X. Wang, P. Zanardi
We study the entanglement of unitary operators on $d_1\times d_2$ quantum systems. This quantity is closely related to the entangling power of the associated quantum evolutions. The entanglement of a class of unitary operators is quantified by the concept of concurrence.
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I explore entanglement dynamics in a three qubit system comparing the ability of entanglement witnesses to detect tri-partite entanglement to the phenomenon of entanglement sudden death (ESD). Using a system subject to dephasing I invoke entanglement witnesses to detect tri-partite GHZ and W-type entanglement and compare the evolution of their detection capabilites with the evolution of the negativity, bi-partite concurrence, and tri-partite negativity. Interestingly, I find a state in which there is no concurrence or tri-partite negativity but there is entanglement. Finally, I utilize a three…
Two pure states of a multi-partite system are alway are related by a unitary transformation acting on the Hilbert space of the whole system. This transformation involves multi-partite transformations. On the other hand some quantum information protocols such as the quantum teleportation and quantum dense coding are based on equivalence of some classes of states of bi-partite systems under the action of local (one-particle) unitary operations. In this paper we address the question: ``Under what conditions are the two states states, $\varrho$ and $σ$, of a multi-partite system locally unitary eq…
A $(k+1)$-partite entanglement measure of $N$-partite quantum states
本文提出N体量子系统的(k+1)体纠缠度量，并利用置换不变部分给出其强界及两种有效可测程度定义，通过具体例子验证了方法的有效性。
We develop protocols for preparing a GHZ state and, in general,a pure multi-partite maximally entangled state in a distributed network with apriori quantum entanglement between agents using classical communication and local operations. We investigate and characterize the minimal combinatorics of the sharing of EPR pairs required amongst agents in a network for the creation of multi-partite entanglement. We also characterize the minimal combinatorics of agents in the creation of pure maximal multi-partite entanglement amongst the set $N$ of $n$ agents in a network using apriori multi-partite en…
We derive hierarchies of separability criteria that identify the different degrees of entanglement ranging from bipartite to genuine multi-partite in mixed quantum states of arbitrary size.
Identifying the $k$-partite entanglement and $k$-nonseparability of general $N$-partite quantum states are fundamental issues in quantum information theory. By use of computable inequalities of nonlinear operators, we present some simple and powerful $k$-partite entanglement and $k$-nonseparability criteria that works very well and allow for a simple and inexpensive test for the whole hierarchy of $k$-partite entanglement and $k$-separability of $N$-partite systems with $k$ running from $N$ down to 2. We illustrate their strengths by considering several examples in which our criteria perform b…
We theoretically show that bright six-partite continuous-variable entanglement can be generated using cascaded four-wave mixing effects of third-order nonlinearity atomic systems above threshold. The six-partite continuous-variable entanglement among the six cavity fields with different frequencies is analyzed by applying optimized inseparability criteria proposed by Van Loock and Furusawa.
We show how multi-partite entanglement, such as of W-state form, can be created in branched spin chain systems. We also discuss the preservation of such entanglement, once created. The technique could be applied to actual spin chain systems, or to other physical systems such as strings of coupled quantum dots, molecules or atoms.
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 is explored for bi-partite and multi-partite pure and mixed states. It is determined analytically for arbitrary two-qubit mixed states, generalized Werner, and isotropic states, and is also applied to certain multi-partite mixed states, including two distinct multi-partite bound entangled states. Moreover, the ground-state entanglement of the XY model in a transverse field is calculated and shown to exhibit singular behavio…
We present a practical classification scheme for the four-partite entangled states under stochastic local operations and classical communication (SLOCC). By transforming a four-partite state into a triple-state set composed of two tripartite and one bipartite states, the entanglement classification is reduced to the classification of tripartite and bipartite entanglements. This reduction method has the merit of involving only the linear constrains, and meanwhile providing an insight into the entanglement character of the subsystems.
