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Continuous-Variable Gaussian Analog of Cluster States

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Continuous-Variable Gaussian Analog of Cluster States PHYSICAL REVIEW A 73, 032318 ͑2006͒ Continuous-variable Gaussian analog of cluster states Jing Zhang1,* and Samuel L. Braunstein2 1State Key Laboratory of Quantum Optics and Quantum Optics Devices, Institute of Opto-Electronics, Shanxi University, Taiyuan 030006, People’s Republic of China 2Computer Science, University of York, York YO10 5DD, United Kingdom ͑Received 21 October 2005; published 16 March 2006͒ We present a continuous-variable ͑CV͒ Gaussian analog of cluster states, a new class of CV multipartite entangled states that can be generated from squeezing and quantum nondemolition coupling HI = ប␹XAXB. The entanglement properties of these states are studied in terms of classical communication and local operations. The graph states as general forms of the cluster states are presented. A chain for a one-dimensional example of cluster states can be readily experimentally produced only with squeezed light and beam splitters. DOI: 10.1103/PhysRevA.73.032318 PACS number͑s͒: 03.67.Hk, 03.65.Ta I. INTRODUCTION ment. The study of entanglement properties of the harmonic chain has aroused great interest ͓12͔, which is in direct anal- Entanglement is one of the most fascinating features of ogy to a spin chain with an Ising interaction. To date, the quantum mechanics and plays a central role in quantum- study and application of CV multipartite entangled states has information processing. In recent years, there has been an mainly focused on CV GHZ-type states. However, a richer ongoing effort to characterize qualitatively and quantitatively understanding and more definite classification of CV multi- the entanglement properties of multiparticle systems and ap- partite entangled states are needed for developing a CV ply them in quantum communication and information. The quantum-information network. In this paper, we introduce a study of multipartite entangled states in the discrete-variable class of CV Gaussian multipartite entangled states, CV clus- regime has shown that there exist different types of entangle- terlike states, which are different from CV GHZ-like states. ment, inequivalent up to local operations and classical com- We propose an interaction model to realize CV clusterlike ͓ ͔͑ ͒ munication 1 LOCC . For example, it is now well known states and compare these states to GHZ-like states in terms of ͑ ͒ that the Greenberger-Horne-Zeilinger GHZ and W states LOCC. It is worth noting that a CV Gaussian cluster state in are not equivalent up to LOCC. Recently, Briegel and Raus- our protocol cannot be used in universal quantum computers sendorf introduced a special kind of multipartite entangled over continuous variables ͓13͔; however, it may be applied in states, the so-called cluster states, which can be created via quantum network communication as a different type of mul- ͓ ͔ an Ising Hamiltonian 2 . It has been shown that via cluster tipartite entanglement. states, one can implement a quantum computer on a lattice of qubits. In this proposal, which is known as a “one-way quan- tum computer,” information is written onto the cluster and II. CV GAUSSIAN CLUSTER STATE then is processed and read out from the cluster by one-qubit measurements ͓3͔. We consider an N-mode N-party system described by a set ͑ ͒ In recent years, quantum communication, or more gener- of quadrature operators R= X1 , P1 ;X2 , P2 ,...;X1 , Pn obey- ͓ ͔ ␦ ally quantum information with continuous variables ͑CVs͒, ing the canonical commutation relation Xl , Pk =i lk. The has attracted a lot of interest and appears to yield very prom- quantum nondemolition ͑QND͒ coupling Hamiltonian is em- ប␹ ising perspectives concerning both experimental realizations ployed in the form HI = XlXk, so in this process and general theoretical insights ͓4͔, due to its relative sim- quadrature-amplitude ͑“position”͒ and quadrature-phase plicity and high efficiency in the generation, manipulation, ͑“momentum”͒ operators are transformed in the Heisenberg and detection of the CV state. The investigation of CV mul- picture according to the following expressions: tipartite entangled states has made significant progress in XЈ = X , PЈ = P + gX , theory and experiment. The quantification and scaling of l l l l k multipartite entanglement has been developed by different Ј Ј ͑ ͒ methods, for example, by determining the necessary and suf- Xk = Xk, Pk = Pk + gXl, 1 ͓ ͔ ficient criteria for their separability 5–7 , and by the en- g ␹t ␹ t ͓ ͔ where =− is the gain of the interaction, and and are tanglement of formation 8 . Multipartite quantum protocols the coupling coefficient and the interaction time, respec- were proposed and performed experimentally, such as quan- ͓ ͔ ͓ ͔ tively. The important feature of this Hamiltonian is that the tum teleportation networks 9 , controlled dense coding 10 , momentum P and P pick up the information of the position and quantum secret sharing ͓11͔ based on tripartite entangle- l k Xk and Xl, respectively, while the position remains un- changed. The QND coupling of light was widely investigated in many previous experiments ͓14͔ and recently has been *Corresponding author. Email address: [email protected]; experimentally observed between light and the collective [email protected] spin of atomic samples ͓15͔. 