# Experimental test of rotational invariance and entanglement of photonic six-qubit singlet state

###### Abstract

We experimentally test invariance properties of the six-photonic-qubits generalization of a singlet state. Our results clearly corroborate with theory. The invariance properties are useful to beat some types of decoherence. We also experimentally detect entanglement in the state using an appropriate witness observable, composed of local observables. Our results clearly indicate that the tested setup, which is very stable, is a good candidate for realization of various multi-party quantum communication protocols. The estimated fidelity of the produced state is very high: it reveals very strong EPR six-photon correlations of error rate below .

###### pacs:

03.67.-a, 03.67.Mn, 03.67.Pp.It is well known that quantum information processing relies on preparation, manipulation, and detection of superpositions of quantum states. Superpositions, however, are very fragile and are easily destroyed by the decoherence processes due to unwanted couplings with the environment Z91 (). Such uncontrollable influences cause noise in the communication, or errors in the outcome of a computation. Several strategies have been devised to cope with decoherence. For instance, if the qubit-environment interaction, no matter how strong, exhibits some symmetry, then there exist quantum states which are invariant under this interaction. These states are called decoherence-free (DF) states, and allow to protect quantum information ZR97a (); LCW98 (). This situation occurs for instance when the spatial (temporal) separation between the carriers of the qubits is small relative to the correlation length (time) of the environment. Experimental efforts investigating features of DF systems have been carried out to demonstrate the properties of a specific two-qubit DF state KBAW00 (), and the existence of three-qubit noiseless subsystems VFPKLC01 (). Bourennane et al. BEGKCW04 () produced the four-photon polarization-entangled state which is a generalization of the singlet, , and demonstrated its invariance under general collective noise and experimentally showed the immunity of a qubit encoded in this state.

To encode an arbitrary two-qubit state Cabello has theoretically constructed DF states formed by six qubits. One of these states is C06 (). It is invariant under transformations which consist of identical unitary transformations of each individual constituentZR97a ():

(1) |

where denotes the tensor product of six identical unitary operators . Besides protecting against collective noise, the DF states are useful for communication of quantum information between two observers who do not share a common reference frame BRS03 (). In such a scenario, any realignment of the receiver’s reference frame corresponds to an application of the same transformation to each of the qubits which were sent. The states can also be used for secure quantum multiparty cryptographic protocols such as the six-party secret sharing protocol HBB99 (); GKBW07 ().

Recently multiphoton interferometry based on parametric down conversion reached the stage at which one can observe genuine six-photon interference. The experiment of ref. LZGGZYGYP07 () a generalization of the schemes suggested in ZHWZ97 () was used. In our recent experiment RWZB09 () we used a generalization of the blueprint of ZW01 (), and its realization GBEKW03 (). We obtained a six-photon invariant entangled state by pulse pumping just one crystal and extracting the third order process. This is done only via suitable filtering, and the interference is observed behind four beamsplitters. The setup is strongly robust, as it faces no alignment problems. The observed six-photon correlation with high fidelity agree with the ones of the theoretical .

In this paper present results of the invariance tests of the experimental correlations attributable to . This is done by sequence measurements of three mutually complementary polarizations at all six detection stations (linear vertical/horizontal, linear diagonal/antidiagonal, circular right/left). The other interesting feature of the state is that it reveals various interesting types of entanglement within the subsystems. This is studied here theoretically, and compared with the data. Finally we present tests aimed at verification of the entanglement of the obtained state. We use the toolbox of entanglement witnesses provided in Toth-MATLAB ().

The state component corresponding to the emission in a PDC process of six photons into two spatial modes in a PDC process is proportional to

(2) |

where () is the creation operator for one horizontal (vertical) photon in mode (), and conversely; is a normalization constant, is a function of pump power, non-linearity and length of the crystal, is the phase difference between horizontal and vertical polarizations due to birefringence in the crystal, and denotes the vacuum state. This is a good description of the initial six-photon state, provided one collects the photons under conditions that allow the indistinguishability between separate two-photon emissions ZZW95 (). A particle interpretation of this term can be obtained through its expansion

(3) |

and is given by the following superposition of photon number states:

(4) |

where e.g. denotes three horizontally polarized photons in mode . The third order PDC is fundamentally and intrinsically different than a product of three entangled pairs. Due to the bosonic nature of photons the emissions of completely indistinguishable photons are favored compared with the ones with orthogonal polarization.

We report experimental observations which are aimed at testing whether the correlations produced in our setup are indeed rotationally invariant. The invariant six-qubit polarization entangled state given by the following superposition of a six-qubit Greenberger-Horne-Zeilinger (GHZ) state and two products of three-qubit W states.

(5) |

The GHZ state is here defined as , and the W-state is defined as . is the spin-flipped , and and denote horizontal and vertical polarization, respectively. This state is obtained from the third order emission of the PDC process eq. (2) with the phase . The emitted photons are beam-split into six modes and one selects the terms with one photon in each mode.

It is easy to see that if one moves into the spin description of the polarization variables, the state is a singlet (total spin equal to zero) of a composite system consisting of six spins .