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…
Complete $k$-partite entanglement measure
本文提出完全k-部分纠缠度量的公理化公设，并展示两类k-部分纠缠度量，其中一类是完全的，另一类在k=2时为统一但非完全的。
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…
Conditions for 3-partite and 4-partite genuine entanglement
本文研究了三体和四体系统中真正纠缠的判定条件，提出基于双体纠缠的额外充分条件，用于检测系统是否为真正纠缠。
We show that vacuum induced coherence in three level atomic systems can lead to preservation of bi-partite entanglement, when two such atoms are prepared as two initially entangled qubits each independently interacting with their respective vacuum reservoirs. We explicitly calculate the time evolution of concurrence for two different Bell states and show that a large amount of entanglement can survive in the long time limit. The amount of entanglement left among the two qubits depends strongly on the ratio of the non- orthogonal transitions in each qubit and can be more than 50%
Following previous work, we distinguish between genuine $N$-partite entanglement and full $N$-partite inseparability. Accordingly, we derive criteria to detect genuine multipartite entanglement using continuous variable (position and momentum) measurements. Our criteria are similar but different to those based on the van Loock-Furusawa inequalities, which detect full $N$-partite inseparability. We explain how the criteria can be used to detect the genuine $N$-partite entanglement of continuous variable states generated from squeezed and vacuum state inputs, including the continuous variable Gr…
Detecting $k$-nonseparability and $k$-partite Entanglement with Generalized Skew Information and Mutually Unbiased Measurements
本文提出基于广义Wigner-Yanase斜信息和互无偏测量的充分判据，用于检测多体系统中的$k$-不可分性与$k$体纠缠，并通过实例验证其检测能力，且可应用于量子网络深度识别。
The Mermin inequality provides a criterion for experimentally ruling out local-realistic descriptions of multiparticle systems. A violation of this inequality means that the particles must be entangled, but does not, in general, indicate whether N-partite entanglement is present. For this, a stricter bound is required. Here we discuss this bound and use it to propose two different schemes for demonstrating N-partite entanglement with atoms. The first scheme involves Bose-Einstein condensates trapped in an optical lattice and the second uses Rydberg atoms in microwave cavities.
Multipartite entanglement detection via generalized Wigner-Yanase skew information
本文利用广义Wigner-Yanase斜信息构造两类不等式判据，实现对N体量子系统从2体到N体纠缠及不可分性的精细层级检测，并通过例子证明其能发现其他方法漏检的纠缠。
We study a set of $L$ spatial bosonic modes localized on a graph $Γ.$ The particles are allowed to tunnel from vertex to vertex by hopping along the edges of $Γ.$ We analyze how, in the exact many-body eigenstates of the system i.e., Bose-Einstein condensates over single-particle eigenfunctions, the bi-partite quantum entanglement of a lattice vertex with respect to the rest of the graph depends on the topology of $Γ.$
We investigate the concentration of multi-party entanglement by focusing on simple family of three-partite pure states, superpositions of Greenberger-Horne-Zeilinger states and singlets. Despite the simplicity of the states, we show that they cannot be reversibly concentrated by the standard entanglement concentration procedure, to which they seem ideally suited. Our results cast doubt on the idea that for each N there might be a finite set of N-party states into which any pure state can be reversibly transformed. We further relate our results to the concept of locking of entanglement of forma…
The ability to deterministically generate genuine multi-partite entanglement is fundamental for the advancement of quantum information science. We show that the interaction between entangled twin beams of light and an atomic ensemble under conditions for electromagnetically induced transparency leads to the generation of genuine hybrid tri-partite entanglement between the two input fields and the atomic ensemble. In such a configuration, the system is driven through dissipation to a steady state given by the hybrid entangled state. To show the presence of the genuine hybrid entanglement, we in…
We present local invariants of multi-partite pure or mixed states, which can be easily calculated and have a straight-forward physical meaning. As an application, we derive a new entanglement criterion for arbitrary mixed states of $n$ parties. The new criterion is weaker than the partial transposition criterion but offers advantages for the study of multipartite systems. A straightforward generalization of these invariants allows for the construction of a complete set of observable polynomial invariants.
We consider a quantum system consisting of N parts, each of which is a "quKit" described by a K dimensional Hilbert space. We prove that in the symmetric subspace, S, a pure state is not globally entangled, if and only if it is a coherent state. It is also shown that in the orthogonal complement all states are globally entangled.
We show that Einstein--Podolsky--Rosen--Bohm (EPR) and Greenberger--Horne--Zeilinger--Mermin (GHZ) states can not generate, through local manipulation and in the asymptotic limit, all forms of three--partite pure--state entanglement in a reversible way. The techniques that we use suggest that there may be a connection between this result and the irreversibility that occurs in the asymptotic preparation and distillation of bipartite mixed states.
