Implementation of quantum state tomography for time-bin entangled photon pairs

Hiroki Takesue; Yuita Noguchi

doi:10.1364/OE.17.010976

1. Introduction

Quantum entanglement is an essential resource in quantum information systems including quantum key distribution (QKD), quantum repeaters, and quantum computation. To realize those systems, establishing a method for evaluating the quality of the entanglement is very important. In past entanglement generation experiments, many evaluation methods have been used including a Bell’s inequality test [1] and a two-photon interference fringe visibility measurement [2]. In 2000, James et al. proposed and demonstrated quantum state tomography (QST) for entangled photon pairs [3]. This method enables us to reconstruct the full density matrix of entangled photon pairs. Using the density matrix, we can obtain various parameters that show the quality of entanglement, such as concurrence, linear entropy, and fidelity. Currently, QST is widely used as the most comprehensive method for evaluating entanglement.

Although QST has been used to evaluate various kinds of polarization entangled photon pair sources [4, 5, 6, 7, 8], it has not been used to measure time-bin entangled photon pairs [9]. A time-bin qubit is a superposed state of two temporal modes, and is more suitable for fiber transmission than a polarization qubit, because it is generally difficult to preserve a polarization state during fiber transmission. In fact, time-bin entanglement has been employed in most long-distance entanglement distribution [10, 11, 12, 13] and entanglement-based QKD experiments [14] over fiber. Therefore, it is important to establish a comprehensive way of evaluating time-bin entangled photon pairs to realize entanglement-based quantum communication over fiber.

It is obvious that we can convert a time-bin qubit into a polarization qubit using linear optics components [15, 16], and so we can implement QST after time-bin qubits are converted into polarization qubits. However, the use of such qubit conversion generally induces additional degradation of the quality of entanglement. Moreover, the insertion of additional loss elements results in an increased measurement time. For example, in the experiment reported in [15], a polarization interferometer was used to convert a time-bin qubit to a polarization qubit. Consequently, the phase fluctuation of the interferometer degraded the quality of the entanglement. In addition, the use of a passive interferometer induced a 6-dB intrinsic loss in [15]. Consequently, QST incorporating time-bin to polarization qubit conversion has not yet been implemented.

In this paper, we describe the QST for time-bin entangled photon pairs. We propose a way to implement QST for time-bin entangled photon pairs using 1-bit delayed interferometers that have been used in previous time-bin entanglement experiments [11, 12, 13, 14]. Then we provide an experimental result, in which we undertook QST for 1.5−µm band time-bin entangled photon pairs generated using spontaneous four-wave mixing (SFWM) [17] in a dispersion shifted fiber.

2. Projection measurements of time-bin qubit using 1-bit delayed interferometer

Let us first consider projection measurements of polarization qubits. We denote the horizontal and vertical polarization states as |H〉 and |V〉, respectively. Then, we can express any polarization state as a point on a Poincaré sphere, as shown in Fig. 1(a). In the QST for polarization entangled photon pairs in [3], they used four projection measurements corresponding to {|H〉, |V〉, |R〉, |+〉} for one photon and five projection measurements corresponding to {|H〉, |V〉, |L〉, |R〉, |+〉} for the other photon. Here, |L〉, |R〉, and |+〉 denote the left hand circular (=(|H〉+i|L〉)/√2), the right hand circular (=(|H〉−i|L〉)/√2) and the diagonal (=(|H〉+|L〉)/√2) polarization states, respectively. The positions of these states on the Poincaré sphere are also shown in Fig. 1(a). They undertook 16 combinations of projection measurements consisting of the above five measurements for entangled pairs. The coincidence counts obtained in this measurement were used to calculate the density matrix of the entangled pairs. Therefore, if we come up with a way to implement these projection measurements for a time-bin qubit, we can use the same procedure to obtain the density matrix of time-bin entangled photon pairs.

A time-bin qubit is defined as a superposed state of two temporal modes. We denote a quantum state in which a photon is in the first (second) temporal position as |1〉 (|2〉). Then, a time-bin qubit is generally expressed as

∣ ϕ 〉 = α ∣ 1 〉 + β ∣ 2 〉,

where α and β are the complex coefficients that satisfy |α|²+|β|²=1. As in the case of a polarization qubit, we can express any time-bin qubit state as a position on the Poincaré sphere shown in Fig. 1(b). We can define states |L〉, |R〉 and |+〉 for time-bin qubit as

$∣ L 〉 = \frac{∣ 1 〉 + i ∣ 2}{\sqrt{2}}$ $∣ R 〉 = \frac{∣ 1 〉 - i ∣ 2 〉}{\sqrt{2}}$ $∣ + 〉 = \frac{∣ 1 〉 + ∣ 2 〉}{\sqrt{2}}$

The positions of the above states on the Poincaré sphere for a time-bin qubit are naturally the same as those on the Poincaré sphere for a polarization qubit. Thus, the projection measurements of the time-bin qubit states {|1〉, |2〉, |L〉, |R〉, |+〉} are equivalent to those of the polarization qubit states {|H〉, |V〉, |L〉, |R〉, |+〉}.

