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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 24 — Nov. 19, 2012
  • pp: 26351–26362
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Experimental characterization of two spatial qutrits using entanglement witnesses

A. J. Gutiérrez-Esparza, W. M. Pimenta, B. Marques, A. A. Matoso, J. L. Lucio M., and S. Pádua  »View Author Affiliations


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

Detection and quantification of entanglement in composite quantum states of arbitrary dimensions is an important and fundamental resource both for theoretical studies and for quantum information applications. In particular, entanglement theory of bipartite systems frames for states of two qubits (2 ⊗ 2) or of one qubit and one qutrit (2 ⊗ 3) the positivity under partial transposition (PPT) of the density matrix (σTB > 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

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

]). For any other dimensionality, however, positivity of the partial transposition is a necessary but not sufficient condition: any bipartite state of (NM) with nonpositive partial transposition (NPT) describes an entangled state, but there are some PPT density matrices describing non-separable states.

A scheme for the determination of entanglement for NPT states must be independent from assumptions on special forms for the state or on its purity. A way to surmount these hurdles is through the measurement of observables called entanglement witnesses (EW), which ensure a necessary and sufficient separability criterion. According to the theory developed in [2

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

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]

], an entanglement witness operator is defined as an hermitian operator 𝒲 = 𝒲 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(𝒲σs) ≥ 0 for all density matrices σs describing separable states and Tr(𝒲σe) < 0 for at least one density matrix σe connected to an entangled state. Existing implementations of the EW method [6

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

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

] in the case of two, three and four qubits have been published.

The present work reports a two-qutrit pure state involving three complex parameters. As far as we know, no experimental work has been reported with the experimental characterization of a qutrit NPT state [10

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]

]. Besides the intrinsic relevance in the entanglement quantification of states of this dimensionality, qutrits are of interest in diverse areas such as security margin and higher dimensional alphabets relevants in criptography [11

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

]. The experiment reported in this work relies on the use of pairs of entangled photons in qutrit states generated by spontaneous parametric down-conversion (SPDC) [12

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

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]

]. Experimentally, such pairs of photons can be propagated and detected separately while preserving information about one another. Such information may exhibit correlations in diverse degrees of freedom –continuous and discrete spectra– such as polarization [13

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

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]

], number of photons [16

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

], orbital angular [17

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]

], longitudinal [18

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

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]

], and transverse momenta [20

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

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]

], energy-time [22

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]

], and time-bin [23

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

, 24

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]

]. On this basis, we have taken advantage of the transverse momentum correlations in order to generate bipartite entangled states belonging to a three-dimensional Hilbert space (spatial qutrits) [25

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

]. This is achieved by sending the entangled photons through apertures with three slits. Each slit defines a possible photon path which is associated with an orthonormal state. The technique has been developed and an optimal control in the generation of qudits using entangled photons pairs as a primary element exists [20

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]

].

An important feature of any entanglement criterion or quantification is the easiness of an experimental implementation, in particular as far as the locality of the measurements is concerned. We consider EW operators decomposed into a linear combination of local operators, involving single photon path states or linear combinations of these states. Our work presents the experimental techniques for the verification and quantification of entanglement in two-qutrit pure states by means of the experimental determination of the expectation values of each of these operators.

This paper proceeds as follows: In Sec. 2 we describe the quantum state of interest and review a general scheme for the construction of EW operators for NPT states. Then, we introduce their decomposition into locally measurable observables, using a basis for the Lie algebra SU(3). In Sec. 3 we describe the experimental setup used for the generation and characterization of path states and the implementation of measurement settings for obtaining the expectation values of witness operators by means of amplitude modulation. We also report the experimental results. In Sec. 4, the results are discussed and the conclusions presented.

2. Theory

In the following, we briefly discuss the theoretical outline for the construction and decomposition of EW operators for a given quantum system. Consider an experimental setup able to produce a two-qutrit state described by
|Ψ=a|02+b|11+c|20|a|2+|b|2+|c|2.
(1)
The state |Ψ〉 is an element of the ℋ𝒜3 ⊗ ℋ3 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.

