# Experimental preparation of eight-partite linear and two-diamond shape cluster states for photonic qumodes

###### Abstract

The preparation of multipartite entangled states is the prerequisite for exploring quantum information networks and quantum computation. In this letter, we present the first experimental demonstration of eight-partite spatially separated CV entangled states. The initial resource quantum states are eight squeezed states of light, through the linearly optical transformation of which two types of the eight-partite cluster entangled states are prepared, respectively. The generated eight entangled photonic qumodes are spatially separated, which provide valuable quantum resources to implement more complicated quantum information task.

###### pacs:

03.67.Bg, 03.67.Lx, 03.65.Ud, 42.50.DvDeveloping quantum information (QI) science have exhibited unusual potentiality Nielsen2000 (); Brau (). Optical QI based on exploiting discrete-variable (DV) of single-photon states (photonic qubits) and continuous-variable (CV) of optical modes (photonic qumodes) plays important role in QI development. The one-way quantum computation(QC) based on multipartite cluster entanglement is initially proposed by Raussendorf and Briegel in the DV modelRaussendorf2001 (), then it is extended to the CV regime by Menicucci et al Menicucci2006 (). For one-way QC model the qubits (qumodes) are initialized in a multipartite cluster entangled state firstly, then a variety of quantum logical operations can be achieved only via the single-qubit (qumode) projective measurement and the classical feedforward of the measured outcomes, in which the order and choices of measurements are determined by the required algorithmRaussendorf2001 (). The basic logical operations of one-way DVQC has been experimentally demonstrated by several groups Walther2005 (); Chen (); Gao ().

Parallelly, the theoretical and experimental explorations on one-way CVQC were also proceeding continually vanLoockJOSA2007 (); Tan2009 (); Gu2009 (); Miwa2009 (); Wang2010 (); Ukai2011 (); Ukai20112 (). In contrast of the probabilistic generation of photonic qubits in most cases, CV cluster states are produced in an unconditional fashion and thus the one-way QC with CV cluster entangled photonic qumodes can be implemented deterministically Su2007 (); Yukawa2008 (); Tan2008 (); Miwa2009 (); Wang2010 (); Ukai2011 (); Ukai20112 (); Matt2011 (). Following the theoretical proposals on one-way CVQC the principally experimental demonstrations of various one-way QC logical operations over CVs were achieved by utilizing bipartite and four-partite cluster entangled photonic qumodes, respectively Miwa2009 (); Wang2010 (); Ukai2011 (); Ukai20112 (). To develop more complicated QC larger cluster states with more numbers of entangled qubits (qumodes) are desired. However, the numbers of spatially separable entangled qumodes generated by experiments still stay below four-partites, so far Su2007 (); Yukawa2008 (); Tan2008 (). In the paper, we present the first experimental achievement on producing CV eight-partite entangled states for photonic qumodes. Using eight squeezed states of light to be the initial resource quantum states and passing through the linearly optical transformation on a specially designed beam-splitter network, the eight-partite linear and two-diamond shape cluster states for photonic qumodes are prepared, respectively. The entanglement feature among the obtained eight space-separated photonic qumodes is confirmed by the fully inseparability criteria of CV multipartite entangled states proposed by van Loock and Furusawa Loock2003 ().

The cluster state is a type of multipartite quantum entangled graph states corresponding to some mathematic graphs Menicucci2006 (); Gu2009 (); Zhang2006 (). The CV cluster quadrature correlations (so-called nullifiers) can be expressed by Gu2009 (); Zhang2006 (); Loock2007 ()

(1) |

where and stand for quadrature-amplitude and quadrature-phase operators of an optical mode , respectively. The subscript a (b) expresses the designated mode (). The modes of denote the vertices of the graph , while the modes of are the nearest neighbors of mode . For an ideal cluster state the left-hand side of equation (1) trends to zero, which stands for a simultaneous zero eigenstate of the quadrature combination Gu2009 (). The CV cluster quantum entanglements generated by experiments are deterministic, but also are imperfect, the entanglement features of which have to be verified and quantified by the sufficient conditions for the fully inseparability of multipartite CV entanglement Su2007 (); Yukawa2008 (); Matt2011 (); Tan2008 (). There are different correlation combinations [left-hand side of equation (1)] in a variety of CV cluster multipartite entangled states, which reflect the complexity and rich usability of these quantum systems. The expressions of the nullifiers for different graph states depend on their graph configurations.

Figure 1 (a) and (b) show the graph representations of CV eight-partite linear (a) and two-diamond (b) shape CV cluster states, respectively, each node of which corresponds to an optical mode and the connection lines between neighboring nodes stand for the interaction between the connected two nodes. From equation (1) and Fig. 1, we can write the nullifiers of the linear and the two-diamond shape CV cluster states, respectively, which are , , , , , , , for the linear states; and , , , , , , , for the two-diamond states, where the subscripts L and D () denote the individual nodes of the linear and the two-diamond shape cluster states, respectively, and express the excess noises resulting from the imperfect quantum correlations. When the variance of () is smaller than the corresponding quantum noise limit (QNL) determined by vacuum noises, the correlations among the combined optical modes is within the quantum region, otherwise the quantum correlations do not exist.