We can utilize the 1-bit delayed Mach-Zehnder interferometer shown in Fig. 2 followed by a single photon detector for time-bin qubit projection measurements.

When a time-bin qubit passes through a 1-bit delayed interferometer, there is the possibility of observing a photon in three time slots. Photon detection at the first or third slot and that at the second time slot corresponds to two non-orthogonal measurement bases. Conventionally, the former is called the “time basis”, while the latter called the “energy basis” [18]. Photon detection at the first (third) slot corresponds to a projection measurement of state |1〉 (|2〉). On the other hand, detection at the second slot corresponds to a projection measurement of a state that is given by a point on the meridian of the Poincaré sphere. When the phase difference between two arms is given by θ, the projected state is given by

\frac{∣ 1 〉 + e^{- i θ} ∣ 2 〉}{\sqrt{2}} .

Therefore, photon detection in the second slot with θ=0 corresponds to a projection to |+〉, while that with θ=−π/2 corresponds to |L〉.

Fig. 1. Poincaré spheres for (a) polarization and (b) time-bin qubits.

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When we use the setup shown in Fig. 2, in which only one detector is connected to an interferometer output, we need to take account of the fact that half of the photon is discarded in the time basis measurement. For example, if a photon in state |1〉 is input into the interferometer, the photon is output from the empty port with a probability of 50%. The situation is different in the energy basis measurement. For example, if a photon in state |+〉 is input into an interferometer with θ=0, the photon is output from one port with 100% probability (if there is no insertion loss in the interferometer). In the coincidence measurements, we need to consider this basis-dependent difference in detection probabilities.

3. Two-photon projection measurement and reconstruction of density matrix

We use the apparatus described in the previous section to perform two-photon projection measurements. Since there are 16 unknown parameters that determine a 4×4 density matrix, we need to undertake at least 16 projection measurements. To achieve this, we choose projection measurements that correspond to {|1〉, |2〉, |+〉, |L〉} for each photon for 16 combinations. The projected two-photon states are summarized in Table 1.

We use the coincidence counts obtained for the two-photon projection measurements listed in Table 1 to construct the density matrix for the photon pair, employing a procedure similar to that in [3]. The fourth column in Table 1 refers to vectors for two-photon states to which photons 1 and 2 are projected. For example, the following vector is obtained for ν=7

$∣ ψ_{7} 〉 = {∣ + 〉}_{s} {∣ + 〉}_{i} = \frac{1}{2} ({∣ 1 〉}_{s} + {∣ 2 〉}_{s}) ({∣ 1 〉}_{i} + {∣ 2 〉}_{i})$ $= \frac{1}{2} (∣ 11 〉 + ∣ 12 〉 + ∣ 21 〉 + ∣ 22 〉),$ ,

where |xy〉≡|x〉_s|y〉_i. Here, we derive density matrices expressed in a time basis (i.e. |1〉 and |2〉 basis).

Fig. 2. Projection measurement using 1-bit delayed interferometer.

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Using the two-qubit Pauli matrices Γ_µ given in Appendix A of [3], we can express the density matrix with the following equation.

\hat{ρ} = Σ_{μ = 1}^{16} {\hat{Γ}}_{μ} r_{μ}

Here, r_µ is the µth element of a 16-element column vector that is given by the following equation.

r_{μ} = T r {{\hat{Γ}}_{μ} \hat{ρ}}

We define n_ν as a measured value obtained in a projection measurement numbered with ν in Table 1. ν_n is given by

n_{v} = C 〈 ψ_{v} ∣ \hat{ρ} ∣ ψ_{v} 〉,

where C denotes a proportional constant that depends on the experimental setup.

Table 1. Tomographic analysis states used in the experimentDThe fourth column refers to the two-photon states to which photons 1 and 2 are projected.

View Table | View all tables in this article

By plugging Eq. (3) into (5), we obtain

(\begin{array}{c} n_{1} \\ n_{2} \\ ⋮ \\ n_{16} \end{array}) = C (\begin{array}{c} B_{1,1} & B_{1,2} & \dots & B_{1,16} \\ B_{2,1} & \dots & \dots & B_{2,16} \\ ⋮ & ⋮ \\ B_{16,1} & \dots & \dots & B_{16,16} \end{array}) (\begin{array}{c} r_{1} \\ r_{2} \\ ⋮ \\ r_{16} \end{array}),

where

B_{x, y} = 〈 ψ_{x} ∣ Γ_{y} ∣ ψ_{x} 〉 .

In the following, we denote a 16×16 matrix whose element is given by B_x,y with BD

Then, using Eq. (6), we can obtain r_µ as

(\begin{array}{c} r_{1} \\ r_{2} \\ ⋮ \\ r_{16} \end{array}) = C B^{- 1} (\begin{array}{c} n_{1} \\ n_{2} \\ ⋮ \\ n_{16} \end{array})

By plugging Eq. (8) into Eq. (3), we obtain the density matrix ρ as

\hat{ρ} = C^{- 1} Σ_{v = 1}^{16} {\hat{M}}_{v} n_{v},

where M̂_ν is given by

{\hat{M}}_{v} = \underset{x}{Σ} Γ_{x} {(B^{- 1})}_{x, ν} .