Let us now consider the EW. Since the two qutrit state ρ = |Ψ〉〈Ψ| is a NPT state, at least one EW operator can be constructed by computing the partial transpose of the eigenvector |ηj〉 that corresponds to some nonpositive eigenvalue εj of ρTB. For a suitable notation purpose, it is convenient to express the amplitudes in polar form a = |a|e2a, b = |b|e2b, and c = |c|e2c. We call the phase differences entering in the photon path states as θj = θrθs (j = 1, 2, 3; r, s = a, b, c). In Table 1, three possible eigenvectors corresponding to the negative part of the spectrum of ρTB are reported. As a result, three EW operators 𝒲j (j = 1, 2, 3) can be constructed. Such witnesses detect the state ρ, when their expectation value give 〈𝒲j〉=Tr(|ηj〉 〈ηj|TB ρ) = εj < 0. The resulting witnesses have similar mathematical structures that can be written as
𝒲j=12(|llll|+|mmmm|e2i(θj)|mllm|e2i(θj)|lmml|),
(2)
where {|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 𝒲j is composed by the path states {|l〉, |m〉}𝒜 ⊗ {|l′〉, |m′〉}. According to Table 1, one can infer that 𝒲1 involves {|0〉, |2〉}𝒜 ⊗ {|0〉, |2〉}, 𝒲2 involves {|1〉, |2〉}𝒜 ⊗ {|0〉, |1〉}, and 𝒲3 involves {|0〉, |1〉}𝒜 ⊗ {|1〉, |2〉}.

Table 1. Nonpositive Eigensystem of ρTB

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Notice that Eq. (2) is not expressed in terms of measurements operators that can be directly interpreted in terms of local measurements. A straightforward way for implementation of non-local observables like 𝒲j = |ηj〉 〈ηj|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:
𝒳lm±=|xlm±xlm±|;|xlm±=12(|l±|m),𝓎lm±=|ylm±ylm±|;|ylm±=12(|l±i|m).
(3)

A decomposition of 𝒲j 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 λγ, which are the generators of the fundamental representation of the SU(3) group [26

26. J. Fuchs and C. Schweigert, Symmetries, Lie algebras and representations: A graduate course for physicists (Cambridge University Press, 1997).

]. SU(3) is generated by the 3 × 3 hermitian, traceless matrices with determinant 1, described by
λ1=(010100000),λ2=(0i0i00000),λ3=(100010000),λ4=(001000100),λ5=(00i000i00),λ6=(000001010),λ7=(00000i0i0),λ8=13(100010002).
(4)

They are normalized such that Tr(λnλm) = 2δnm. Since both λ3 and λ8 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
λ1=𝒳01+𝒳01,λ2=𝓎01+𝓎01,λ3=|00||11|,λ4=𝒳02+𝒳02,λ5=𝓎02+𝓎02,λ6=𝒳12+𝒳12,λ7=𝓎12+𝓎12,λ8=13(|00|+|11|2|22|).
(5)

Using the same basis, the Gell-Mann matrices along with the identity λ9 = form an orthogonal basis for the complex Hilbert space of all 3 × 3 matrices. One can therefore decompose any single operator 𝒪Q3 defined in one qutrit Hilbert space ℋ = 3 as
𝒪Q3(d)=12k=19λkTr(λk𝒪Q3),
(6)
and decompose an operator 𝒪nQ3 defined in n-qutrit Hilbert space =1323n3 in terms of tensor products as
𝒪nQ3(d)=12nk1=19kn=19λk1λknTr(λk1λkn𝒪nQ3).
(7)

Thus, EW operators 𝒲j for two-qutrit systems, i.e., 9 × 9 matrices, can be decomposed into a sum of tensor products of Gell-Mann matrices of the form
𝒲j(d)=14k1=19k2=19λk1λk2Tr(λk1λk2𝒲j).
(8)

In the following, we will restraint to the case a = b = c = 1 where the phase differences entering in the photon path states are the same. This assumption reduce the witness structure to the first and second terms only, obtaining as a result, a local decomposition into 10 tensor products of operators. In this configuration, the resulting witnesses are given by
𝒲setup(j)=14(2|ll||ll|+2|mm||mm|𝒳lm+𝒳lm++𝒳lm+𝒳lm+𝒳lm𝒳lm+𝒳lm𝒳lm𝓎lm+𝓎lm++𝓎lm+𝓎lm+𝓎lm𝓎lm+𝓎lm𝓎lm).
(12)
Each witness involves correlated measurements in 8 settings of the form 𝒪lm+𝒪lm+ for path superposition operators and two settings |l〉〈l| ⊗ |l′〉〈l′| for single path operators. They only differ in the path states involved for each local measurement. The theoretical estimate of their expectation values gives 𝒲setup(j)th=1/3 for j = (1, 2, 3) to a maximally entangled state.