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.
The laws of quantum mechanics allow for the distribution of a secret random key between two parties. Here we analyse the security of a protocol for establishing a common secret key between N parties (i.e. a conference key), using resource states with genuine N-partite entanglement. We compare this protocol to conference key distribution via bipartite entanglement, regarding the required resources, achievable secret key rates and threshold qubit error rates. Furthermore we discuss quantum networks with bottlenecks for which our multipartite entanglement-based protocol can benefit from network c…
We study the entanglement properties of random pure stabilizer states in spin-1/2 particles. For two contiguous groups of spins of arbitrary size we obtain a compact and exact expression for the probability distribution for the entanglement values between the two sets. This allows for exact derivations of the average entanglement and the degree of concentration of measure around this average. We also give simple bounds on these quantities. We find that for large systems the average entanglement is near maximal and the measure is concentrated around it.
We show that there is a unique maximal decomposition of a pure multi-partite (N>2) quantum state into a sum of states which are "locally orthogonal" in the sense that the local reduced state for a term in the sum lives in its own orthogonal subspace for each subsystem. Observers can make local measurements on any subsystem and determine which "branch" they are on. The Shannon entropy of the resulting branch weights defines a new measure of global, GHZ-like entanglement, which is insensitive to local pairwise entangling operations and vanishes when there is no piece of information recorded at e…
Bi-partite entanglement in multi-qubit systems cannot be shared freely. The rules of quantum mechanics impose bounds on how multi-qubit systems can be correlated. In this paper we utilize a concept of entangled graphs with weighted edges in order to analyze pure quantum states of multi-qubit systems. Here qubits are represented by vertexes of the graph while the presence of bi-partite entanglement is represented by an edge between corresponding vertexes. The weight of each edge is defined to be the entanglement between the two qubits connected by the edge, as measured by the concurrence. We pr…
Advantage of multi-partite entanglement for quantum cryptography over long and short ranged networks
本文证明了基于GHZ态变体的秘密共享协议在面对一般攻击时的安全性，并发现量子中继器能解锁多体纠缠在长距离量子密码学中相对点对点方法的性能优势。
Geometric genuine N-partite entanglement measure for arbitrary dimensions
本文提出了适用于任意维度多体系统的真正多体纠缠（GME）度量，基于concurrence正多边形棱锥体积构建度量方法，验证了GHZ态比W态更纠缠，并证明该方法能更好地表征多体纠缠。
We show how to create maximally entangled dressed states of a weakly interacting multi-partite quantum system by suitably tuning an external, periodic driving field. Floquet theory allows us to relate, in a transparent manner, the occurrence of entanglement resonances to avoided crossings in the spectrum of quasi-energies, tantamount of well-defined conditions for the controlled, resonant interaction of particles. We demonstrate the universality of the phenomenon for periodically driven, weakly interacting two-level systems, by considering different interaction mechanisms and driving profiles.…
We introduce inequalities for multi-partite entanglement, derived from the geometry of spin vectors. The criteria are constructed iteratively from cross and dot products between the spins of individual subsystems, each of which may have arbitrary dimension. For qubit ensembles the maximum violation for our inequalities is larger than that for the Mermin-Klyshko Bell inequalities, and the maximally violating states are different from Greenberger-Horne-Zeilinger states. Our inequalities are violated by certain bound entangled states for which no Bell-type violation has yet been found.
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 present a protocol for performing entanglement swapping with intense pulsed beams. In a first step, the generation of amplitude correlations between two systems that have never interacted directly is demonstrated. This is verified in direct detection with electronic modulation of the detected photocurrents. The measured correlations are better than expected from a classical reconstruction scheme. In the entanglement swapping process, a four--partite entangled state is generated. We prove experimentally that the amplitudes of the four optical modes are quantum correlated 3 dB below shot nois…
We study quantum entanglement distribution on networks with full-rank bi-partite mixed states linking qubits on nodes. In particular, we use entanglement swapping and purification to partially entangle widely separated nodes. The simplest method consists of performing entanglement swappings along the shortest chain of links connecting the two nodes. However, we show that this method may be improved upon by choosing a protocol with a specific ordering of swappings and purifications. A priori, the design that produces optimal improvement is not clear. However, we parametrize the choices and find…
In this paper, we focus on two different kinds of multipartite correlation, $k$-nonseparability and $k$-partite entanglement, both of which can describe the essential characteristics of multipartite entanglement. We propose effective methods to detect $k$-nonseparability and $k$-partite entanglement in terms of quantum Fisher information. We illustrate the significance of our results and show that they identify some $k$-nonseparability and $k$-partite entanglement that cannot be identified by known criteria by several concrete examples.