For the projection states shown in Table 1, it can be shown that Tr{M̂_ν}=1 for ν=1,2,3,4 and Tr{M̂_ν}=0 for other ν values. Therefore, the parameter C can be obtained as

C = Σ_{ν = 1}^{4} n_{ν}

Thus, we finally obtain the density matrix as

\hat{ρ} = (Σ_{ν = 1}^{16} {\hat{M}}_{ν} n_{ν}) ⁄ (Σ_{ν = 1}^{4} n_{ν}) .

4. Experiments

Based on the scheme described in the previous section, we undertook QST on time-bin entangled photon pairs generated using SFWM in a dispersion shifted fiber (DSF) [19, 11]. The experimental setup is shown in Fig. 3. A 1551.1-nm continuous light from an external-cavity laser diode was modulated into double pulses using an optical intensity modulator (IM). The repetition frequency, pulse interval and pulse width were 100 MHz, 2 ns and 100 ps, respectively. The peak power of the pump pulse was ~80 mW. The IM was driven by a signal from a pulse generator, which also generated a trigger signal that was synchronized with the pump pulses and was used as a start pulse for a time interval analyzer (TIA). The double pulses were amplified by an erbium-doped fiber amplifier (EDFA), passed through optical filters to suppress amplified spontaneous emission noise from the EDFA, and launched into a 500-m DSF with a zero dispersion wavelength of 1551.1 nm. The DSF was cooled with liquid nitrogen to suppress the noise photons caused by spontaneous Raman scattering (SpRS) [11]. Through SFWM in the DSF, a time-bin entangled photon pair state is generated whose state is given by the following equation.

∣ Ψ 〉 = \frac{1}{\sqrt{2}} ({∣ 1 〉}_{s} {∣ 1 〉}_{i} + {∣ 2 〉}_{s} {∣ 2 〉}_{i})

Here, the subscripts s and i denote the signal and idler photons, respectively.

The photons from the DSF were input into a polarizer to suppress the SpRS noise photons whose polarization state is orthogonal to that of the photon pairs. The photons output from the polarizer were passed through a fiber Bragg grating (FBG) to eliminate the pump photons, and launched into an arrayed waveguide grating (AWG), by which we separated the signal and idler photons with a 0.2 nm bandwidth. The wavelengths of the signal and idler channels were 1547.9 and 1554.3 nm, respectively. Each photon was then passed through an optical bandpass filter to further suppress the pump photons, and then input into a 1-bit delayed interferometer. The filter loss for each channel, including the losses of the polarizer, the FBG, the AWG and the bandpass filter, was approximately 6 dB. The pump suppression with respect to the signal and idler channel was larger than 110 dB, and so the effect of the unsuppressed pump photons was negligible in the coincidence count measurement. The interferometers used in this experiment were silica waveguide devices fabricated using planar lightwave circuit (PLC) technology [19]. These interferometers were very stable over a long period without any feedback control other than the temperature control. The phase difference between two arms was adjusted by controlling the temperature of the substrate: we can change the phase difference by π radians by changing the temperature by 0.180°C.

The photons output from one port of each interferometer were received by a gated-mode single photon detector based on a InGaAs/InP avalanche photodiode (APD). We denote the signal and idler detectors as D0 and D1, respectively. The APDs were operated at a 500-MHz gate frequency using the sine wave gating technique [20, 21]. The quantum efficiencies were 8% (D0) and 13% (D1). The dark count rate was 8×10⁻⁵ for both detectors. To reduce the afterpulse probabilities, we inserted a 1.5-µs dead time for both detectors. As a result, the afterpulse probabilities were 5.2% (D0) and 1.1% (D1). The detection signal from D0 was directly input into an OR logic gate, whose output was connected to the TIA as a stop pulse. The signal from D1 was delayed by about 200 ns before being launched into the OR logic gate, so that coincidence events between the two detectors were recognized as two separate detection events by the TIA. The detection signals from the two detectors were also input into the TIA as “tag” signals, with which we can determine whether each stop signal was from D0 or D1. By counting two consecutive detection events observed between start pulses using the TIA, we obtained the numbers of coincidence events that correspond to n_ν. Here, we applied a 1.4-ns time window to the data obtained by the TIA to reduce the effect of dark counts.

In the experiment, we set the average photon pair number per qubit (i.e. the average photon pair number in two pulses) at 0.02. We implemented |+〉 and |L〉 projection measurements with the energy basis by setting the interferometer temperature at 21.80 and 21.89°C for the signal and at 32.54 and 32.63°C for the idler, respectively. The scheme for the initial alignment of the interferometers is described in Appendix A.

Fig. 3. Experimental setup.

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A typical histogram of the photon arrival time for D1 (before applying a time window) is shown in Fig. 4. We can clearly observe several triple peaks. The center peaks correspond to the projection measurements with the energy basis, while the side peaks correspond to the time basis measurements. Thus, we can achieve both energy and time basis projection measurements simultaneously, and so we can obtain all the n_ν values with only four TIA measurements, as we describe next.