3. Experimental setup

The outline of the experiment for generation of a two-qutrit pure state and implementation of EW operators is illustrated in Fig. 1. In the experiment, a 50 mW diode laser beam operating at 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 λs,i = 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.

Fig. 1 Experimental setup for the generation and entanglement detection of a two-qutrit state. An entangled photon source is made from type I SPDC by pumping a Bismuth Borate (BiBO) crystal using a CW blue laser beam. The two-qutrit state is produced by placing a 3-slit aperture into the photon paths. A spatial light modulator (SLM) together with a polarizing beam splitter (PBS) are used for the characterization and entanglement witness implementation of the state. The measurement operators are implemented by using convergent lenses Ls and Li when a three-slit image or an interference pattern are created at the detector planes Ds and Di. C is a photon counter which records single and coincidence counts.

In order to generate qutrit states we use an aperture with three slits of width 2a = 100 μm with a slit spacing of d = 250 μm 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.

A reflective SLM is placed right after the aperture, 1.5 mm behind it, to avoid aperture diffraction at the SLM plane. This modulator is a Holoeye Photonics LC-R 2500 SLM, with a pixel resolution 1024 × 768 (each pixel consists in a 19 × 19 μm 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]

]. Additional polarizers are necessary for this device to operate properly [28

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]

]. In this sense, the PBS works as two polarizers, horizontal transmission for the incoming beams, vertical reflexion for the outgoing ones. In this work the SLM and the PBS are arranged for doing amplitude-only modulation. At this stage it is worth mentioning that depending upon the gray level configuration of the LCD, we will perform the characterization of the two-qutrit state and the realization of measurements of single and superposition path operators. Such important roles of the SLM will be discussed in detail in the following sections.

After being reflected by the SLM and passed through both the three-slit aperture and the PBS once again, the two qutrits are still linearly polarized, but now with a vertical polarization direction. Then, the qutrits are sent towards a source characterization module that will be used for EW implementation. This module includes convergent lenses Ls and Li, both with focal length f = 200 mm, fixed at a distance zL from the detectors Ds and Di, which are kept fixed at a distance of 800 mm from the aperture. In front of each detector there is a 100-μm-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

The preparation of the spatial two-qutrit state is achieved when the two entangled photons are transmitted by the three-slit aperture. Due to the transverse correlations [29

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]

] of the photon pairs at the three-slit aperture, they will pass only through symmetrically opposite slits, generating thus the two-qutrit entangled state of the form of Eq. (1). This preparation is ensured by putting a lens Lp of focal length fp = 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.

A schematic representation is exemplified in Fig. 2(a). Horizontally polarized incoming photons, which are transmitted by the PBS and the three-slit aperture, strike the LCD, addressed previously with certain gray level configuration, such that the reflected photons may have their polarization modified. Due to the changes in polarization, the reflected photons may exhibit amplitude modulation after being reflected by the PBS. This modulation is done by changing the gray level of the LCD at the pixels where the two qutrits have arrived on the LCD. In this way, the SLM-PBS arrangement is used to modulate the amplitude of both prepared qutrits, independently. Different amplitudes can be obtained by means of an interface which allows to control the gray level of the SLM just behind each slit from a computer. The SLM configuration that allows the characterization of the two-qutrit state at the minimum attenuation of the SLM-PBS arrangement, analogous to a mirror, is achieved by setting a gray level=240 using the calibration method presented in Ref. [28

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]

].

Fig. 2 Experimental setups to demonstrate the (a) characterization with a minimum modulation (gray level=240), and the selection of path states (b) {|0〉, |2〉}s ⊗ {|0〉, |2〉}i, (c) {|1〉, |2〉}s ⊗ {|0〉, |1〉}i, and (d) {|0〉, |1〉}s ⊗ {|1〉, |2〉}i, by setting the gray levels 240 and 0 for minimum and maximum attenuations, respectively, just behind the slits at the LCD.

Coincidence measurements are performed onto the basis |0〉, |1〉, and |2〉, selecting the three photon path states in the following way: the three slits in each photon path are imaged by the lenses Ls and Li on the detector planes Ds and Di, respectively. Such images are achieved when the lenses are placed into the halfway between the aperture and the detectors, at a distance 2f after the aperture and zL = 2f from the detector plane. At the image plane, Ds is kept fixed at the maximum single counts for each slit |0〉, |1〉, and |2〉, while the idler image is scanned by the detector Di 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.

Table 2. Probabilities for all basis states of 𝒟 = 3.

