1. Introduction

Characterizing a quantum state is a fundamental step in quantum information science. To achieve this goal it is used a technique known as quantum state tomography, which consists of a series of measurements on a large number of identically prepared copies of a system using a special set of measurement operators. The experimental results are used together with the maximum likelihood method to obtain a meaningful density matrix that represents the physical state under study. This technique has been used to reconstruct the density matrix of different quantum systems such as the spin and the energy levels of trapped ions [1, 2], molecules [3], and photons [4].

Spontaneous Parametric Down Conversion (SPDC) [5] is a natural source of entangled systems because the photon pairs are easy to produce and manipulate. Another advantage of this system is that entanglement occurs in many degrees of freedom of the photon pairs, e. g., polarization [5, 6]; transversal [7], longitudinal [8, 9], and orbital angular momenta [10, 11]; energy-time [12], and time bin [13,14]. The entanglement can even exist simultaneously, in two or more different degrees of freedom of the photon pairs, the so called hyperentanglement [15–18].

The quantum tomographic technique has been used to reconstruct the system density matrix in the various photon’s degrees of freedom: polarization [4, 19], transversal momenta [11, 20, 21], orbital angular momenta [10], and in more than one degree of freedom in hyperentangled photon pairs [22]. Research in improving the efficiency of quantum state estimation techniques is an area of intense theoretical study [23–25]. For many applications it is essential to make the fastest possible density matrix determination. In particular, Řeháček et al. [26] proposed a method for state estimation of polarization-based single qubits, which requires a minimum number of measurements. In [27], the authors implement this theoretical method in a tomographic reconstruction of twin-photons polarization states.

The tomographic reconstruction is based in measurements on a quorum of measurement operators. In order to implement them, we need to control phases and amplitudes. G. Lima and collaborators [28] have shown that a Spatial Light Modulator (SLM), can be used to perform this kind of state control. Recently, the SLM was used for tomographic purposes in the photon’s polarization degree of freedom by Cialdi et al. [29]. In [30], the authors used the SLM to observe the quantum entanglement of down converted photon pairs. It was also used to demonstrate a Bell’s inequality violation in the orbital angular momentum of photon pairs [31].

In this work, we use a SLM to reconstruct the state’s density matrix of two photonic spatial qubits. Two spatial qubits were produced in the experiment when two photons generated by SPDC cross a double-slit. The method of minimal quantum state tomography proposed in [26] is implemented for obtaining the density matrix of two qubit photon states. The minimal tomography technique is applied for the first time for obtaining the density matrix of photon pairs entangled in the spatial variables.

The paper is organized as follows: in section 2, we describe briefly, the mathematical theory used in a minimal quantum state tomography protocol; the experimental results, are presented in section 3, together with a discussion of these measurements; section 4 is dedicated to a discussion in which we characterize and quantify entanglement, using our principal result, the system density matrix. In this section, we also make a correspondence between the experimental results and the density matrix. Section 5 is dedicated to the paper’s conclusion.

2. Theory

Let us start in abstract terms by considering only one qubit. A source generates identically prepared qubits, described by a density matrix ρ. Our problem is to obtain ρ. With the usual {|0〉, |1〉} basis, suppose we have a detection apparatus which clicks with a probability proportional to Tr (ρP), with P = |+〉 〈+| the projector over $|+〉=\frac{1}{\sqrt{2}}(|0〉+|1〉)$. Moreover, suppose we are able to implement alternative evolution maps, on the density operator given by the matrices:

In this sense, for each of the four evolution maps (always assuming the same source), the detector counts of the evolved density operator $E_j\rho E_j^{†}(j=1,\dots ,4)$ will be given by

An important property of the Π_j measurement operators that can be easily verified is that their sum is equal to the identity operator. Since each Π_j is a positive operator and they add up to the identity, {Π_j} can be considered as a positive operator valued measure (POVM). The most important implication of it is that the four counting ratios c_i are minimal for the one-qubit state reconstruction. One simple way to recognize it is to write the density matrix ρ in terms of parameters, and to recognize that the system given by (2) allows for a unique solution (one needs four ratios, since relations (2) are not equations, but proportions). This tomographic strategy can be visualized with a tetrahedron in the Bloch sphere, which is represented in Fig. 1 (see [26]).