In the following, we denote the two-photon projection state with |XY〉, where X and Y are the projected states of signal and idler photons, respectively. We also assume that the interferometer phase is adjusted so that the projected state with the energy basis becomes |E〉 (E={+,L}). Then, with one TIA measurement, we can implement two photon projection measurements for the following states: |11〉, |12〉, |21〉, |22〉, |1E〉, |2E〉, |E1〉, |E2〉, and |EE〉. Thus, we can simultaneously implement nine projection measurements. However, as explained in section 2, a single photon projection measurement with the time basis using our setup suffers from an intrinsic 50% loss. Therefore, to obtain nν, we need to double the coincidence measurement time when a photon is projected to the time basis and the other is projected to the energy basis. Similarly, the measurement time should be increased fourfold when both photons are projected to the time basis. As we show with the following experimental results, these requirements are automatically fulfilled by undertaking four TIA measurements with four combinations of energy-basis projection measurements to states |++〉, |+L〉, |L+〉, and |LL〉.

Fig. 4. Typical histogram of photon arrival time for D1.

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Table 2. Experimentally obtained coincidence counts. A dash (-) indicates that coincidences were not obtained for the corresponding projection measurement. The 4th, 5th, 6th and 7th columns correspond to the obtained coincidence counts for four different energy-basis projection measurement settings, and the 8th column gives the total coincidence counts denoted as n_ν.

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We undertook four TIA measurements each for 60 s. The coincidence counts obtained in the four TIA measurements are summarized in Table 2. For all measurements, we were able to obtain coincidence counts for ν=1,2,3, 4, in which both photons were projected to the time basis. Therefore, by summing up the coincidence counts obtained in the four measurements, we can obtain the values for n ₁, n ₂, n ₃ and n ₄. Similarly, when a photon is projected to the time basis and the other to the energy basis, two out of four measurements gave the coincidence counts, and thus we can obtain n_ν for ν=5,6,9,10,12,13,15,16 by adding the two results together. Needless to say, n ₇, n ₁₄, n ₈ and n ₁₁ are obtained in one shot in each TIA measurement. Thus, we were able to obtain all n_ν values using the data obtained in the four TIA measurements. The n_ν values are shown in the rightmost column in Table 2.

We can calculate the density matrix ρ of the two photons using Eq. (12) with the n_ν values shown in Table 2. As a result, we obtained the following density matrix.

ρ = (\begin{array}{c} 0.445626 & 0.011447 + i 0.013083 & 0.012674 + i 0.010221 & 0.421504 - i 0.107931 \\ 0.011447 - i 0.013083 & 0.040065 & 0.059689 - i 0.012674 & - 0.05070 + i 0.048242 \\ 0.012674 - i 0.010221 & 0.059689 + i 0.012674 & 0.026165 & - 0.022486 + i 0.004497 \\ 0.421504 + i 0.107931 & - 0.05070 - i 0.048242 & - 0.022486 - i 0.004497 & 0.488144 \end{array})

The real and imaginary parts of this matrix are shown graphically in Fig. 5(a) and 5(b), respectively.

Fig. 5. Density matrix of time-bin entangled photon pairs obtained by QST. (a) and (b) show the real and the imaginary parts of the matrix obtained by linear tomography, while (c) and (d) are those obtained by maximum likelihood estimation.

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Density matrices for all the physical states must have eigenvalues that lie in the interval [0,1] [3]. However, the eigenvalues of Eq. (14) are 0.904522, 0.132136, 0.003150, and -0.039808, which implies that the obtained density matrix is unphysical. This problem is caused by experimental inaccuracies and statistical fluctuations of coincidence counts, and has often been observed in previous QST measurements. Therefore, we applied the maximum likelihood estimation approach described in [3] to the matrix obtained by linear tomography, which provides a physically legitimate density matrix that best describes the state. As a result, we obtained the following matrix.

ρ_{p} = (\begin{array}{c} 0.461266 & 0.006138 + i 0.014805 & - 0.002069 + i 0.021371 & 0.395186 - i 0.090409 \\ 0.006138 - i 0.014805 & 0.039398 & 0.019510 + i 0.013286 & - 0.034332 + i 0.028899 \\ - 0.002069 - i 0 . 021371 & 0.019510 - i 0.013286 & 0.027424 & - 0.000301 + i 0.001415 \\ 0.395186 + i 0.090409 & - 0.034332 - i 0.028899 & - 0.000301 - i 0.001415 & 0.471911 \end{array})

The real and imaginary parts of this matrix are shown in Fig. 5(c) and 5(d), respectively. The above matrix has eigenvalues of 0.872964, 0.102097, 0.023512, and 0.001427, and thus provides a physically legitimate density matrix.

Using the density matrix ρ_p obtained with maximum likelihood estimation, we can derive the various parameters that indicate the quality of the entanglement. The procedures are similar to those given in [3], except for a slight modification required in the error analysis. The error analysis method used in our calculation is described in Appendix B. The obtained fidelity for the entangled pure state given by Eq. (13), von Neumann entropy, linear entropy, and concurrence are summarized in Table 3.

