally equivalent if and only if all these invariants have the same values.
We show how two distrustful parties, "Bob" and "Charlie", can share a secret key with the help of a mutually trusted "Alice", counterfactually - that is with no information-carrying particles travelling between any of the three parties.
Owing to their long-lifetimes at cryogenic temperatures, mechanical oscillators are recognized as an attractive resource for quantum information science and as a testbed for fundamental physics. Key to these applications is the ability to prepare, manipulate and measure quantum states of mechanical motion. Through an exact formal solution to the Schrodinger equation, we show how tripartite optomechanical interactions, involving the mutual coupling between two distinct optical modes and an acoustic resonance enables quantum states of mechanical oscillators to be synthesized and interrogated.
The upper bound of the fine-grained uncertainty relation is different for classical physics, quantum physics and no-signaling theories with maximal nonlocality (supper quantum correlation), as was shown in the case of bipartite systems [J. Oppenheim and S. Wehner, Science 330, 1072 (2010)]. Here, we extend the fine-grained uncertainty relation to the case of tripartite systems. We show that the fine-grained uncertainty relation determines the nonlocality of tripartite systems as manifested by the Svetlichny inequality, discriminating between classical physics, quantum physics and super quantum…
We address the balance of quantum correlations for continuous variable (CV) states. In particular, we consider a class of feasible tripartite CV pure states and explicitly prove two Koashi-Winter-like conservation laws involving Gaussian entanglement of formation, Gaussian quantum discord and sub-system Von Neumann entropies. We also address the class of tripartite CV mixed states resulting from the propagation in a noisy environment, and discuss how the previous equalities evolve into inequalities.
A global measure of quantum correlations for tripartite nonorthogonal states is presented. It is introduced as the overall average of the pairwise correlations existing in all possible partitions. The explicit expressions for the global measure are derived for squared concurrence, entanglement of formation, quantum discord and its geometric variant. As illustration, we consider even and odd three-mode Schrödinger cat states based on Glauber coherent states. We also discuss limitations to sharing quantum correlations known as monogamy relations.
We show that for tripartite quantum pure states of qubits, all the kinds of entanglement in terms of SLOCC classification are experimentally measurable by simple projective measurements, provided that four copies of the composite quantum system are available. In particular, the entanglement of reduced density matrices, even though they are mixed states, can be exactly determined in experiment. Concurrence of assistance is also shown to be measurable by introducing an interesting equations with explicit physical meanings.
We review some current ideas about tripartite entanglement, the case representing the next level of complexity beyond the simplest one (though far from trivial), namely the bipartite. This kind of entanglement has an essential role in the understanding of foundations of quantum mechanics. Also, it allows several applications in the fields of quantum information processing and quantum computing. In this paper, we make a revision about the main foundational aspects of tripartite entanglement and we discuss the possibility of using it as a resource to execute quantum protocols. We present some ex…
We investigate schemes for quantum secret sharing and quantum dense coding via tripartite entangled states. We present a scheme for sharing classical information via entanglement swapping using two tripartite entangled GHZ states. In order to throw light upon the security affairs of the quantum dense coding protocol, we also suggest a secure quantum dense coding scheme via W state in analogy with the theory of sharing information among involved users.
Quantum Zeno Effect on Genuine Tripartite Nonlocality and Entanglement in Quantum Dissipative System
本文通过数值方法高精度量化任意三量子比特态的真正三体非定域性，以 W 态为例研究其与真正三体纠缠的动力学及互补性，并提出利用量子芝诺效应在退相干下保护这些关联的策略。
The notion of entanglement is the most well-known nonclassical correlation in quantum mechanics and a fundamental resource in quantum information and computation. This correlation, which is displayed by certain classes of quantum states, is of utmost importance when dealing with protocols such as quantum teleportation, cryptography, and quantum key distribution. In this paper, we exploit a classification of tripartite entanglement by introducing the concepts of entangled hypergraph and evolving entangled hypergraph as data structures suitable to model quantum protocols which use entanglement. …
We demonstrate a multipartite protocol to securely distribute and reconstruct a quantum state. A secret quantum state is encoded into a tripartite entangled state and distributed to three players. Any two of the three players are able to reconstruct the state, whilst individual players obtain nothing. We characterize this (2,3) threshold quantum state sharing scheme in terms of fidelity, signal transfer and reconstruction noise. We demonstrate a fidelity averaged over all reconstruction permutations of 0.73, which is achievable only using quantum resources.
