## Experimental characterization of two spatial qutrits using entanglement witnesses |

Optics Express, Vol. 20, Issue 24, pp. 26351-26362 (2012)

http://dx.doi.org/10.1364/OE.20.026351

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### Abstract

We present an experimental technique for a complete characterization of entanglement in a two-qutrit state generated using transverse spatial correlations of two parametric down-converted photons. We verify entanglement for a particular case *via* entanglement witness operators which are decomposed into a sum of local observables of single path and superposition projection operators. Experimentally, these operators are accomplished by using a spatial light modulator and a polarizing beam splitter which allow to modulate the amplitude of individually chosen path states. The quantification of entanglement is computed by the negativity obtained from the expectation values of the entanglement witnesses implemented.

© 2012 OSA

## 1. Introduction

*σ*

^{TB}> 0) is a necessary and sufficient condition for separability (Peres-Horodecki criterion [1

1. A. Peres, “Separability criterion for density matrices,” Phys. Rev. Lett. **77**, 1413–1415 (1996). [CrossRef] [PubMed]

2. M. Horodecki, P. Horodecki, and R. Horodecki, “Separability of mixed states: necessary and sufficient conditions,” Phys. Lett. A **223**, 1–8 (1996). [CrossRef]

*N*⊗

*M*) with nonpositive partial transposition (NPT) describes an entangled state, but there are some PPT density matrices describing non-separable states.

2. M. Horodecki, P. Horodecki, and R. Horodecki, “Separability of mixed states: necessary and sufficient conditions,” Phys. Lett. A **223**, 1–8 (1996). [CrossRef]

5. O. Gühne, P. Hyllus, D. Bruss, A. Ekert, M. Lewenstein, C. Macchiavello, and A. Sanpera, “Experimental detection of entanglement via witness operators and local measurements,” J. Mod. Opt. **50**, 1079–1102 (2003). [CrossRef]

^{†}whose expectation value is positive for all separable states, and negative for at least one non-separable state. Mathematically, 𝒲 must fullfill the properties (i) 〈

*i*,

*j*|𝒲|

*i*,

*j*〉 ≥ 0 for all product vectors, (ii) 𝒲 is indefinite, and (iii) Tr(𝒲) = 1. In accordance with these prescribed conditions, the expectation value of 𝒲 is Tr(𝒲

*σ*) ≥ 0 for all density matrices

_{s}*σ*describing separable states and Tr(𝒲

_{s}*σ*) < 0 for at least one density matrix

_{e}*σ*connected to an entangled state. Existing implementations of the EW method [6

_{e}6. M. Barbieri, F. De Martini, G. Di Nepi, P. Mataloni, G. D’Ariano, and C. Macchiavello, “Detection of entanglement with polarized photons: experimental realization of an entanglement witness,” Phys. Rev. Lett. **91**, 227901 (2003). [CrossRef] [PubMed]

9. O. Gühne and G. Tóth, “Entanglement detection,” Physics Reports **474**, 1–75 (2009). [CrossRef]

10. After submission of this work we became aware of an entanglement witness measurement in high dimensional orbital angular momentum states: M. Agnew, J. Leach, and R.W. Boyd, “Observation of entanglement witnesses for orbital angular momentum states,” Eur. Phys. J. D **66**, 1–4 (2012). [CrossRef]

11. S. Gröblacher, T. Jennewein, A. Vaziri, G. Weihs, and A. Zeilinger, “Experimental quantum cryptography with qutrits,” New J. Phys. **8**, 75 (2006). [CrossRef]

12. R. Ghosh, C. K. Hong, Z. Y. Ou, and L. Mandel, “Interference of two photons in parametric down conversion,” Phys. Rev. A **34**, 3962–3968 (1986). [CrossRef] [PubMed]

14. Y. H. Shih and C. O. Alley, “New type of einstein-podolsky-rosen-bohm experiment using pairs of light quanta produced by optical parametric down conversion,” Phys. Rev. Lett. **61**, 2921–2924 (1988). [CrossRef] [PubMed]