Table 3. Quantities derived from the density matrix

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The probable main source for the limited fidelity is the accidental coincidences caused by multi-pair emission. In particular, when we use a fiber-based entanglement source, the SpRS noise photons enhance the number of accidental coincidences [11].

As explained above, the merit of the presented scheme is that we can obtain a full density matrix with only four TIA measurements, while the QST for polarization entanglement requires 16 measurements with different orientations of the polarization optics. This merit arises from the fact that two non-orthogonal basis measurements are inherently implemented with the projection measurement using a 1-bit delayed interferometer.

A possible drawback of this scheme results from the loss difference between the time and energy basis measurements. When the effect of the dark counts is not negligible, a larger number of false detections caused by the dark counts are observed in the time basis measurement than in the energy basis measurement, because of the intrinsic 50% loss in the time basis measurement. We can overcome this problem by using single photon detectors whose dark count probabilities are negligible compared with the photon detection probability. In the 1.5-µm band, we can use high signal-to-noise-ratio detectors such as an up-conversion detector [22] or a superconducting single photon detector [23] for this purpose.

5. Conclusion

We described QST for time-bin entangled photon pairs. We first proposed a scheme for undertaking projection measurements for a time-bin qubit using a 1-bit delayed interferometer followed by a single photon detector. Then, we explained the concrete procedure for QST using the proposed projection measurements. We successfully demonstrated QST on time-bin entangled photons using a 1.5-µm band entanglement source based on SFWM in a DSF and 500-MHz gated mode single photon detectors. We also showed that we can construct a density matrix with only four measurements using a TIA, which means that we can realize the simple and fast evaluation of the quality of entanglement using the proposed scheme. We hope that this scheme will contribute to the realization of quantum communication networks over fiber based on time-bin entangled photons.

Appendix A: Alignment of 1-bit delayed interferometers

In theory, the phase differences induced in the two 1-bit delayed interferometers and the relative phase term in the expression of a time-bin entangled state can be set at zero. However, in an actual experiment, these parameters are extremely difficult to control. In our experiment, we aligned the phase differences of the interferometers in the following way, which is the same procedure used in the interferometer alignment of QKD employing time-bin entanglement.

If we neglect the terms caused by multi-pair emission, a time-bin entangled photon pair actually obtained in the experiment is given by the following expression.

∣ Ψ 〉 = \frac{1}{\sqrt{2}} (∣ {1 〉}_{s} ∣ {1 〉}_{i} + e^{iϕ} ∣ {2 〉}_{s} {2 〉}_{i})

Here, the relative phase ϕ is determined by the phase difference between the two pump pulses. By passing through a 1-bit delayed interferometer, the state |k〉_x (k: integer, x=s,i) is coverted to $({| k 〉}_{x} + e^{i ϕ_{x}} {| k + 1 〉}_{x}) ⁄ 2$ , where ϕ_x is the phase difference induced in the interferometer. Then, the two photon state after passage through the interferometer is given by

∣ Ψ 〉 \to ∣ {1 〉}_{s} ∣ {1 〉}_{i} + e^{i ϕ_{i}} ∣ {1 〉}_{s} ∣ {2 〉}_{i} + e^{i ϕ_{s}} ∣ {2 〉}_{s} ∣ {1 〉}_{i} + (e^{i (ϕ_{s} + ϕ_{i})} + e^{iϕ}) ∣ {2 〉}_{s} ∣ {2 〉}_{i}

where the normalization term is not shown for simplicity. The 4th term on the right hand side of the above equation corresponds to the case where both photons are projected to the energy basis. In the experiment, we adjusted the phase differences ϕ_s and ϕ_i so that we obtained a coincidence peak, i.e. ϕ_s+ϕ_i=ϕ, by adjusting the temperatures of the interferometers. Then, we defined these phase difference values ϕ, ϕ_s and ϕ_p as the reference points of the phase differences. For the sake of simplicity, these terms are not shown in the explanation in Section 4. Therefore, in a projection measurement to a state (|1〉_x+e^iθ|2〉_x)/√2, the actual phase difference induced in the interferometer is ϕ_x−θ. It is easily shown that the inclusion of the values ϕ, ϕ_s and ϕ_i does not change the results of coincidence measurements that involve energy-basis projection measurements, and thus does not change the density matrix reconstruction result.

Appendix B: Errors analysis

Here, we define s_ν as n_ν/C. The density matrix is specified by s_ν(ν=1-16). Therefore, we can analyze the errors in quantities derived from the density matrix by estimating the errors of s_ν.

Possible reasons for the errors of sν are coincidence count statistics, uncertainties of phase differences induced in the 1-bit delayed interferometers, detector dark counts, and detector timing jitter. Of these, phase difference uncertainties only affect the energy basis projection measurements, while timing jitter results in errors in the time basis projection measurements. We assume that the effect of the dark counts was negligible in our experiment. The detector timing jitter was ~500 ps, and thus its effect on time basis measurement is also negligible. Therefore, we consider the errors caused by coincidence count statistics and phase difference uncertainties.