 

Fig. 1 Schematic vector representation of the tomographic strategy. Measurement operators Π_j, are proportional to projectors |ψ_j〉 〈ψ_j|, with the states |ψ_j〉 being the tetrahedron vertices of the Bloch sphere.

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For two-qubits one only needs to use the sixteen-operator POVM given by {Π_j ⊗ Π_k}. Again, state reconstruction is straightforward, in the ideal case.

3. Experimental Setup

The experimental setup used for obtaining the density matrix of the two-qubit state in path variables is shown in Fig. 2. A 50 mW He-Cd laser, operating at λ = 325 nm, is used to pump a 2-mm-thick Lithium Iodate crystal and generate, by type I SPDC, degenerate non-collinear photon pairs. Signal and idler (λ_s,i = 650 nm) beams passes through a polarizer P₁, before they cross a double-slit placed perpendicular to the signal and idler beams plane at a distance of 250 mm from the crystal. Considering the pump beam direction as the z-direction, the double-slit plane is in the x-y plane with its smaller dimension in the x-direction. The slits are 2a = 100 μm wide and have a separation of 2d = 250 μm.

 

Fig. 2 (Color online) Experimental setup scheme for quantum tomography, in the transverse path degrees of freedom of two-qubit photon states. The L₁ lens focuses the pump beam in the double slit’s plane; lenses L_s1 and L_i1 are used to detect the signal and idler beams at the Fourier plane, while lenses L_s2 and L_i2 are used to project the double slits images in the detectors planes. A half-wave plate is placed right after the crystal and polarizers P_i and P_s are positioned in front of APD’s detectors. CNC is a coincidence counter and SLM is the Spatial Light Modulator.

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A 350 mm focal length lens, L₁, placed at 100 mm before the crystal is used for focusing the pump beam at the double-slit plane such that an entangled two-photon state in transverse path variables is generated, after the double-slit [7]. The two-qubit quantum states are created after the photon pairs cross the double-slit plane. The state determination starts with the SLM positioned just behind the double-slit, at 2.0 mm from it, for preventing diffraction. Signal and idler photons are reflected by the SLM, returning through the slit’s paths. They are detected at the image or Fourier plane where there are two single photon module detectors as shown in Fig. 2.

Two 200 mm focal length lenses, L_s1 and L_i1, placed at the focal distance from the detectors plane, are used to obtain an interference pattern at the detectors plane, which, in this case, is the lenses Fourier plane or two 125 mm focal length lenses, L_s2 and L_i2, placed at 2f = 250 mm from the detectors plane, are used for forming the double-slit’s image on the detectors. All measurements necessary to reconstruct the density matrix, ρ, are made in the Fourier plane. By detecting at the origin of the interference pattern, at the Fourier Plane, we are able to produce the detection projector P = |+〉 〈+|, with $|+〉=\frac{1}{\sqrt{2}}(|0〉+|1〉)$. Here {|0〉, |1〉} are the photon path states provided by the slits. Polarizers P_i and P_s are placed attached to the detectors because the SLM modify the photon polarizations differently depending on the applied grey scale. The pump beam is blocked right before the double-slit.

Single-slits, S_i and S_s are placed in front of the detectors and have a width of 100 μm. Their planes (xy_i,s planes) are aligned perpendicular to the propagation direction of the idler and signal beam (z_i,s direction), respectively. The small direction, of each slit, is parallel to the corresponding x-direction. The used SLM is a Holoeye Photonics LC-R 2500, which has a 1024 × 768 pixel resolution (each pixel consists in a 19 × 19 μm square) and it is controlled by a computer. Signal and idler beams were focused on the detectors with a microscope objective lens and two interference filters, centered at 650 nm, with 10 nm Full Width Half Maximum (FWHM), were kept before the objective lenses. Pulses from the detectors are sent to a photon-counter and a coincidence detection setup with 5.0 ns resolving time.