13. Z. Y. Ou and L. Mandel, “Violation of bells inequality and classical probability in a two-photon correlation experiment,” Phys. Rev. Lett. **61**, 50–53 (1988). [CrossRef] [PubMed]

15. P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, “New high-intensity source of polarization-entangled photon pairs,” Phys. Rev. Lett. **75**, 4337–4341 (1995). [CrossRef] [PubMed]

16. I. Afek, O. Ambar, and Y. Silberberg, “High-NOON states by mixing quantum and classical light,” Science **328**, 879–881 (2010). [CrossRef] [PubMed]

17. A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature (London) **412**, 313–316 (2001). [CrossRef]

18. J. G. Rarity and P. R. Tapster, “Experimental violation of bells inequality based on phase and momentum,” Phys. Rev. Lett. **64**, 2495–2498 (1990). [CrossRef] [PubMed]

19. A. Rossi, G. Vallone, A. Chiuri, F. De Martini, and P. Mataloni, “Multipath entanglement of two photons,” Phys. Rev. Lett. **102**, 153902 (2009). [CrossRef] [PubMed]

20. L. Neves, G. Lima, J. G. Aguirre Gómez, C. H. Monken, C. Saavedra, and S. Pádua, “Generation of entangled states of qudits using twin photons,” Phys. Rev. Lett. **94**, 100501 (2005). [CrossRef] [PubMed]

21. M. V. Fedorov, M. A. Efremov, P. A. Volkov, E. V. Moreva, S. S. Straupe, and S. P. Kulik, “Anisotropically and high entanglement of biphoton states generated in spontaneous parametric down-conversion,” Phys. Rev. Lett. **99**, 063901 (2007). [CrossRef] [PubMed]

22. J. Brendel, N. Gisin, W. Tittel, and H. Zbinden, “Pulsed energy-time entangled twin-photon source for quantum communication,” Phys. Rev. Lett. **82**, 2594–2597 (1999). [CrossRef]

23. J. D. Franson, “Bell inequality for position and time,” Phys. Rev. Lett. **62**, 2205–2208 (1989). [CrossRef] [PubMed]

24. A. Rossi, G. Vallone, F. De Martini, and P. Mataloni, “Generation of time-bin-entangled photons without temporal postselection,” Phys. Rev. A **78**, 012345 (2008). [CrossRef]

25. L. Neves, S. Pádua, and C. Saavedra, “Controlled generation of maximally entangled qudits using twin photons,” Phys. Rev. A **69**, 042305 (2004). [CrossRef]

20. L. Neves, G. Lima, J. G. Aguirre Gómez, C. H. Monken, C. Saavedra, and S. Pádua, “Generation of entangled states of qudits using twin photons,” Phys. Rev. Lett. **94**, 100501 (2005). [CrossRef] [PubMed]

## 2. Theory

_{𝒜}

^{3}⊗ ℋ

_{ℬ}

^{3}Hilbert space spanned by the basis states |0〉, |1〉, and |2〉. |

*l*〉 (

*l*= 0, 1, 2) represents the photon path state corresponding to the photon passing through the “

*l*” slit.

*ρ*= |Ψ〉〈Ψ| is a NPT state, at least one EW operator can be constructed by computing the partial transpose of the eigenvector |

*η*〉 that corresponds to some nonpositive eigenvalue

_{j}*ε*of

_{j}*ρ*

^{TB}. For a suitable notation purpose, it is convenient to express the amplitudes in polar form

*a*= |

*a*|

*e*

^{2iθa},

*b*= |

*b*|

*e*

^{2iθb}, and

*c*= |

*c*|

*e*

^{2iθc}. We call the phase differences entering in the photon path states as

*θ*=

_{j}*θ*−

_{r}*θ*(

_{s}*j*= 1, 2, 3;

*r, s*=

*a*,

*b*,

*c*). In Table 1, three possible eigenvectors corresponding to the negative part of the spectrum of