Strong quantum nonlocality and unextendibility without entanglement in $N$-partite systems with odd $N$
针对多体系统中无纠缠强非定域正交集是否存在的问题，本文利用$N$维超立方体分解，为所有奇数$N\geq 3$构造强非定域正交集和不可扩展乘积基，并应用于量子秘密共享等场景。
In this paper we show that sufficient multi-partite quantum entanglement helps in fair and unbiased election of a leader in a distributed network of processors with only linear classical communication complexity. We show that a total of $O(\log n)$ distinct multi-partite maximally entanglement sets (ebits) are capable of supporting such a protocol in the presence of nodes that may lie and thus be biased. Here, $n$ is the number of nodes in the network. We also demonstrate the difficulty of performing unbiased and fair election of a leader with linear classical communication complexity in the a…
Quantifying genuine entanglement is a crucial task in quantum information theory.In this work, we give an approach of constituting genuine $m$-partite entanglement measures from any bipartite entanglement and any $k$-partite entanglement measure, $3\leq k<m$. In addition, as a complement to the three-qubit concurrence triangle proposed in [Phys. Rev. Lett., 127, 040403], we show that the triangle relation is also valid for any continuous entanglement measure and system with any dimension. We also discuss the tetrahedron structure for the four-partite system via the triangle relation associated…
The theory of the asymptotic manipulation of pure bipartite quantum systems can be considered completely understood: The rates at which bipartite entangled states can be asymptotically transformed into each other are fully determined by a single number each, the respective entanglement entropy. In the multi-partite setting, similar questions of the optimally achievable rates of transforming one pure state into another are notoriously open. This seems particularly unfortunate in the light of the revived interest in such questions due to the perspective of experimentally realizing multi-partite …
We construct a method for verifying mode entanglement of N-mode W states. The ideal W state contains exactly one excitation symmetrically shared between N modes, but our method takes the existence of higher numbers of excitations into account, as well as the vacuum state and other deviations from the ideal state. Moreover, our method distinguishes between full N-party entanglement and states with M-party entanglement with M<N, including mixtures of the latter. We specialize to the case N=4 for illustrative purposes. In the optical case, where excitations are photons, our method can be implemen…
We present a tripartite three-level state that allows a secret sharing protocol among the three parties, or a quantum key distribution protocol between any two parties. The state used in this scheme contains entanglement even after one system is traced out. We show how to utilize this residual entanglement for quantum key distribution purposes, and propose a realization of the scheme using entanglement of orbital angular momentum states of photons.
Variational M-Partite Geometric Entanglement Algorithm
本文提出变分M部几何纠缠算法，通过定制单、多量子比特方的变分ansatz电路优化目标态与M部可分态重叠，可在近期设备上灵活量化任意划分的多体纠缠。
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.