4. Experimental Results

In Eq. (1), we have defined the evolution maps E_j (j = 1, . . . , 4) used in our tomographic reconstruction. Our experimental setup is arranged in a way that a photon passing through the inferior slit of the double-slit corresponds to state |0〉, while a photon that passes through the superior slit corresponds to state |1〉. To prepare these maps, we must be able to modify the amplitude and phase of each of these states. It was shown [32] that a SLM plus the input and output polarizers, can be properly calibrated to obtain this goal. Our state engineering method will be described in [33]. The liquid crystal display of the SLM is divided in four regions for the two qubit tomography, each region attenuating and/or adding a phase to the photon path state defined by the slits. A SLM gray level is associated with each one of the display’s regions. Different gray levels introduce a relative phase between the photon path states. By also adjusting correctly the half-wave plate and polarizers P_i and P_s angles, in the two-particle tomography (Fig. 2), the necessary relative path attenuation is also obtained. The evolution maps E_j (j = 1, . . . , 4) are implemented when the correct phase difference and amplitude attenuation are introduced in the photon paths by the SLM and the polarizers. In our setup, the SLM can not perform state rotations (|0〉 → (a |0〉+b |1〉)), so, the implemented SLM maps are diagonal. In Fig. 3 and Fig. 4 we measured the phase difference and attenuation, respectively, and the maps E_i might be fully determined.

 

Fig. 3 Double slit interference patterns. In (a), the conditional interference pattern is presented. Idler detector is kept fixed at x_i = 0 (closed circles) or at x_i = 250 μm (open circles), while signal detector is scanned. This result is obtained with the SLM turned off. Graphs (b) and (c), show the patterns obtained for phase differences of 0 and π rad, added by the SLM, between states |0〉 and |1〉, respectively, and the relative amplitude ratio necessary for implementing the evolution maps. Plot with closed circles in (a) is reproduced in (b) and (c), as a reference.

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Fig. 4 Double slit images recorded with the idler detector D_i fixed at x_i = 0, while D_s is scanned in the x direction. Lenses L_i2 and L_s2, described in the experimental apparatus, were used. In (a), we have the same experimental parameters used to obtain the conditional interference patterns, shown in Fig. 3(a). The phase difference imposed, by the SLM, on states |0〉 and |1〉 were: (b) 0, (c) π rad The states relative probability attenuation, imposed by the SLM and the polarizers was one half. Closed circles are single counts, and open circles coincidence detections.

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Fig. 3 shows interference patterns obtained when the idler detector, D_i, is kept fixed, while D_s is scanned in the x direction. This measurement is made using lenses L_i1 and L_s1, and removing lenses L_i2 and L_s2 in our experimental apparatus, in order to obtain the double slit’s interference pattern. In particular, the interference patterns, shown in Fig. 3 (a) were obtained when the SLM was turned off. In this graph, it is shown the two double-slit conditional interference pattern [34, 35], i. e., the fringe pattern depends on the position of both photon detectors. In the closed circles graph, detector D_i is fixed at x_i = 0, while for the open circles, detector D_i is fixed at x_i = 250μm (the position of the first minimum in the closed circles interference pattern). We can see that the interference pattern changes drastically: a maximum in the first measurement transforms to a minimum, in the second one. The closed circles plot is shown in every graph of the figure for comparison purposes. In Fig. 3 (b) and Fig. 3 (c), we show the interferences patterns obtained when two different gray levels are generated at the SLM, each in one of the two regions at the signal side where the signal photon is reflected. By choosing the right gray levels and polarizer angles we are able to implement the maps E₁(Δϕ = 0) and E₂(Δϕ = π rad), respectively, as given in Eq. (1). In the plots of Fig. 3 (b) and Fig. 3 (c), the same gray levels are applied to the two-regions of the SLM in the idler side (path phase difference equal to zero) and the idler detector is kept fixed at x_i = 0 while signal detector is scanned.

Our SLM was not able to implement maps E₃ and E₄, directly. To avoid this problem, we have used a alternative experimental approach. Different gray levels at the SLM and angles of the input and output polarizers were chosen to implement the evolution map $E\prime =\left[\begin{array}{ll}1\hfill & 0\hfill \\ 0\hfill & \frac{1}{\sqrt{2}}\hfill \end{array}\right]$ in the signal and idler sides. On the other hand the detectors were not kept fixed at x = 0 as above but at the position x = ±125μm in order to implement the detector projector P_±π/2 = |±π/2〉 〈±π/2| with $|\pm \pi /2〉=\frac{1}{\sqrt{2}}(|0〉\pm i|1〉)$. It can be shown that ${\Pi }_j=\frac{2}{3}{E\prime }^{†}{P}_{\pm \pi /2}E\prime$, with j = 3 for plus and j = 4, for the minus sign.