*ρ*

^{TB}are reported. As a result, three EW operators 𝒲

*(*

_{j}*j*= 1, 2, 3) can be constructed. Such witnesses detect the state

*ρ*, when their expectation value give 〈𝒲

*〉=Tr(|*

_{j}*η*〉 〈

_{j}*η*|

_{j}^{TB}

*ρ*) =

*ε*< 0. The resulting witnesses have similar mathematical structures that can be written as where {|

_{j}*l*〉, |

*m*〉} and {|

*l*′〉, |

*m*′〉} denote the photon path states for the subsystems 𝒜 and ℬ, respectively. It should be noted that the witnesses do not depend on the amplitude but phase differences of path states. Our notation states that a witness 𝒲

*is composed by the path states {|*

_{j}*l*〉, |

*m*〉}

_{𝒜}⊗ {|

*l*′〉, |

*m*′〉}

_{ℬ}. According to Table 1, one can infer that 𝒲

_{1}involves {|0〉, |2〉

*}*

_{𝒜}⊗ {|0〉, |2〉}

_{ℬ}, 𝒲

_{2}involves {|1〉, |2〉}

_{𝒜}⊗ {|0〉, |1〉}

_{ℬ}, and 𝒲

_{3}involves {|0〉, |1〉}

_{𝒜}⊗ {|1〉, |2〉}

_{ℬ}.

*= |*

_{j}*η*〉 〈

_{j}*η*|

_{j}^{TB}in our case is a very difficult task. Thus, in order to measure these witnesses it is necessary to decompose them into observables that can be measured locally, such as operators like |

*l*〉〈

*l*| ⊗ |

*m*〉〈

*m*|. Henceforth, it is important to point out that associated to the detection behind the

*l*-th slit we have the single path measurement operator |

*l*〉〈

*l*|. Besides these operators we introduce the path superposition operators, corresponding to the superposition in only two out of three path states defined for each subsystem. These operators can be expressed in the following way:

*can be computed using a basis of linearly independent operators. We choose to use a set of traceless hermitian operators acting in the path state space, known as Gell-Mann matrices*

_{j}*λ*, which are the generators of the fundamental representation of the SU(3) group [26]. SU(3) is generated by the 3 × 3 hermitian, traceless matrices with determinant 1, described by

_{γ}*λ*) = 2

_{n}λ_{m}*δ*. Since both

_{nm}*λ*

_{3}and

*λ*

_{8}are the only diagonal operators, it is necessary to represent them in terms of physical observables. For this purpose, all Gell-Mann matrices associated with a single subsystem can be rewritten using the computational basis {|0〉, |1〉, |2〉} in terms of single and superposition operators as

*λ*

_{9}= form an orthogonal basis for the complex Hilbert space of all 3 × 3 matrices. One can therefore decompose any single operator 𝒪

_{Q3}defined in one qutrit Hilbert space ℋ = 3 as and decompose an operator 𝒪

_{nQ3}defined in

*n*-qutrit Hilbert space

## 3. Experimental setup

*λ*= 405 nm, is sent through a 2-mm-long BiBO crystal cut for type I SPDC. Inside this nonlinear crystal, degenerated non-collinear pairs of entangled photons at

*λ*= 810 nm are created by the SPDC process. The remaining pump beam, after the crystal, is blocked by a dichroic mirror (DM) to avoid fluorescence emission at the optical components. Due to vertical polarization of the pump laser beam and the type I phase matching, the polarization of the entangled photons is horizontal.

_{s,i}*a*= 100

*μ*m with a slit spacing of

*d*= 250

*μ*m in the x-direction. Down-converted photons are transmitted by a PBS and then are passed through a three-slit aperture placed after the PBS. At this point, two-qutrit states are generated.