The schemes of generating CV multipartite entangled states commonly used in experiments are to achieve a linearly optical transformation of input squeezed states on a specific beam-splitter network Loock2007 (). Assuming and stand for the input squeezed states and the unitary matrix of a given beam-splitter network, respectively, the output optical modes after the transformation are given by , where the subscripts l and k express the designated input and output modes, respectively. In our experiment, four quadrature-amplitude -squeezed states, , and four quadrature-phase -squeezed states, , are applied, where and denote the quadrature-amplitude and the quadrature-phase operators of the corresponding vacuum field, respectively, is the squeezing parameter to quantify the squeezing level, and correspond to the two cases of no squeezing and the ideally perfect squeezing,respectively. The unitary matrix for generating the CV eight-partite linear cluster state by combining eight squeezed states on optical beam splitters equals to (see Supplementary Material)

(2) |

The unitary matrix in equation (2) expresses an optical transformation on a beam-splitter network consisting of seven beam splitters and can be decomposed into , where denotes the Fourier transformation of mode , which corresponds to a 90 rotation in the phase space; stands for the linearly optical transformation on the jth beam-splitter with the transmission of (), where , , , and are elements of beam-splitter matrix. corresponds to a 180 rotation of mode in the phase space.

Figure 2 shows the schematic of the experimental set-up for preparing the eight-partite CV linear cluster state. The four -squeezed and four -squeezed states are produced by four NOPAs pumped by a common laser source, which is a CW intracavity frequency-doubled and frequency-stabilized Nd:YAP/LBO(Nd-doped YAlO perorskite/lithium triborate) with both outputs of the fundamental and the second-harmonic waves WangIEEE2010 (). The output fundamental wave at 1080 nm wavelength is used for the injected signals of NOPAs and the local oscillators of the balanced homodyne detectors (BHDs), which are applied to measure the quantum fluctuations of the quadrature-amplitude and the quadrature-phase of the output optical modes Su2007 (). The second-harmonic wave at 540 nm wavelength serves as the pump field of the four NOPAs, in which through an intracavity frequency-down-conversion process a pair of signal and idler modes with the identical frequency at 1080 nm and the orthogonal polarizations are generated Li2002 (); Wang20102 (). Since the amplitude and the phase quadratures of the signal and the idler modes are entangled each other, the two coupled modes of them at polarization directions both are the squeezed states Su2007 (); Yun2000 (). In our experiment, the four NOPAs are operated at the parametric deamplification situation, i.e. the phase difference between the pump field and the injected signal is ( is an integer). Under this condition, the coupled modes at +45 and -45polarization directions are the quadrature-amplitude and the quadrature-phase squeezed states, respectively Su2007 (); Yun2000 (). When the transmissions of the seven beam splitters are chosen as , , , , the eight output optical modes are in a eight-partite CV linear cluster state. The quadrature-amplitude and quadrature-phase of each are measured by eight BHDs, respectively. The nullifiers of the eight output modes depend on the squeezing parameters of the resource squeezed states. For our experimental system all four NOPAs have the identical configuration (the construction of NOPA is described in the Supplementary Material) and are operated under the same conditions. Each of NOPAs is also adjusted to produce two balanced squeezed states. So, the eight initial squeezed states own the same squeezed parameter . In this case we can easily calculate the excess noises of the nullifiers for the eight-partite linear CV cluster state consisting of the eight output modes (), which are , , , , , , and , respectively.

The unitary matrix of the two-diamond cluster state equals to , with (see Supplementary Material), thus the two-diamond shape cluster state can be prepared from the linear cluster state via local Fourier transforms and phase rotations. The excess noise terms of the nullifiers of the two-diamond shape cluster state are expressed by , , , , , , , and , respectively. According to the inseparability criteria for CV multipartite entangled states proposed by van Loock and Furusawa Loock2003 (), we deduced the inseparability criterion inequalities for CV eight-partite linear and two-diamond shape cluster states, which are given by equations (3a)-(3g) and equations (4a)-(4i), respectively (see Supplementary Material).

(3a) | |||||

(3b) | |||||

(3c) | |||||

(3d) | |||||

(3e) | |||||

(3f) | |||||

(3g) |

and

(4a) | |||||

(4b) | |||||

(4c) | |||||

(4d) | |||||

(4e) | |||||

(4f) | |||||

(4g) | |||||

(4h) | |||||

(4i) |

where left-hand sides and right-hand sides of these inequalities are the combination of variances of nullifiers and the boundary, respectively. In the Supplementary Material, we numerically calculated the dependencies of the combinations of the correlation variances in the left-hand sides of equations (3) and equations (4) on the squeezing factor for and ( is the optimal gain factor), respectively. It can be seen, when the optimal gain factors are used the correlation variance combinations are always smaller than the boundary for all values of , i.e. the eight-partite CV cluster entanglement always can be realized by the presented system whatever how low the squeezing of the initial squeezed states is. However, when , a lower limitation of is required to result in the correlation variances to be smaller than the boundary.