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Only the full determination of the two-qutrit state is achieved by performing a quantum-state tomography. However, by looking at the interference patterns we can know that the state purity is high and the phases between the slits are approximately zero [30

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]

]. Thus, the state obtained experimentally is approximately
|Ψ(0.57±0.02)|02+(0.59±0.02)|11+(0.56±0.02)|20,
(13)
observed with a fidelity of 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.

It has been shown for an entangled pure state that the coincidences counts exhibit an interference pattern with conditional fringes while for an incoherent mixture there is no interference [31

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

]. Therefore, in order to show the prepared two-photon state after the three-slit aperture is indeed not an incoherent mixture, coincidence measurements are performed at the Fourier plane. Interference patterns are obtained by inserting the lenses, after the apertures, in each arm of the two photons at a distance zL = f from the detector plane. Coincidence measurements are made by scanning the Fourier plane in the x-direction with the Di while keeping fixed the detector Ds at xs = 0 mm and then at xs = 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

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]

] for two ququarts.

3.2. Local measurement operators of the EW operators

EW operators are decomposed into local operators involving ten specific measurement settings in single and superposition operators. These measurements are achieved in the following way: local measurement onto single path operators |0〉 〈0|, |1〉 〈1|, and |2〉 〈2| correspond to a measurement through a single slit when the three slits are imaged on the detectors. Such measurement settings are identical to those performed in the characterization of the state, using the SLM con-figuration illustrated in Fig. 2(a). The experimental values for single path operators are reported in Table 2. On the other hand, the experimental realization of operators 𝒳lm± and 𝓎lm± requires superpositions of lm and lm′ paths for signal and idler arms, respectively, with specific phase differences.

First, the superposition task is done by the attenuation of a single photon path to achieve the transmission in only two possible photon paths to the detectors. Such attenuation is implemented by using specific configurations of the SLM-PBS arrangement.

A three-slit image, proportional in size to the three-slit aperture, is transferred to the SLM by a computer interface. Each slit image on the display is located at a specific pixel area with an associated gray level. It is possible to change the gray level of a single slit image and obtain, as a result, amplitude attenuation for individually chosen paths. In Fig. 2(b), (c), and (d) it is illustrated how gray level configurations of the LCD are used to obtain only two-path interference. For instance, a two-slit image illustrated in Fig. 3(a), is obtained when the central slit |1〉 is attenuated using a gray level=0, corresponding to the setup configuration shown in Fig. 2(b). The attenuation percentage achieved is around 93%, showing that we are able to implement almost full atenuation in order to implement the superposition operators.

Fig. 3 At the image plane, (a) a double slit image is recorded by idler detector when the photon path state |1〉 is attenuated by using a gray level=0 of the SLM. As a result, (b) a two-slit interference pattern is obtained at the Fourier plane with Di scanning in the x-direction at the idler arm, while Ds is kept fixed at xs = 0 mm. The path phase differences of ±π and ±π/2 with respect to the central maximum give the states related to the measurement operators 𝒳02± and 𝓎02±. Continuous curves represent the theoretical fits of the experimental data.

At the Fourier plane, coincidence measurements are made in an analogous way to those obtained with the procedure used for the state characterization. As a result, interference patterns of lm and lm′ paths are obtained for signal and idler detectors, respectively. This procedure, which consists in scanning the two-slit pattern with the Di detector while keeping fixed the detector Ds, is illustrated in Fig 3(b).

Each operator 𝒪lm± has a defined phase difference between the 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 |xlm+. Thus, a detector fixed at the first minimum on the right side of the interference pattern corresponds to a phase difference ϕ = π, and for the left side corresponds to ϕ = −π. In both cases, the detector is selecting only photons in the state |xlm. The phase difference 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 |ylm+ while at ϕ = −π/2 in the state |ylm. This procedure is illustrated in Fig. 3(b) in which a two-slit pattern is recorded when only two paths are transmitted to the detector Di 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 𝒪lm±𝒪lm± are implemented by placing the two detectors at the points where the phase differences are such that only photons in the states |xlm± and |ylm± are detected.

In Table 3 we report the experimental values of the probabilities for a set of eight superposition operators for three different path superpositions, each associated to one of the EW operators. These probabilities are coincidences counts normalized by the sum of coincidences counts for all basis states of the operators 𝒳lm± and 𝓎lm±. Thus, the experimental expectation values of the witnesses 𝒲setup(j) can be computed by adding the experimental results reported in Tables 2 and 3 with the corresponding weights given in Eq. (12). Experimentally it is found Tr(𝒲setup(1)ρ)=0.34±0.08, Tr(𝒲setup(2)ρ)=0.40±0.07, and Tr(𝒲setup(3)ρ)=0.23±0.08. These results clearly prove the entanglement of the detected state.