*μ*m square) which address 8-bit images (256 values of gray level) and works with specific polarized incoming/outgoing beams and can operate as a phase and/or amplitude modulator [27

27. W. M. Pimenta, B. Marques, M. A. Carvalho, M. R. Barros, J. G. Fonseca, J. Ferraz, M. Terra Cunha, and S. Pádua, “Minimal state tomography of spatial qubits using a spatial light modulator,” Opt. Express **18**, 24423–24433 (2010). [CrossRef] [PubMed]

28. I. Moreno, P. Velásquez, C. Fernández-Pousa, M. Sánchez-López, and F. Mateos, “Jones matrix method for predicting and optimizing the optical modulation properties of a liquid-crystal display,” J. Appl. Phys. **94**, 3697–3702 (2003). [CrossRef]

*L*and

_{s}*L*, both with focal length

_{i}*f*= 200 mm, fixed at a distance

*z*from the detectors

_{L}*D*and

_{s}*D*, which are kept fixed at a distance of 800 mm from the aperture. In front of each detector there is a 100-

_{i}*μ*m-width single slit, oriented parallel to the three slits, followed by an interference filter of 10 nm bandwidth centered at 810 nm and a microscope objective focusing on their detection active area. The output pulses are sent to counters C, which registers single and coincidence counts, with a resolving time of 5 ns.

### 3.1. State preparation

29. S. P. Walborn, C. H. Monken, S. Pádua, and P. H. Souto Ribeiro, “Spatial correlations in parametric down-conversion,” Physics Reports , **495**, 87–139 (2010). [CrossRef]

*L*of focal length

_{p}*f*= 300 mm into the pump-beam path, 50 mm before the crystal such that the pump beam profile is focused at the three-slit plane.

_{p}28. I. Moreno, P. Velásquez, C. Fernández-Pousa, M. Sánchez-López, and F. Mateos, “Jones matrix method for predicting and optimizing the optical modulation properties of a liquid-crystal display,” J. Appl. Phys. **94**, 3697–3702 (2003). [CrossRef]

*L*and

_{s}*L*on the detector planes

_{i}*D*and

_{s}*D*, respectively. Such images are achieved when the lenses are placed into the halfway between the aperture and the detectors, at a distance 2

_{i}*f*after the aperture and

*z*= 2

_{L}*f*from the detector plane. At the image plane,

*D*is kept fixed at the maximum single counts for each slit |0〉, |1〉, and |2〉, while the idler image is scanned by the detector

_{s}*D*in the x-direction. Each scanning selects three of the nine basis states. The normalized coincidence counts obtained from the three measured coincidence numbers give the probabilities for all basis states. In Table 2, the probabilities for all basis states 𝒟⊗𝒟 of 𝒟 = 3 are reported.

_{i}30. E. J. S. Fonseca, J. C. Machado da Silva, C. H. Monken, and S. Pádua, “Controlling two-particle conditional interference” Phys. Rev. A **61**, 023801 (2000). [CrossRef]

*F*= 0.98 ± 0.03 relative to the theoretical state in Eq. (1) with

*a*=

*b*=

*c*= 1. The fidelity was obtained by means of the amplitudes derived from the experimental probabilities (Table 2). These measurements agree with the theoretical proposed state.

31. D. Greenberger, M. Horne, and A. Zeilinger, “Multiparticle interferometry and the superposition principle,” Phys. Today **46**, 22–22 (1993). [CrossRef]

*z*=

_{L}*f*from the detector plane. Coincidence measurements are made by scanning the Fourier plane in the x-direction with the

*D*while keeping fixed the detector

_{i}*D*at

_{s}*x*= 0 mm and then at

_{s}*x*= 0.20 mm in a second scan. As a result, a three-slit interference pattern with conditional fringes were observed analogously to those already shown in Ref. [20

_{s}20. L. Neves, G. Lima, J. G. Aguirre Gómez, C. H. Monken, C. Saavedra, and S. Pádua, “Generation of entangled states of qudits using twin photons,” Phys. Rev. Lett. **94**, 100501 (2005). [CrossRef] [PubMed]

### 3.2. Local measurement operators of the EW operators

*lm*and

*l*′

*m*′ paths for signal and idler arms, respectively, with specific phase differences.