The relative phase difference, in the signal beam, was verified fitting the experimental data. This fitting is shown as solid lines in each graph of Fig. 3. For the closed circles data of Fig. 3(a), which is our reference measurement and has no additional phase imposed by the SLM, we have found a phase difference of Δϕ = (–0.8 ± 0.2) rad, in the experimental fit. Using the same procedure for Figs. 3 (b) and (c), we obtain Δϕ₀ = (−1.0±0.3) rad and Δϕ_π = (−4.0±0.2) rad, when external phases of zero and π rad are added by the SLM, respectively. In this way, one can see that the SLM can provide these two phases presented in our evolution maps, within the experimental error, and together with the alternative experimental approach, described in the previous paragraph, we are able to obtain each of the necessary phases for the evolution maps.

In Eq. (1), we see that besides the phase control, we need to modify the relative amplitude between the basis states in order to produce the evolution maps. Notice that in all maps, in this equation, there is a relative probability attenuation of one-half between these states. As mentioned above, different gray scales for each slit path and polarizer angles are carefully chosen, in a way that the necessary phase difference and amplitude attenuation are added to the photon paths. Fig. 4 shows double-slit’s images, with the same experimental parameters used in Fig. 3, except that in order to obtain the images, we changed the lenses L_i1 and L_s1 to L_i2 and L_s2, in the experimental apparatus. The closed circles in the graphs, refers to single counts, while the open circles are coincidence counts. Again, the idler detector was fixed in x_i = 0. All measurements were made in 10 seconds. The left peak (negative position values in the x_s scan) corresponds to the base state |0〉, while state |1〉 is characterized by the right side peak in the graphs. In this figure, we verify the amplitude control necessary to implement the maps described by Eq. (1). In Fig. 4 (a), the SLM was turned off, which corresponds to the same experimental conditions used in Fig. 3 (a). The area under each peak was obtained using a double gaussian fit and has a value of A₀ = (27 ± 1) in arbitrary units for the |0〉 state, while |1〉 state has a area of A₁ = (28 ± 1) in arbitrary units. Since no relative attenuation is added by the SLM, this measurement shows that the prepared photon path base state in the signal side are balanced. A similar graph is obtained for the idler side. Fig. 4 (b) and (c) show the signal double-slit images when we employ the same gray levels as the ones used to obtain the phase difference of 0 and π rad, respectively. The areas in the coincidence plot of Fig. 4 (b) were A₀ = (15 ± 1) and A₁ = (9 ± 1), while in Fig. 4 (c), A₀ = (22 ± 1) and A₁ = (13 ± 1), all areas are in arbitrary units.

In order to prepare the spatially entangled two-qubit states, the pump beam was focused in the double-slit’s plane [36], using a lens with 300 mm of focal distance, shown in Fig. 2 as L₁. Double slit’s image measurements, shown in Fig. 5 (a) and (b), were done for testing the state preparation. Closed circles corresponds to single counts, and open circles to coincidences. All measurements were made in 10 seconds. Once again detector D_i is kept fixed, while the detector D_s is scanned in the x direction. On Fig. 5 (a), the idler detector position is fixed in x_i = 120 μm, which corresponds to the central position of the superior slit image, and on (b) in x_i = −120 μm, that corresponds to the central position of the inferior slit image. L. Neves et al. showed that the twin-photon state in spatial variables prepared with an analogous experimental setup is correlated, in a way that if the idler photon passes through one of the slits, the signal photon will pass through the symmetrically opposite one [7, 37]. In our case, as we are dealing with a double slit, it indicates that if idler photon passes through the superior (inferior) slit, the signal photon will pass through the inferior (superior) one. Fig. 5 (a) and (b) shows the results. In these measurements, the SLM’s gray level was the same in all regions of the SLM display behind the slits. This correlation, known as conditional images, together with the conditional fringes, shown in Fig. 3 (a), are usually considered as entanglement signatures [34–36]. However, each result can be, independently, simulated with separable states. This reinforces the utility of quantum state tomography, as a tool, for unambiguously decision on entanglement issues. To verify the conditional aspect of this measurement, we have calculated the areas under each peak of Fig. 5. In Fig. 5 (a), the peak in the coicidence plot that corresponds to state |0〉 has an area of A_0C = (43.8 ± 0.1), while state |1〉 has an area of A_1C = (0.4 ± 0.1). In Fig. 5 (b) the areas are A_0C = (0.9 ± 0.1) and A_1C = (46.3 ± 0.1). Once again the areas are in arbitrary units.