Table 3. Expectation values of path superposition operators

table-icon
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Since the negativity 𝒩(ρ) is defined as the absolute sum of the negative eigenvalues of ρTB, 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]

]. The negativity can be computed by adding the experimental expectation values of the three witnesses yielding a result 𝒩(ρ)=0.97 ± 0.23. This result agrees with the theoretical prediction (𝒩=1) for the negativity of a two-qutrit maximally entangled state.

Finally, there is a connection between the mean value of an EW operator and the fidelity ℱψ of the desired state |ψ〉 [9

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

]. The state we are considering in Eq. (1) with (a = b = c = 1) can be written in terms of the identity and the three EW operators 𝒲setup(j). Thus, its fidelity is obtained by means of the experimental mean values of all witnesses 𝒲setup(j) as
Ψ=1323(𝒲setup(1)+𝒲setup(2)+𝒲setup(3)),
(14)
since the witnesses 𝒲setup(j) are detecting the same prepared state |Ψ〉. The fidelity is defined as ℱΨ = 〈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 ℱΨ = 0.98 ± 0.15.

A quantum state tomography for the prepared two-qutrit state would require exactly 81 measurements. Its reconstruction provides all the information regarding the quantum state, but only when all measurements have been performed. The entanglement witness method gives only limited information, but requires fewer measurements; with only 10 we can verify that the state is entangled and with 30 measurements we can quantify the entanglement and fidelity.

4. Conclusion

Acknowledgments

We acknowledge Leonardo Neves for very useful discussions and suggestions for feasibility of this project. This work is part of the Brazilian National Institute for Science and Technology for Quantum Information and was supported by the Brazilian agencies CNPq, CAPES, and FAPEMIG. Also, we acknowledge financial support from CONACyT project 136186 (Mexico).

References and links

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]

3.

B. Terhal, “Bell inequalities and the separability criterion,” Phys. Lett. A 271, 319–326 (2000). [CrossRef]

4.

M. Lewenstein, B. Kraus, J. I. Cirac, and P. Horodecki, “Optimization of entanglement witnesses,” Phys. Rev. A 62, 052310 (2000). [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]

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]

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. 92, 87902 (2004). [CrossRef]

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. 95, 210502 (2005). [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]

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]

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]

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]

26.

J. Fuchs and C. Schweigert, Symmetries, Lie algebras and representations: A graduate course for physicists (Cambridge University Press, 1997).

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]

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]

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]

31.

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

32.

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

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

  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. A223, 1–8 (1996). [CrossRef]
  3. B. Terhal, “Bell inequalities and the separability criterion,” Phys. Lett. A271, 319–326 (2000). [CrossRef]
  4. M. Lewenstein, B. Kraus, J. I. Cirac, and P. Horodecki, “Optimization of entanglement witnesses,” Phys. Rev. A62, 052310 (2000). [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]
  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]
  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.92, 87902 (2004). [CrossRef]
  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.95, 210502 (2005). [CrossRef] [PubMed]
  9. O. Gühne and G. Tóth, “Entanglement detection,” Physics Reports474, 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. D66, 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. A34, 3962–3968 (1986). [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]
  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]
  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,” Science328, 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. A78, 012345 (2008). [CrossRef]
  25. L. Neves, S. Pádua, and C. Saavedra, “Controlled generation of maximally entangled qudits using twin photons,” Phys. Rev. A69, 042305 (2004). [CrossRef]
  26. J. Fuchs and C. Schweigert, Symmetries, Lie algebras and representations: A graduate course for physicists (Cambridge University Press, 1997).
  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. Express18, 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]
  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]
  30. E. J. S. Fonseca, J. C. Machado da Silva, C. H. Monken, and S. Pádua, “Controlling two-particle conditional interference” Phys. Rev. A61, 023801 (2000). [CrossRef]
  31. D. Greenberger, M. Horne, and A. Zeilinger, “Multiparticle interferometry and the superposition principle,” Phys. Today46, 22–22 (1993). [CrossRef]
  32. G. Vidal and R. F. Werner, “Computable measure of entanglement,” Phys. Rev. A65, 032314 (2002). [CrossRef]

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