*lm*and

*l*′

*m*′ paths are obtained for signal and idler detectors, respectively. This procedure, which consists in scanning the two-slit pattern with the

*D*detector while keeping fixed the detector

_{i}*D*, is illustrated in Fig 3(b).

_{s}*l*and

*m*paths. Such phase differences are given when the detectors are placed fixed at relative positions to the central maximum of the interference patterns, which corresponds to the relative phase

*ϕ*= 0. At this position, a detector is selecting only photons in the state

*ϕ*=

*π*, and for the left side corresponds to

*ϕ*= −

*π*. In both cases, the detector is selecting only photons in the state

*ϕ*=

*π*/2 is achieved when a detector is placed at the right side of the interference pattern in the halfway between the positions corresponding to

*ϕ*= 0 and

*ϕ*=

*π*. Analogously, but in a symmetrical position to the left side a phase difference

*ϕ*= −

*π*/2 is obtained. A detector at

*ϕ*=

*pi*/2 is selecting only photons in the state

*ϕ*= −

*π*/2 in the state

*D*using the setup configuration of Fig. 2(b). Moreover, in this interference pattern it is shown the detector’s positions for four specific phase differences (dotted lines) with respect to the central maximum (solid line) for zero phase. In this way, the operators

_{i}*ρ*) is defined as the absolute sum of the negative eigenvalues of

*ρ*

^{TB}, it is possible to obtain a quantitative measurement of entanglement from the witnesses [32

32. G. Vidal and R. F. Werner, “Computable measure of entanglement,” Phys. Rev. A **65**, 032314 (2002). [CrossRef]

*ρ*)=0.97 ± 0.23. This result agrees with the theoretical prediction (𝒩=1) for the negativity of a two-qutrit maximally entangled state.

*of the desired state |*

_{ψ}*ψ*〉 [9

9. O. Gühne and G. Tóth, “Entanglement detection,” Physics Reports **474**, 1–75 (2009). [CrossRef]

*a*=

*b*=

*c*= 1) can be written in terms of the identity and the three EW operators

_{Ψ}= 〈Ψ|

*ρ*|Ψ〉 where

*ρ*stands for the density matrix of the state proposed and Ψ is the state measured. The fidelity measures how close the experimental state is with respect to the theoretical state. The resulting experimental value is ℱ

_{Ψ}= 0.98 ± 0.15.

## 4. Conclusion

## Acknowledgments

## References and links

1. | A. Peres, “Separability criterion for density matrices,” Phys. Rev. Lett. |

2. | M. Horodecki, P. Horodecki, and R. Horodecki, “Separability of mixed states: necessary and sufficient conditions,” Phys. Lett. A |

3. | B. Terhal, “Bell inequalities and the separability criterion,” Phys. Lett. A |

4. | M. Lewenstein, B. Kraus, J. I. Cirac, and P. Horodecki, “Optimization of entanglement witnesses,” Phys. Rev. A |

5. | O. Gühne, P. Hyllus, D. Bruss, A. Ekert, M. Lewenstein, C. Macchiavello, and A. Sanpera, “Experimental detection of entanglement via witness operators and local measurements,” J. Mod. Opt. |

6. | M. Barbieri, F. De Martini, G. Di Nepi, P. Mataloni, G. D’Ariano, and C. Macchiavello, “Detection of entanglement with polarized photons: experimental realization of an entanglement witness,” Phys. Rev. Lett. |

7. | M. Bourennane, M. Eibl, C. Kurtsiefer, S. Gaertner, H. Weinfurter, O. Gühne, P. Hyllus, D. Bru, M. Lewenstein, and A. Sanpera, “Experimental detection of multipartite entanglement using witness operators,” Phys. Rev. Lett. |

8. | N. Kiesel, C. Schmid, U. Weber, G. Tóth, O. Gühne, R. Ursin, and H. Weinfurter, “Experimental analysis of a four-qubit photon cluster state,” Phys. Rev. Lett. |