Quantum state estimation of multi-partite single photon path entanglement via local measurements
本文提出一种仅用局域测量估计任意多体单光子路径纠缠态的方法，并通过重构3量子比特W态密度矩阵实验验证其可行性。
We show that all multi-partite pure states can, under local operations, be transformed into bi-partite pairwise entangled states in a "lossless fashion": An arbitrary distinguished party will keep pairwise entanglement with all other parties after the asymptotic protocol - decorrelating all other parties from each other - in a way that the degree of entanglement of this party with respect to the rest will remain entirely unchanged. The set of possible entanglement distributions of bi-partite pairs is also classified. Finally, we point out several applications of this protocol as a useful primi…
Why many-partite entanglement is essential for holography
本文论证多体纠缠在全息和全息量子纠错码中普遍存在，基于新度量真实多熵，并揭示了体IR重构与大量边界子区域上多体纠缠的联系。
The paper focuses on aspects of the measurement problem introducing quantum states (q-states) for measured and measuring systems. The link between non-interacting and interacting quantum systems is first look at. For two independent partite systems logical sums A+B stand for non-interacting q-systems; while a direct product space AxB gathers interacting states. However this latter should support physical q-states with base states that do not separately belong to either A nor B; the latter correspond to bridge states, namely entangled states that can perform as links (bridges) between A+B and A…
With the complexity of information tasks, the bipartite and tripartite entanglement can no longer meet our needs, and we need more entangled particles to process relativistic quantum information. In this paper, we study the genuine N-partite entanglement and distributed relationships for Dirac fields in the background of dilaton black holes. We present the general analytical expression including all physically accessible and inaccessible entanglement in curved spacetime. We find that the accessible N-partite entanglement exhibits irreversible decoherence as the increase of black hole's dilaton…
We show how to make event-ready multi-partite entanglement between qubits which may be encoded on photons or matter systems. Entangled states of matter systems, which can also act as single photon sources, can be generated using the entangling operation presented in quant-ph/0408040. We show how to entangle such sources with photon qubits, which may be encoded in the dual rail, polarization or time-bin degrees of freedom. We subsequently demonstrate how projective measurements of the matter qubits can be used to create entangled states of the photons alone. The state of the matter qubits is in…
Entanglement distillation is the process of concentrating entanglement from a given quantum state. We present a technique for distillation of bi-partite polarization entanglement using interferometry. This technique can be optimized to extract maximal entanglement from any pure or mixed entangled state. A model for this method is presented and in particular we present experimental results for pure states when using polarizing Mach-Zehnder interferometers. These experimentally distilled states always demonstrate an increased violation of Bell's inequality.
The problem of noise incidence on qubits taking part of bipartite entanglement-based protocols is addressed. It is shown that the use of a three-partite GHZ state and measurements instead of their EPR counterparts allows the experimenter to detect $2/3$ of the times whenever one of the qubits involved in the measurement is affected by bit-flip noise through the mere observation of unexpected outcomes in the teleportation and superdense coding protocols when compared to the ideal case. It is shown that the use of post-selection after the detection of noise leads to an enhancement in the efficie…
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 …
We construct a simplex for multipartite qubit states of even number n of qubits, which has the same geometry concerning separability, mixedness, kind of entanglement, amount of entanglement and nonlocality as the bipartite qubit states. We derive the entanglement of the class of states which can be described by only three real parameters with the help of a multipartite measure for all discrete systems. We prove that the bounds on this measure are optimal for the whole class of states and that it reveals that the states possess only n-partite entanglement and not e.g. bipartite entanglement. We…
We apply the axiomatic approach to thermodynamics presented by Giles to derive a unique measure of entanglement for bi-partite pure states. This implies that local manipulations of entanglement in quantum information theory and adiabatic transformations of states in thermodynamics have the same underlying mathematical structure. We discuss possible extensions of our results to mixed and multi-partite states.
A classification of N-partite states, based on K-way negativities (K=2 to N), is proposed. The K-way partial transpose with respect to a subsystem is defined so as to shift the focus to K-way coherences instead of K subsystems of the composite system. For an N-partite system, the fraction of K-way negativity contributing to global negativity, is obtained. After minimizing K-way negativities through local unitary qubit rotations, a combined analysis of 2-way, 3-way and global negativities is shown to provide distinct measures of genuine tripartite, W-state like and bipartite entanglement, for t…
We consider bi-linear analogues of $s$-positivity for linear maps. The dual objects of these notions can be described in terms of Schimdt ranks for tri-tensor products and Schmidt numbers for tri-partite quantum states. These tri-partite versions of Schmidt numbers cover various kinds of bi-separability, and so we may interpret witnesses for those in terms of bi-linear maps. We give concrete examples of witnesses for various kinds of three qubit entanglement.
We propose to produce five-partite entanglement via cascaded four-wave mixing in a high-Q microresonator that may become a key to future one-way quantum computation on chip. A theoretical model is presented for the underlying continuous-variable entanglement among the generated comb modes that is expansible to more complicated scenarios. We analyze the entanglement condition when the van Loock and Furusawa criteria are violated, and discuss the device parameters for potential experimental realization that may be utilized to build an integrated compact five-partite entanglement generator. The p…
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