The errors caused by coincidence count statistics are obtained with the procedure described in [3]. Since the measured coincidence counts n_ν are statistically independent Poissonian random variables, the errors in s_ν is given by

\overset{ˉ}{δ S_{μ} δ S_{ν}} = \frac{S_{ν} δ_{μ, ν}}{C} .

Next we consider the errors caused by uncertainties of the phase differences induced in the interferometers. While we should take account of the angle uncertainties of the four waveplates in the QST for polarization entanglement [3], only two phase differences have to be considered in the proposed QST scheme. We assume that the projected states for the energy basis are given by $(∣ {1 〉}_{x} + e^{i θ_{x}} ∣ 2 〉) ⁄ \sqrt{2} (x = s, i)$ . Then, the two-photon projected states |ψ_ν〉 are expressed as a function of (θ_s,θ_i). For example, |ψ ₇〉 is given by $(1 ⁄ 2, e^{i θ_{i}} ⁄ 2, e^{i θ_{s} ⁄ 2}, e^{i (θ_{s} + θ_{i})} ⁄ 2)$ , with θ_s=θ_i=0. Using a similar procedure to derive Eq. (5.10) in [3], we obtain the following expression for the errors in sν caused by phase difference settings.

\overset{ˉ}{δ S_{μ} δ S_{μ}} = δ_{ν, μ} \underset{x = s, i ε,}{Σ} Σ_{λ = 1}^{16} f_{ν, ε}^{(x)} f_{ν, λ}^{(x)} S_{ε} S_{λ} {(Δ θ)}^{2}

Here, Δθ is the uncertainty in the setting of the phase difference, and f ^(x) _ν,µ is given by

f_{ν, μ}^{(x)} = \{\frac{\partial}{{\partial θ}_{ν, x}} 〈 ψ_{ν} ∣\} {\hat{M}}_{μ} ∣ ψ_{ν} 〉 + 〈 ψ_{ν} ∣ {\hat{M}}_{μ} {\frac{\partial}{\partial θ_{ν, x}} ∣ ψ_{ν} 〉},

where θ_ν,x is the phase difference induced in the interferometer for photon x(=s, i) in the measurement of n_ν.

Combining Eqs. (18) and (19), we obtain the following formula for the total error in the quantities s_ν:

\overset{ˉ}{δ S_{ν} δ S_{μ}} = δ_{ν, μ} Λ_{ν}

where

Λ_{ν} = [\frac{s_{v}}{C} + \underset{x = s, i}{Σ} Σ_{ε, λ = 1}^{16} f_{ν, ε}^{(x)} f_{ν, λ}^{(x)} S_{ε} S_{λ} {(Δ θ)}^{2}] .

In our experiment, we were able to change the phase difference by π by changing the interferometer temperature by 0.180°C, and the uncertainty of the temperature control was estimated to be 0.005°C. Therefore, in the error calculation presented in Section 5, we used Δθ=π/36.

Using Eq. (21), we can calculate the errors in the various quantities derived from the reconstructed density matrix. As an example, we describe the error of the fidelity F of the state for the entangled pure state given by Eq. (13). F is given by

F = 〈 ψ ∣ ρ ∣ ψ 〉 .

The error in this quantity is given by

{(Δ F)}^{2} = Σ_{ν = 1}^{16} {(\frac{\partial F}{\partial s_{ν}})}^{2} Λ_{ν} .

Using the relationship $ρ = \sum_{ν = 1}^{16} M_{ν} s_{ν}, \frac{\partial F}{\partial s_{ν}}$ is obtained as

\frac{\partial F}{\partial s_{v}} = \frac{1}{2} {〈 11 ∣ M_{v} ∣ 11 〉 + 〈 11 ∣ M_{v} ∣ 22 〉 + 〈 22 ∣ M_{v} ∣ 11 〉 + 〈 22 ∣ M_{v} ∣ 22 〉}

Using Eqs. (24) and (25), we can calculate the error in the fidelity obtained from the reconstructed density matrix. For the experimental result given above, F=0.862±0.044.

We can calculate other quantities such as the von Neumann entropy, linear entropy, and concurrence using the procedure described in [3].

Acknowledgements

We thank S. Odate and F. Morikoshi for fruitful discussions, and B. Miquel for his assistance with the experiments.

References and links

1. A. Aspect, P. Grangier, and G. Roger, “Experimental realization of Einstein-Podolsky-Rosen-Bohm gedanken-experiment: a new violation of Bell’s inequalities,” Phys. Rev. Lett. 49, 91–94 (1982). [CrossRef]

2. P. G. Kwiat, A. M. Steinberg, and R. Y. Chiao, “High-visibility interference in a Bell-inequality experiment for energy and time,” Phys. Rev. A 47, R2472–R2475 (1993). [CrossRef] [PubMed]

3. D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, “Measurement of qubits,” Phys. Rev. A 64, 052312 (2001). [CrossRef]

4. Y. Nambu, K. Usami, Y. Tsuda, K. Matsumoto, and K. Nakamura, “Generation of polarization-entangled photon pairs in a cascade of two type-I crystals pumped by femtosecond pulses,” Phys. Rev. A 66, 033816 (2002) [CrossRef]