9. | O. Gühne and G. Tóth, “Entanglement detection,” Physics Reports |

10. | After submission of this work we became aware of an entanglement witness measurement in high dimensional orbital angular momentum states: M. Agnew, J. Leach, and R.W. Boyd, “Observation of entanglement witnesses for orbital angular momentum states,” Eur. Phys. J. D |

11. | S. Gröblacher, T. Jennewein, A. Vaziri, G. Weihs, and A. Zeilinger, “Experimental quantum cryptography with qutrits,” New J. Phys. |

12. | R. Ghosh, C. K. Hong, Z. Y. Ou, and L. Mandel, “Interference of two photons in parametric down conversion,” Phys. Rev. A |

13. | Z. Y. Ou and L. Mandel, “Violation of bells inequality and classical probability in a two-photon correlation experiment,” Phys. Rev. Lett. |

14. | Y. H. Shih and C. O. Alley, “New type of einstein-podolsky-rosen-bohm experiment using pairs of light quanta produced by optical parametric down conversion,” Phys. Rev. Lett. |

15. | P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, “New high-intensity source of polarization-entangled photon pairs,” Phys. Rev. Lett. |

16. | I. Afek, O. Ambar, and Y. Silberberg, “High-NOON states by mixing quantum and classical light,” Science |

17. | A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature (London) |

18. | J. G. Rarity and P. R. Tapster, “Experimental violation of bells inequality based on phase and momentum,” Phys. Rev. Lett. |

19. | A. Rossi, G. Vallone, A. Chiuri, F. De Martini, and P. Mataloni, “Multipath entanglement of two photons,” Phys. Rev. Lett. |

20. | L. Neves, G. Lima, J. G. Aguirre Gómez, C. H. Monken, C. Saavedra, and S. Pádua, “Generation of entangled states of qudits using twin photons,” Phys. Rev. Lett. |

21. | M. V. Fedorov, M. A. Efremov, P. A. Volkov, E. V. Moreva, S. S. Straupe, and S. P. Kulik, “Anisotropically and high entanglement of biphoton states generated in spontaneous parametric down-conversion,” Phys. Rev. Lett. |

22. | J. Brendel, N. Gisin, W. Tittel, and H. Zbinden, “Pulsed energy-time entangled twin-photon source for quantum communication,” Phys. Rev. Lett. |

23. | J. D. Franson, “Bell inequality for position and time,” Phys. Rev. Lett. |

24. | A. Rossi, G. Vallone, F. De Martini, and P. Mataloni, “Generation of time-bin-entangled photons without temporal postselection,” Phys. Rev. A |

25. | L. Neves, S. Pádua, and C. Saavedra, “Controlled generation of maximally entangled qudits using twin photons,” Phys. Rev. A |

26. | J. Fuchs and C. Schweigert, |

27. | W. M. Pimenta, B. Marques, M. A. Carvalho, M. R. Barros, J. G. Fonseca, J. Ferraz, M. Terra Cunha, and S. Pádua, “Minimal state tomography of spatial qubits using a spatial light modulator,” Opt. Express |

28. | I. Moreno, P. Velásquez, C. Fernández-Pousa, M. Sánchez-López, and F. Mateos, “Jones matrix method for predicting and optimizing the optical modulation properties of a liquid-crystal display,” J. Appl. Phys. |

29. | S. P. Walborn, C. H. Monken, S. Pádua, and P. H. Souto Ribeiro, “Spatial correlations in parametric down-conversion,” Physics Reports , |

30. | E. J. S. Fonseca, J. C. Machado da Silva, C. H. Monken, and S. Pádua, “Controlling two-particle conditional interference” Phys. Rev. A |

31. | D. Greenberger, M. Horne, and A. Zeilinger, “Multiparticle interferometry and the superposition principle,” Phys. Today |

32. | G. Vidal and R. F. Werner, “Computable measure of entanglement,” Phys. Rev. A |