1050-2947/2006/73͑3͒/032318͑5͒/$23.00032318-1 ©2006 The American Physical Society J. ZHANG AND S. L. BRAUNSTEIN PHYSICAL REVIEW A 73, 032318 ͑2006͒ Considering first a one-dimensional example of a chain of product state. From this point of view, it is impossible to N modes which are numbered from 1 to N with next- destroy all the entanglement of the cluster if fewer than neighbor interaction. Initially, all modes are prepared in the int ͑N/2͒ parties are traced out ͑discarded͒. But only if one +r ͑0͒ −r ͑0͒ quadrature-phase squeezed state Xl =e Xl , Pl =e Pl , party for the GHZ state is traced, will the remaining state be where r is the squeezing parameter and the superscript ͑0͒ completely unentangled. Note that this is not true in the case denotes initial vacuum modes. Applying QND coupling to of finite squeezing. For example, if three weakly squeezed next-neighbor modes of a chain at different or the same time vacuum states are used to generate a tripartite GHZ state, the yields a CV clusterlike state in the form state is a fully inseparable tripartite entangled state, but the remaining bipartite state after tracing out one of the three C +r ͑0͒ C −r ͑0͒ +r ͑0͒ X1 = e X1 , P1 = e P1 + e X2 , subsystems is still entangled ͓9,10͔. ӇӇ Next, we investigate the entanglement properties of the ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ quantum teleportation network. In other words, we will an- XC = e+rX 0 , PC = e−rP 0 + e+rX 0 + e+rX 0 , ͑2͒ i i i i i−1 i+1 swer the question of how many parties need to be measured ӇӇ from an N-partite entangled state to make bipartite entangle- C +r ͑0͒ C −r ͑0͒ +r ͑0͒ ment between any two of the N parties “distilled,” which XN = e XN , PN = e PN + e XN−1. enables quantum teleportation. We first show that the parties Here, without loss of generality, we put the gain of the inter- at the ends of the chain, i.e., parties 1 and N, can be brought action g=1. For simplicity we discuss the properties of CV into bipartite entanglement by measuring the parties clusterlike states in the ideal case r→ϱ corresponding to 2,...,N−1. After measurement of the momentum of party 2, infinite squeezing. When N=2, one obtains a state with total the remaining parties are identical to an entangled chain of C C → C C → position X1 +X2 0 and relative momentum P1 − P2 0by length N−1 when one displaces the position of party 1 as ؠ C C C applying a local −90 rotation transformation X → P , P XЈC =XC − PC and measuring the result PC on party 2, then C 1 1 2 2 →−X on mode 2, which corresponds to Einstein-Podolsky- applies a local rotation transformation XC →−PC, PC →XC Rosen or two-mode squeezed states. The entanglement of a on parties 1. We can repeat this procedure and measure party state is not affected by local unitary transformation. It obvi- 3, and so on. At the end, party 1 and N are brought into a ously exhibits maximum bipartite entanglement. Similarly, bipartite entanglement. To bring any two parties j,k ͑jϽk͒ C C one obtains the state for N=3 with total position X1 +X2 from the chain ͕1,2,...,N͖ into bipartite entanglement, we C → C C → ͑ ͒ +X3 0 and relative momentum Pi − Pj 0 i, j=1,2,3 , first measure the position of the “outer” parties j−1 and k which corresponds to a CV three-mode GHZ-like state ͓9͔. ЈC +1, then displace the momentum of parties j and k as Pj However, quantum correlation of position and momentum of = PC −XC and PЈC = PC −XC , which projects the parties C C C → C j j−1 k k k+1 the state for the N=4 case become X1 +X2 +X3 0, X3 j, j+1,...,k into an entangled chain of length k− j+1. This C → C C → C C C → +X4 0 and P1 − P2 0, P2 − P3 + P4 0 by applying a lo- ;”ؠ process that breaks chain into parts is called “disconnection cal −90 rotation transformation on modes 2 and 4. Clearly, a for example, one breaks a chain into two independent chains CV four-partite clusterlike state is not equivalent to a GHZ- when measuring the position of party j and displacing the → like state with total position X1 +X2 +X3 +X4 0 and relative momentum of parties j−1 and j+1 as PЈ C = PC −XC and → ͑ ͒ j−1 j−1 j momentum Pi − Pj 0 i, j=1,2,3,4 .
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