5. K. Edamatsu, G. Oohata, R. Shimizu, and T. Itoh, “Generation of ultraviolet entangled photons in a semiconductor,” Nature 431, 167–170 (2004). [CrossRef] [PubMed]

6. N. A. Peters, J. B. Altepeter, D. Branning, E. R. Jeffrey, T.-C. Wei, and P. G. Kwiat, “Maximally entangled mixed states: creation and concentration,” Phys. Rev. Lett. 92, 133601 (2004). [CrossRef] [PubMed]

7. R. M. Stevenson, R. J. Young, P. Atkinson, K. Cooper, D. A. Ritchie, and A. J. Shields, “A semiconductor source of triggered entangled photon pairs,” Nature 439, 179–182 (2006) [CrossRef] [PubMed]

8. S. Odate, A. Yoshizawa, and H. Tsuchida, “Polarisation-entangled photon-pair source at 1550 nm using 1 mm-long PPLN waveguide in fibre-loop configuration,” Electron. Lett. 43, 1376–1377 (2007). [CrossRef]

9. 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]

10. I. Marcikic, H. de Riedmatten, W. Tittel, H. Zbinden, M. Legre, and N. Gisin, “Distribution of time-bin entangled qubits over 50 km of optical fiber,” Phys. Rev. Lett. 93, 180502 (2004). [CrossRef] [PubMed]

11. H. Takesue, “Long-distance distribution of time-bin entanglement generated in a cooled fiber,” Opt. Express 14, 3453–3460 (2006). [CrossRef] [PubMed]

12. T. Honjo, H. Takesue, H. Kamada, Y. Nishida, O. Tadanaga, M. Asobe, and K. Inoue, “Long-distance distribution of time-bin entangled photon pairs over 100 km using frequency up-conversion detectors,” Opt. Express 15, 13957–13964 (2007). [CrossRef] [PubMed]

13. J. Dynes, H. Takesue, Z. L. Yuan, A. W. Sharpe, K. Harada, T. Honjo, H. Kamada, O. Tadanaga, Y. Nishida, M. Asobe, and A. J. Shields, “Ultra-long distance and efficient entanglement distribution over 200 km,” submitted to Opt. Express.

14. T. Honjo, S. W. Nam, H. Takesue, Q. Zhang, H. Kamada, Y. Nishida, O. Tadanaga, M. Asobe, B. Baek, R Hadfield, S. Miki, M. Fujiwara, M. Sasaki, Z. Wang, K. Inoue, and Y. Yamamoto, “Long-distance entanglement-based quantum key distribution over optical fiber,” Opt. Express 16, 19118–19126 (2008). [CrossRef]

15. H. Takesue, K. Inoue, O. Tadanaga, Y. Nishida, and M. Asobe, “Generation of pulsed polarization-entangled photon pairs in a 1.55-µm band with a periodically poled lithium niobate waveguide and an orthogonal polarization delay circuit,” Opt. Lett. 30, 293–295 (2005). [CrossRef] [PubMed]

16. G. Berlin, G. Brassard, F. Bussieres, N. Godbout, J. A. Slater, and W. Tittel, “Flipping quantum coins,” arXiv:0904.3946.

17. M. Fiorentino, P. L. Voss, J. E. Kumar, and P. Sharping, “All-fiber photon-pair source for quantum communications,” IEEE Photon. Technol. Lett. 14, 983–985 (2002). [CrossRef]

18. W. Tittel, J. Brendel, H. Zbinden, and N. Gisin, “Quantum cryptography using entangled photons in energy-time Bell states,” Phys. Rev. Lett. 84, 4737–4740 (2000). [CrossRef] [PubMed]

19. H. Takesue and K. Inoue, “Generation of 1.5-µm band time-bin entanglement using spontaneous fiber four-wave mixing and planar lightwave circuit interferometers,” Phys. Rev. A 72, 041804(R) (2005). [CrossRef]

20. N. Namekata, S. Sasamori, and S. Inoue, “800 MHz single-photon detection at 1550-nm using an InGaAs/InP avalanche photodiode operated with a sine wave gating,” Opt. Express 14, 10043–10049 (2006). [CrossRef] [PubMed]

21. B. Miquel and H. Takesue, “Observation of 1.5 µm band entanglement using single photon detectors based on sinusoidally gated InGaAs/InP avalanche photoidodes,” New J. Phys. 11, 045006 (2009). [CrossRef]

22. C. Langrock, E. Diamanti, R. V. Roussev, Y. Yamamoto, M. M. Fejer, and H. Takesue, “Highly efficient single-photon detection at communication wavelengths by use of upconversion in reverse-proton-exchanged periodically poled LiNbO₃ waveguides,” Opt. Lett. 30, 1725–1727 (2005). [CrossRef] [PubMed]

23. G. N. Gol’tsman, O. Okunev, G. Chulkova, A. Lipatov, A. Semenov, K. Smirnov, B. Voronov, A. Dzardanov, C. Williams, and R. Sobolewski, “Picosecond superconducting single-photon optical detector” Appl. Phys. Lett. 79, 705–707 (2001). [CrossRef]