**OCIS Codes**

(270.0270) Quantum optics : Quantum optics

(270.5565) Quantum optics : Quantum communications

(270.5585) Quantum optics : Quantum information and processing

**ToC Category:**

Quantum Optics

**History**

Original Manuscript: August 21, 2012

Revised Manuscript: October 19, 2012

Manuscript Accepted: October 20, 2012

Published: November 7, 2012

**Citation**

A. J. Gutiérrez-Esparza, W. M. Pimenta, B. Marques, A. A. Matoso, J. L. Lucio M., and S. Pádua, "Experimental characterization of two spatial qutrits using entanglement witnesses," Opt. Express **20**, 26351-26362 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-24-26351

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### References

- A. Peres, “Separability criterion for density matrices,” Phys. Rev. Lett.77, 1413–1415 (1996). [CrossRef] [PubMed]
- M. Horodecki, P. Horodecki, and R. Horodecki, “Separability of mixed states: necessary and sufficient conditions,” Phys. Lett. A223, 1–8 (1996). [CrossRef]
- B. Terhal, “Bell inequalities and the separability criterion,” Phys. Lett. A271, 319–326 (2000). [CrossRef]
- M. Lewenstein, B. Kraus, J. I. Cirac, and P. Horodecki, “Optimization of entanglement witnesses,” Phys. Rev. A62, 052310 (2000). [CrossRef]
- O. Gühne, P. Hyllus, D. Bruss, A. Ekert, M. Lewenstein, C. Macchiavello, and A. Sanpera, “Experimental detection of entanglement via witness operators and local measurements,” J. Mod. Opt.50, 1079–1102 (2003). [CrossRef]
- M. Barbieri, F. De Martini, G. Di Nepi, P. Mataloni, G. D’Ariano, and C. Macchiavello, “Detection of entanglement with polarized photons: experimental realization of an entanglement witness,” Phys. Rev. Lett.91, 227901 (2003). [CrossRef] [PubMed]
- M. Bourennane, M. Eibl, C. Kurtsiefer, S. Gaertner, H. Weinfurter, O. Gühne, P. Hyllus, D. Bru, M. Lewenstein, and A. Sanpera, “Experimental detection of multipartite entanglement using witness operators,” Phys. Rev. Lett.92, 87902 (2004). [CrossRef]
- N. Kiesel, C. Schmid, U. Weber, G. Tóth, O. Gühne, R. Ursin, and H. Weinfurter, “Experimental analysis of a four-qubit photon cluster state,” Phys. Rev. Lett.95, 210502 (2005). [CrossRef] [PubMed]
- O. Gühne and G. Tóth, “Entanglement detection,” Physics Reports474, 1–75 (2009). [CrossRef]
- After submission of this work we became aware of an entanglement witness measurement in high dimensional orbital angular momentum states: M. Agnew, J. Leach, and R.W. Boyd, “Observation of entanglement witnesses for orbital angular momentum states,” Eur. Phys. J. D66, 1–4 (2012). [CrossRef]
- S. Gröblacher, T. Jennewein, A. Vaziri, G. Weihs, and A. Zeilinger, “Experimental quantum cryptography with qutrits,” New J. Phys.8, 75 (2006). [CrossRef]
- R. Ghosh, C. K. Hong, Z. Y. Ou, and L. Mandel, “Interference of two photons in parametric down conversion,” Phys. Rev. A34, 3962–3968 (1986). [CrossRef] [PubMed]
- Z. Y. Ou and L. Mandel, “Violation of bells inequality and classical probability in a two-photon correlation experiment,” Phys. Rev. Lett.61, 50–53 (1988). [CrossRef] [PubMed]
- Y. H. Shih and C. O. Alley, “New type of einstein-podolsky-rosen-bohm experiment using pairs of light quanta produced by optical parametric down conversion,” Phys. Rev. Lett.61, 2921–2924 (1988). [CrossRef] [PubMed]
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