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arXiv:quant-ph/0205068

Multipartite entanglement

Peter van Loock, Samuel L. Braunstein

Published May 13, 2002• 2 citations
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Abstract

First, we show how the quantum circuits for generating and measuring multi-party entanglement of qubits can be translated to continuous quantum variables. We derive sufficient inseparability criteria for $N$-party continuous-variable states and discuss their applicability. Then, we consider a family of multipartite entangled states (multi-party multi-mode states with one mode per party) described by continuous quantum variables and analyze their properties. These states can be efficiently generated using squeezed light and linear optics.

DOI
10.48550/arxiv.quant-ph/0205068

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First, we show how the quantum circuits for generating and measuring multi-party entanglement of qubits can be translated to continuous quantum variables. We derive sufficient inseparability criteria for NNN-party continuous-variable states and discuss their applicability. Then, we consider a family of multipartite entangled states (multi-party multi-mode states with one mode per party) described by continuous quantum variables and analyze their properties. These states can be efficiently generated using squeezed light and linear optics.
It is known that even a single one-mode squeezed state incident on a beamsplitter is sufficient to produce bipartite (pure-state) entanglement with respect to the field quadratures. We show that such entanglement enables quantum teleportation with a fidelity better than could be achieved in any classical scheme. Furthermore, using a continuous-variable teleportation protocol involving more than two parties, we show that a single one-mode squeezed state and linear optics is sufficient to produce a truly N-partite entangled state for arbitrarily many parties, provided that a minimal amount of squeezing is available. This multipartite entanglement may be made arbitrarily good using more and stronger one-mode squeezed states.
We show that one single-mode squeezed state distributed among N parties using linear optics suffices to produce a truly N-partite entangled state for any nonzero squeezing and arbitrarily many parties. From this N-partite entangled state, via quadrature measurements of N-2 modes, bipartite entanglement between any two of the N parties can be "distilled," which enables quantum teleportation with an experimentally determinable fidelity better than could be achieved in any classical scheme.
Multipartite entanglement detection is crucial for the develop of quantum information science and quantum computation, communication, simulation and metrology tasks. In contrast to experiments, where several handreds of qubits have been entangled to build a quantum computer or a quantum simulator, theoretically only for lower dimensional quantum system such as two qubit system or two-mode Gaussian state and some special quantum states, efficient criteria have been developed to detect the entanglement. An efficient and practical entanglement detection method is anticipated to appear for current scalable quantum system. Based on the matched entanglement witness scheme and integrated uncertainty relations, we propose a quite general entanglement detection method for all kinds of multipartite entanglement of multimode continuous variable systems. With the covariance matrix of any randomly generated multimode continuous variable quantum state, our criterion can detect entanglement, genuine entanglement and other kinds of inseparabilities almost imidiately.
We address the problem of the persistence of entanglement of quantum light under mode transformations, where orthogonal modes define the parties between which quantum correlations can occur. Since the representation of a fixed photonic quantum state in different optical mode bases can substantially influence the entanglement properties of the said state, we devise a constructive method to obtain families of states with the genuine feature of remaining entangled for any choice of mode decomposition. In the first step, we focus on two-photon states in a bipartite system and optimize their entanglement properties with respect to unitary mode transformations. Applying a necessary and sufficient entanglement witness criteria, we are then able to prove that the class of constructed states is entangled for arbitrary mode decompositions. Furthermore, we provide optimal bounds to the robustness of the mode-independent entanglement under general imperfections. In the second step, we demonstrate the power of our technique by showing how it can be straightforwardly extended to higher-order photon numbers in multipartite systems, together with providing a generally applicable and rigorous definition of mode-independent separability and inseparability for mixed states.
Quantum entanglement shared by more than two partites is the essential bases for developing quantum communication networks and quantum computation. The multipartite entanglement of continuous variables can be generated by distributing or combining squeezed states of light using linear optics. We experimentally obtained a tripartite entangled state of bright optical field by means of a nondegenerate optical parametric amplifier (NOPA) and optical beam-splitters. The NOPA is operated in the state of parametric deamplification by locking the relative phase between the pump laser and the injected seed wave to (2n+1)π, so the generated Einstein-Podolsky-Rosen (EPR) beam for continuous variables is an entangled state of light with the quantum correlation of phase quadratures and the quantum anticorrelation of amplitude quadratures [1]. The output EPR beam consists of two frequency-degenerate optical modes with horizontal and vertical polarizations which are distributed to three parties with two linear beam-splitters. We have theoretically and experimentally demonstrated the three parties are in a tripartite entangled state with three-mode “position” (amplitude quadrature) correlation and relative “momentum” (phase quadrature) correlation [2]-[4]. The obtained GHZ-like state is a fully inseparable tripartite continuous-variables state violating the three inequalities used for the criterion of the inseparability [5][6]. The controlled dense coding quantum communication is experimentally accomplished by exploiting the tripartite entanglement. The three entangled modes are sent to a sender (Alice), a receiver (Bob) and a controller (Claire), respectively. The information transmission capacity of the quantum channel between Alice and Bob is controlled by Claire. The channel capacity completed under Claire’s help is always larger than that without her help.
We suggest two schemes to generate bipartite entangled states by means of a quantum measurement at a third party. The two parties to be entangled have separate entangled states with the third party in modes C1 and C2. In our first scheme we generate entanglement between the two remote parties by considering the modes C1 and C2 indistinguishable. However, in the second scheme we generate entangled states by considering the two modes to be distinguishable. We discuss that the first scheme of remote entanglement generation can be extended to any N number of parties. On making a quantum measurement on this system, we develop quantum networks, based on W-states and other multipartite symmetric entangled states. 1