We propose a more general method for detecting a set of entanglement measures, i.e. negativities, in an \emph{arbitrary} tripartite quantum state by local operations and classical communication. To accomplish the detection task using this method, three observers, Alice, Bob and Charlie, do not need to perform the partial transposition maps by the structural physical approximation; instead, they are only required to collectively measure some functions via three local networks supplemented by a classical communication. With these functions, they are able to determine the set of negativities rela…
To quantify non-Markovianity of tripartite quantum states from an operational viewpoint, we introduce a class $Ω^*$ of operations performed by three distant parties. A tripartite quantum state is a free state under $Ω^*$ if and only if it is a quantum Markov chain. We introduce a function of tripartite quantum states that we call the non-Markovianity of formation, and prove that it is a faithful measure of non-Markovianity, which is continuous and monotonically nonincreasing under a subclass $Ω$ of $Ω^*$. We consider a task in which the three parties generate a non-Markov state from scratch by…
The study of classical and quantum correlations in bipartite and multipartite systems is crucial for the development of quantum information theory. Among the quantifiers adopted in tripartite systems, the genuine tripartite quantum discord (GTQD), estimating the amount of quantum correlations shared among all the subsystems, plays a key role since it represents the natural extension of quantum discord used in bipartite systems. In this paper, we derive an analytical expression of GTQD for three-qubit systems characterized by a subclass of symmetrical X-states. Our approach has been tested on b…
We establish relations between tripartite pure state entanglement and additivity properties of the bipartite relative entropy of entanglement. Our results pertain to the asymptotic limit of local manipulations on a large number of copies of the state. We show that additivity of the relative entropy would imply that there are at least two inequivalent types of asymptotic tripartite entanglement. The methods used include the application of some useful lemmas that enable us to analytically calculate the relative entropy for some classes of bipartite states.
Quantumness for a bipartite unsteerable quantum correlation is operationally characterized by the notion of super-unsteerability. Super-unsteerability refers to the requirement of a larger dimension of the random variable that the steering party has to preshare with the party to be steered in the classical simulation protocol to generate an unsteerable correlation than the local Hilbert space dimension of the quantum states (reproducing the given unsteerable correlation) at the steering party's side. In the present study, this concept of super-unsteerability is generalized by defining the noti…
Usually, the three-tangle of a tripartite pure state of qubits can be directly measured with the simultaneous preparation of a not-less-than-four-fold copy of the state. We show that the exact genuine tripartite entanglement for ($2\otimes2\otimes n$)-dimensional pure quantum states can be measured in a similar manner, provided that only two simultaneous copies of the state are available. Lower bounds are also proposed for more convenient experimental operations. As an example, a comprehensive demonstration of the scheme is provided for the three-tangle of a three-qubit state.
Observation of Non-Markovian Evolution of Tripartite Quantum Steering
研究通过混合态观测三粒子量子引导的非马尔可夫演化，揭示其死亡与复兴过程及非对称引导结构，为多粒子量子引导的层次和方向结构提供基础见解。
The influence of thermal noise on bipartite and tripartite quantum steering induced by a short laser pulse in a hybrid three-mode optomechanical system is investigated. The calculation is carried out under the bad cavity limit, the adiabatic approximation of a slowly varying amplitude of the cavity mode, and with the assumption of driving the cavity mode with a blue detuned strong laser pulse. Under such conditions, explicit expressions of the bipartite and tripartite steering parameters are obtained, and the concept of collective tripartite quantum steering, recently introduced by He and Reid…
Quantum scrambling via accessible tripartite information
本文指出标准三体信息因依赖量子互信息而无法可靠量化量子信息置乱，提出基于可及互信息的可及三体信息作为替代，并通过实例证明其有效性，为深入理解量子置乱开辟了新方向。
In quantum resource theories (QRTs), there exists evidences of intrinsic connections among different measures of quantum resources, including entanglement, coherence, quantum steering, and so on. However, building the relations among different quantum resources is a vital yet challenging task in multipartite quantum systems. Here, we focus on a unified framework of interpreting the interconversions among different quantum resources in tripartite systems. In particular, an exact relation between the generalized geometric measure and the genuinely multipartite concurrence are derived for tripart…
For a tripartite pure state superposed by two individual states, the bipartitely shared entanglement can always be achieved by local measurements of the third party. Consider the different aims of the third party, i.e. maximizing or minimizing the bipartitely shared entanglement, we find bounds on both the possible bipartitely shared entanglement of the superposition state in terms of the corresponding entanglement of the two states being superposed. In particular, by choosing the concurrence as bipartite entanglement measure, we obtain calculable bounds for tripartite $(2\otimes 2\otimes n)$ …
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