The experimentally measured initial squeezing degrees of the output fields from four NOPAs are dB below the QNL which corresponds to the squeezing parameter (see Supplementary Material). During the measurements the pump power of NOPAs at nm wavelength is mW, which is below the oscillation threshold of mW, and the intensity of the injected signal at nm is mW. The phase difference on each beam-splitters are locked according to the requirements. The light intensity of the local oscillator in all BHDs is set to around 5 mW. The measured QNL is about 20 dB above the electronics noise level, which guarantees that the results of the homodyne detections are almost not affected by electronic noises.

The correlation variances measured experimentally are shown in figure 3 for the linear cluster and figure 4 for the two-diamond cluster. They are dB, dB, dB, dB, dB, dB, dB, dB and dB, dB, dB, dB, dB, dB, dB, dB, dB, dB. From these measured results we can calculated the combinations of the correlation variances in the left-hand sides of the inequalities (3a)-(3g) and (4a)-(4i), which are , , , , , , , for the linear cluster and , , , , , , , , for the two-diamond cluster, respectively. All these values are smaller than the boundary. It means that the prepared two types of CV cluster states satisfy the inseparability criteria for verifying multipartite CV entanglement, so the spatially separated eight-partite entangled states of photonic qumodes are experimentally obtained. In the experiment we detected the correlation variances under , and . For our system, the total transmission efficiency of squeezed beams are about and the detection efficiency is about , which lead to the efficient squeezing parameter is which is smaller than the initially measured squeezing parameter. When the gain factors except are taken as 1 and only is utilized, all inequalities in equation (3) and equation (4) are satisfied. If , the unite gain factor of can be chosen also (see Supplementary Material).

In the conclusion, we have experimentally prepared two types of spatially separated eight-partite CV cluster entangled states for photonic qumodes by using eight quadrature squeezed states of light and a specifically designed optical beam-splitter network. The multipartite entangled states are the essential resources to construct a variety of CVQI networks. So far, the single-mode squeezed states over 12.7 dB Eberle2010 () and the two-mode squeezed states over 8.1 dB Yan2012 () have been experimentally generated, respectively, based on which and using the presented scheme the CV cluster states with more space-separable qumodes and higher entanglement can be obtained. The complexity and versatility of CV multipartite entanglement for photonic qumodes not only offer richly potential applications in QC and QI, but also provide the basic and handleable entangled quantum states which can be an important tool for further studying the amazing and attractive quantum entanglement phenomena.

This research was supported by the National Basic Research Program of China (Grant No. 2010CB923103) and NSFC (Grant Nos. 11174188, 61121064).

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Supplementary Information for “Experimental preparation of eight-partite

linear and two-diamond shape cluster states for photonic qumodes”

Xiaolong Su, Yaping Zhao, Shuhong Hao, Xiaojun Jia, Changde Xie, and Kunchi Peng*

[0pt] State Key Laboratory of Quantum Optics and Quantum Optics Devices, Institute of Opto-Electronics, Shanxi University, Taiyuan, 030006, People’s Republic of China

[0pt] *e-mail:kcpeng@sxu.edu.cn

## Appendix A Unitary matrix for generating CV eight-partite cluster states

According to the proposal of Peter van Loock et al Loock2007 (), CV cluster states of photonic qumodes can be created via a general linear-optics transformation of -squeezed input modes. If and a unitary matrix stand for the annihilation operator of the input modes and the linear-optical transformation respectively, the output modes after the transformation are expressed by , which are the CV cluster states Loock2007 (); Zhang2006 (). The CV cluster states satisfy Im[]Re[] in the limit of infinite squeezing, where is the identity matrix, is the matrix of input states, is the adjacency matrix Menicucci (). So we have ImRe and the unitary matrix is obtained

(5) |

Based on the unitarity of matrix , , we have

(6) |

In this case, we can obtain Re and from the adjacency matrix .

For n-partite cluster state, assuming

(7) |

where is a real vectors. According to Eq. (2), we have (), where the numbers of these equations are according to the symmetry of matrix. Since there are unknown numbers in all these equations, we need conditions to solve the equations. For simplicity and without lossing generality, some unknown numbers in the equations are chosen to be 0 when we solve the equations.

For CV eight-partite linear cluster state, the adjacency matrix can be written as

(8) |

we have

(9) |

The matrix elements in Eq. (3) are obtained by the following way. Considering the symmetry, we start from the middle row, . We apply seven initial conditions, , then it becomes According to equations , first unknown number is obtained. Then we apply six conditions, on , we have . Using equations and , two unknown numbers , are gotten, and so on.

After all the unknown numbers in Eq. (3) are obtained, the unitary matrix is given from Eq. (1). The unitary matrix for eight-partite linear cluster state with eight -squeezed states to be the input states are expressed

(10) |

In the experiment, we prepared four