In our experiment we use a frequency-doubled Ti:Sapphire laser ( Mhz repetition rate, fs pulse length) yielding UV pulses with a central wavelength at nm and an average power of mW. The pump beam is focused to a m waist in a mm thick BBO (-barium borate) crystal. Half wave plates and two mm thick BBO crystals are used for compensation of longitudinal and transversal walk-offs. The third order emission of non-collinear type-II PDC is then coupled to single mode fibers (SMF), defining the two spatial modes at the crossings of the two frequency degenerated down-conversion cones. Leaving the fibers the down-conversion light passes narrow band ( nm) interference filters (F) and is split into six spatial modes by ordinary beam splitters (BS), followed by birefringent optics to compensate phase shifts in the BS’s. Due to the short pulses, narrow band filters, and single mode fibers the down-converted photons are temporally, spectrally, and spatially indistinguishable ZZW95 (), see Fig. 1. The polarization is being kept by passive fiber polarization controllers. Polarization analysis is implemented by a half wave plate (HWP), a quarter wave plate (QWP), and a polarizing beam splitter (PBS) in each mode. The outputs of the PBS’s are lead to single photon silicon avalanche photo diodes (APD) through multi mode fibers. The APD’s electronic responses, following photo detections, are being counted by a multi channel coincidence counter with a ns time window. The coincidence counter registers any coincidence event between the APD’s as well as single detection events.

Fig. 2a shows the probabilities to obtain each of the possible sixfold coincidences with one photon detection in each spatial mode, measuring all qubits in basis. The peaks are in very good agreement with theory: half of the detected sixfold coincidences are to be found as and , and the other half should be evenly distributed among the remaining events with three and three detections. This is a clear effect of the bosonic interference (stimulated emission) in the BBO crystal giving higher probabilities for emission of indistinguishable photons.

The six-photon state is invariant under identical (unitary) transformations in each mode. Experimentally this can be shown by using identical settings of all polarization analyzers: no matter what the setting is, the results should be similar. Our results for measurements in (Diagonal/Antidiagonal, ) and (Left/Right, ) polarization bases are presented in fig. 2.b and 2.c. The invariance of the probabilities with respect the joint changes of the measurement basis in all modes is clearly visible. We clearly observe, in the results of these three different settings measurements, the small and uniform noise contribution.

Another property of the is that it exhibit perfect EPR correlations between measurement results in different modes. We obtain the corrections , , and . which are close the theoretical value of . From these results and the approximation that our noise to white noise we have estimate the fidelity where is the experimental the six-photon density. The estimated fidelity clearly shows that the setup is able to produce correlations due to six photon entangled states with unprecedented precision (error rate below ).

Conditioning on a detection of one photon in a specific state we have also obtained four different five-photon entangled states. In the computational basis the projection of the last qubit on leads to

(6) |

A similar projection on results in

(7) |

We have also performed such measurements related with the operator in the mode , which has as its eigenstates , while the other five photons are measured in the basis. The projection on gives

(8) |

where and the projection on gives

(9) |

where

Fig. 3 shows the results (obtained in the observation bases) for these five photon conditional polarization states. In Fig. 3.a and 3.b, we clearly see the terms , and respectively. The terms and are evident in both Fig. 3.c and 3.d. All these results are in agreement with theoretical predictions.

is a six-qubit entangled state, meaning that each of its qubits is entangled with all the remaining ones. In order to show that our experimental correlations reveal a six qubit entanglement we use the entanglement witness method. An entanglement witness is an observable yielding a negative value only for entangled states, the most common being the maximum overlap witness (), which is the best witness with respect to noise tolerance witness (). The maximum overlap witness optimized for has the form

(10) |

where the factor 2/3 is the maximum overlap of with any biseparable state. The witness detects six-partite entanglement with a noise tolerance around , but it also demands a large amount of measurement settings. Since it would be an experimentally very demanding task to perform all these measurements, we have developed a reduced witness that can be implemented using only three measurement settings (this was done using the tools provided by Toth TG05 (); Toth-MATLAB ()). Our reduced witness , is given by

(11) |

where denotes the same terms as in the sum but with and interchanged. This is obtained from the maximum overlap witness as follows. First the maximum overlap witness is decomposed into direct products of Pauli and identity matrices, secondly only terms that are products of one type of Pauli matrices and identity matrices are selected (all terms that include products of at least two different Pauli matrices are deleted, remaining only non-mixed terms), e.g. , . Finally, the constant in front of in the first term of eq. (11) is chosen to be the smallest possible such that all entangled states found by the reduced witness are also found by the maximal overlap witness. Our reduced witness detects sixpartite entanglement of with a noise tolerance of . The theoretical expectation value and our experimental result is , showing entanglement with 2.0 standard deviations.

In summary, we have experimentally tested the property of rotational invariance of the six-photon state produced by our setup. The state is indeed entangled, and various different entangled states can be obtained out of it with the use of projective measurements of one of the qubits. We would like to note that the interference contrast is high enough for our setup to be used in demonstrations of various six-party quantum informational applications (quantum reduction of communication complexity of some joint computational tasks, secret sharing, etc.).

Acknowledgements This work was supported by
Swedish Research Council (VR). M.Ż. was supported by Wenner
Gren Foundations
and by the EU programme QAP (Qubit Applications, No. 015858).

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