Fidelity	0.862±0.044
Von Neumann entropy	0.648±0.247
Linear entropy	0.303±0.101
Concurrence	0.777±0.043

Fidelity	0.862±0.044
Von Neumann entropy	0.648±0.247
Linear entropy	0.303±0.101
Concurrence	0.777±0.043

Implementation of quantum state tomography for time-bin entangled photon pairs

Abstract

1. Introduction

2. Projection measurements of time-bin qubit using 1-bit delayed interferometer

3. Two-photon projection measurement and reconstruction of density matrix

4. Experiments

5. Conclusion

Appendix A: Alignment of 1-bit delayed interferometers

Appendix B: Errors analysis

Acknowledgements

References and links

Cited By

Figures (5)

Tables (3)

Equations (26)

Optics Express

ν	Photon 1	Photon 2	\|ψ_ν 〉=(\|11〉, \|12〉, \|21〉, \|22〉)
1	\|1〉	\|1〉	(1,0,0,0)
2	\|1〉	\|2〉	(0,1,0,0)
3	\|2〉	\|1〉	(0,0,1,0)
4	\|2〉	\|2〉	(0,0,0,1)
5	\|2〉	\|+〉	(0,0 $\frac{1}{\sqrt{2}}$ , $\frac{1}{\sqrt{2}}$ )
6	\|1〉	\|+〉	( $\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}$ ,0,0)
7	\|+〉	\|+〉	( $\frac{1}{2}, \frac{1}{2}, \frac{1}{2}, \frac{1}{2}$ )
8	\|L〉	\|+〉	( $\frac{1}{2}, \frac{1}{2}, \frac{i}{2}, \frac{i}{2}$ )
9	\|L〉	\|1〉	( $\frac{1}{\sqrt{2}}, 0, \frac{i}{\sqrt{2}}, 0$ )
10	\|L〉	\|2〉	( $0, \frac{1}{\sqrt{2}}, 0, \frac{i}{\sqrt{2}}$ )
11	\|L〉	\|L〉	( $\frac{1}{2}, \frac{i}{2}, \frac{i}{2}, - \frac{1}{2}$ )
12	\|1〉	\|L〉	( $\frac{1}{\sqrt{2}}, \frac{i}{\sqrt{2}}, 0, 0$ )
13	\|2〉	\|L〉	( $0, 0, \frac{1}{\sqrt{2}}, \frac{i}{\sqrt{2}}$ )
14	\|+〉	\|L〉	( $\frac{1}{2}, \frac{i}{2}, \frac{1}{2}, \frac{i}{2}$ )
15	\|+〉	\|1〉	( $\frac{1}{\sqrt{2}}, 0, \frac{1}{\sqrt{2}}, 0$ )
16	\|+〉	\|2〉	( $0, \frac{1}{\sqrt{2}} 0, \frac{1}{\sqrt{2}}$ )

ν	Photon 1	Photon 2	++	+L	L+	LL	n_ν
1	\|1〉	\|1〉	139	142	130	134	545
2	\|1〉	\|2〉	7	14	14	14	49
3	\|2〉	\|1〉	9	7	9	7	32
4	\|2〉	\|2〉	149	144	153	151	597
5	\|2〉	\|+〉	146	-	141	-	287
6	\|1〉	\|+〉	151	-	160	-	311
7	\|+〉	\|+〉	570	-	-	-	570
8	\|L〉	\|+〉	-	-	337	-	337
9	\|L〉	\|1〉	-	-	128	148	276
10	\|L〉	\|2〉	-	-	144	120	264
11	\|L〉	\|L〉	-	-	-	38	38
12	\|1〉	\|L〉	-	156	-	125	281
13	\|2〉	\|L〉	-	160	-	149	309
14	\|+〉	\|L〉	-	330	-	-	330
15	\|+〉	\|1〉	154	150	-	-	304
16	\|+〉	\|2〉	135	126	-	-	261

ν	Photon 1	Photon 2	++	+L	L+	LL	n_ν
1	\|1〉	\|1〉	139	142	130	134	545
2	\|1〉	\|2〉	7	14	14	14	49
3	\|2〉	\|1〉	9	7	9	7	32
4	\|2〉	\|2〉	149	144	153	151	597
5	\|2〉	\|+〉	146	-	141	-	287
6	\|1〉	\|+〉	151	-	160	-	311
7	\|+〉	\|+〉	570	-	-	-	570
8	\|L〉	\|+〉	-	-	337	-	337
9	\|L〉	\|1〉	-	-	128	148	276
10	\|L〉	\|2〉	-	-	144	120	264
11	\|L〉	\|L〉	-	-	-	38	38
12	\|1〉	\|L〉	-	156	-	125	281
13	\|2〉	\|L〉	-	160	-	149	309
14	\|+〉	\|L〉	-	330	-	-	330
15	\|+〉	\|1〉	154	150	-	-	304
16	\|+〉	\|2〉	135	126	-	-	261