 

Fig. 5 Double slit conditional images. Closed circles are single counts, and open circles are coincidence detections. In (a) detector D_i is fixed in the superior slit, in (b) D_i is fixed in the inferior slit, while detector D_s is scanned in the x-direction at the image plane. In these measurements the SLM was turned off.

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We finally turn to the tomographic measurements. In order to obtain the to-mographic reconstruction of a two-qubit state, we must be able to implement the sixteen measurement operators, given by {Π_i ⊗ Π_j}, as discussed in section 2. This is done implementing the evolution maps, in both signal and idler photons paths. Our measured coincidence vector was given by: V⃗_C = (c_1,1, c_1,2, ..., c_4,4) = (921, 475, 102, 921, 271, 1332, 700, 404, 229, 392, 634, 108, 843, 513, 69, 875), the experimental errors associated with the measured coincidences given by the square root. In the calculation used to obtain the density matrix, we have subtracted the accidental coincidences counts, which has the maximum value of 5, for each V⃗_C element. The obtained density matrix for the two-qubit state, on the {|00〉, |01〉, |10〉, |11〉} basis, is:

Fig. 6 shows, schematically, the modulus of the real and imaginary parts of ρ̂ elements. All the measurements, for two-qubits, were made using 1500 seconds as the acquisition time.

 

Fig. 6 (Color online) Tomographic reconstruction of the output state for two-qubits. The figure represents the modulus of the real and imaginary parts of each density matrix measured element

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5. Discussion

We begin this section with a brief discussion regarding the robustness of the experimentally implemented maps, in comparison with the theoretical ones, shown in Eq. (1). To perform the operators Π_i we have used the SLM to implement the maps E_i. We construct the experimental maps using the relative attenuation and phases for the photon paths, shown in Fig. 3 and Fig. 4. One way of quantifying the robustness of the technique is to calculate the fidelity, which is defined by $F(\sigma ,\rho )=\sqrt{\mathit{Tr}\left(\sqrt{\sigma \rho }\sqrt{\sigma }\right)}$, between theoretical and implemented operators:

Since the projector implementation P and P_±π is well established [36], the experimentally implemented and theoretical operators Π_i differ mainly because of the maps. The operators Π₃ and Π₄ use the map E′ to be implemented and the fidelities are equal to $F\left({\Pi }_1^{\mathit{exp}},{\Pi }_1^{\mathit{theory}}\right)$. In this way, we see that the fidelity between our experimental and theoretical maps are very close to one, which ensures the technique robustness.

We now turn our discussion to the tomographic result in itself, namely the state density matrix presented in Eq. (3). The state fidelity, associated with the Bell state |Ψ⁺〉 is 0.86. We can see, that the population elements, corresponding to states |00〉 and |11〉, are close to, but different from zero. This can be visualized experimentally, if one looks at the conditional image results in Fig. 5. The areas A_1C in Fig. 5 (a) and A_0C in Fig. 5 (b), should be zero in the ideal case. But our experimental setup was not able to provide this, so we have population reminiscences, which is consistent with the non-nullity of terms ρ₁₁ e ρ₄₄, in the density matrix Eq. (3).

The state obtained has purity P (ρ) = Tr[(ρ̂)²] = 0.83, and concurrence [38] C = 0.80, which certifies it’s entanglement.

6. Conclusion

In this work, we demonstrated that spatial light modulators can be used to perform quantum state tomography in the transverse path degree of freedom of the photon pairs, prepared when photon pairs generated by spontaneous parametric down conversion cross a double-slit. In particular, we have used the minimal tomographic technique in spatial variables, which reduces the number of measurements required to reconstruct the state’s density matrix.

The experimental apparatus is easy to manipulate, and can be used in fundamental tests of quantum mechanics, such as Bell’s inequalities violations, and characterization and quantification of entanglement. The SLM can be used to manipulate and control photon pairs in hyperentangled states or hybrid states, which gives us the perspective of doing a quantum state tomography of hyperentangled system in polarization and transverse path, for example. Another application of this experimental setup is in quantum computation. In [39], the authors reveled a redundancy in the existing form of the Deutsch-Jozsa algorithm. Our experimental setup was used to implement the suggested scheme, and in doing so, we have confirmed its validity [40].