1. Introduction

Quantum key distribution (QKD) allows two separate parties, the sender Alice and the receiver Bob, to share a secret key [1, 2]. Unconditional security of QKD is guaranteed by the laws of quantum mechanics. Since the first demonstration of QKD [3], extensive efforts have been devoted to extending transmission distance [4, 5, 6]. Most of the experiments were implemented on an optical fiber based on the phase-coding BB84 protocol. Recently, single photon interference after 150 km transmission [7] and QKD over 122 km [8] have been reported. However, the channel loss in optical fibers is one of the main obstacles for long distance QKD transmission. The loss reduces the rate of secure key generation, or even makes it impossible by degrading signal-to-noise ratio. The transmission distance is limited to about 250–300 km with a perfect single photon source and the best single photon detectors currently reported [9]. Longer QKD transmission would require a quantum repeater, which is based on entanglement sharing. Though the successful experiment with time-bin entanglement has been performed over a long distance fiber [10], it would take long before the realization of a practical quantum repeater.

Free space transmission may provide a solution for the QKD over a long distance. It has the advantages of essentially no birefringence and low absorption of the atmosphere in certain wavelength ranges. We can employ the polarization states of visible photons for free space transmission, where highly sensitive photon detectors are available. Recently, successful experiments on the free space QKD systems have been reported [11,12,13,14]. For free space QKD, however, one has to consider optical effects of the atmosphere, in particular, the decoherence due to the atmospheric disturbance. A new protocol robust to collective channel noise over long propagation distance has been proposed [15]. Unfortunately, it is unrealizable for the current technology to implement the protocol, which needs to encode three or four-photon polarization states.

Recently, Xiang-bin Wang has proposed a QKD protocol which is fault tolerant with respect to the collective random unitary channel noise [16]. This protocol only needs four two-qubit polarization encoding states and is realizable in the experiment. In our paper, we will report our experimental verification with the protocol for the first time [17]. In the experiment, we will use spontaneous parametric down conversion(SPDC) to produce four polarization encoded states $\mid \mathit{HV}〉,\mid \mathit{VH}〉,\mid {\psi }^{\pm }〉=\frac{1}{\sqrt{2}}\left(\mid \mathit{HV}〉\pm \mid \mathit{VH}〉\right)$. Alice prepares the four states $\mid \mathit{HV}〉,\mid \mathit{VH}〉,\mid {\psi }^{\pm }〉=\frac{1}{\sqrt{2}}\left(\mid \mathit{HV}〉\pm \mid \mathit{VH}〉\right)$, and chooses one of the states randomly to send to Bob. For the states |HV〉 and |ψ ⁺〉, they denote the bit value as 1. For the states |VH〉 and |ψ ^-〉, they denote the bit value as 0. For every photon, Bob can do the measurement in Z basis ({|H〉, |V〉}), X basis ({|±45°〉}), and Y basis $\left(\left\{\frac{1}{\sqrt{2}}\left(\mid H〉\pm i\mid V〉\right)\right\}\right)$. Bob measures the four two-qubit states in the basis Z ⊗ Z, X ⊗ X, and Y ⊗ Y randomly. After measurement, Alice announces her prepared basis {|HV〉, |VH〉} or {|ψ ^±〉}. Bob announces his measurement basis. Through public communication, they will discard all the results in different measurement basis. For the states in the subspace S = {|HV〉, |VH〉}, they only keep the results of measurement in the basis Z ⊗ Z, and they discard all the results which Bob used different basis of Z ⊗ Z. To get final keys, they public all the survive keys from states |ψ ^±〉 and some part of keys from states |HV〉 and |VH〉 to check error rate.

For polarization encoding entangled state QKD experiment, Alice and Bob get share key by doing nonlocal measurement. The security of QKD is guaranteed by checking Bell inequality[13, 14]. In the practical system like ground to satellite QKD, the stability of polarization angle of measurement is crucial for the two parties [18]. Any relative rotation of measurement basis will obviously increase error rate for their final keys. In our experiment, we only need local measurement. Our experiment will show that the protocol is robust to collective noise compare to other QKD protocols, such as single photon polarization encoding QKD and polarization entangled state QKD. The system was insensitive to the polarization angle of measurement. For the ground to satellite QKD, the advantage of this protocol will reduce the complex of techniques to correct stability of a QKD system.

 

Fig. 1. Experimental setup used to investigate fault tolerant QKD.

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2. Experimental setup

Our experimental setup is shown in Fig. 1. Two-photon pairs were produced via SPDC. A pulse laser with central wavelength of 400nm, which was produced by frequency double of a femtosecond laser from a mode-locked Ti:Sapphire laser(Coherent:Vitesse), was used to pump two adjacent nonlinear beta-barium-borate (BBO) crystals [19]. Two BBO crystals were cut to 8 × 8 × 0.13mm for type I phase matching. The planes of their optic axis were orthogonal to each other. The average power of the pump laser was about 840 mw, and repetition rate was 80MHz. Two photon states |HH〉 and |VV〉 were generated by V-polarized and H-polarized pump beams, respectively. A 1.5mm quartz plate was put before the crystals to compensate the dispersion and birefringence effects of the first and the second crystals on the photon pairs. After the BBO crystals, a half wave plate for wavelength of 800nm (HWP3) rotated the polarization of one of SPDC beams 90° in order to produce the states |HV〉 and |VH〉. Alice prepared the states |HV〉 and |VH〉 by setting the polarization of pump pulses to H and V respectively, with a half wave plate (HWP4). To prepare the states $\mid {\psi }^{\pm }〉=\frac{1}{\sqrt{2}}\left(\mid \mathit{HV}〉\pm \mid \mathit{VH}〉\right)$, the HWP4 was used to set the polarization of the pump beam at 45°, and a quarter wave plate (QWP3) was rotated 90° to select two Bell states. The phase factor was adjusted by tilting the quartz a little. At Bob’s side, half wave plate and quarter wave plate(HWP1, HWP2, QWP1, and QWP2) were used in order to measure the states | |ψ ⁺〉, |ψ ^-〉 in the basis X ⊗ X, Y ⊗ Y, and Z ⊗ Z. Two broadband filters were put before the polarization beam splitters(PBS) to cut off scattering light. Finally, SPDC beams were coupled to four Si:APD detectors by multi-mode optical fibre, and the signals from detectors were sent to two-fold coincidence circuits. Decoherence in our experiment was introduced by inserting a half wave plate or a quarter wave plate in the two arms. Because two arms have passed exactly the same wave plate, we can consider it as collective decoherence which is similar like two arms are close together enough.

3. Results and discussion

The error rate can be estimated from the results of Bob’s measurement for the states |HV〉, |VH〉, |ψ ^±〉. Because of collective polarization rotation, there is a finite possibility of bit flipping to produce wrong bit value. For the protocol, the error rate comes from bit flipping in the subspace. For example, if Alice sends a qubit |HV〉 to Bob, there are four possibilities for Bob’s measurement result to obain |HV〉, |HH〉, |VV〉, and |VH〉. As we know that, the states |HH〉 and |VV〉 are out of the subspace S, and will be rejected. The result of the state |VH〉 is the error of the sent state |HV〉. For the collective rotation, bit flipping of qubit in subspace S only comes from the subspace S and can not change from outside the subspace S. The flipping rate between |HV〉 and |VH〉 is

where θ is the angle of collective polarization rotation. As Fig. 2, for the sent states |HV〉 and |VH〉 the error rate are about 0.7%.

 

Fig. 2. Experimental measurement for the four two-qubit states |HV〉, |VH〉, and |ψ ^±〉.

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In the protocol, the parity measurement is required in the basis Z ⊗ Z, X ⊗ X, and Y ⊗ Y to check error rate of the channel. Through the measurement in the three basis, Bob can estimate the probabilities p _ψ⁺, p _ψ^-, p _ϕ⁺, and p _ϕ^- of obtaining the four Bell states |ψ ⁺〉, |ψ ^-〉, |ϕ ⁺〉, |ϕ ^-〉, respectively. The flipping rate of |ψ ^-〉 → |ψ ⁺〉 is given by $\frac{p_{{\psi }^{+}}}{p_{{\psi }^{+}}+p_{{\psi }^{-}}}$. Define ε_z, ε_x, ε_y for the rate of wrong outcome for |ψ ^-〉 in the measurement basis Z ⊗ Z, X ⊗ X and Y ⊗ Y respectively. The net flipping rate from |ψ ^-〉 to |ψ ⁺〉 is [16]

Similarly, the net flipping rate from |ψ ⁺〉 to |ψ ^-〉 is

Where ε′_x,y,z are rate of wrong outcome in local measurement basis X ⊗ X, Y ⊗ Y, Z ⊗ Z, respectively, to all codes originally in |ψ ⁺〉. In theory, the flipping rate between |ψ ⁺〉 and |ψ ^-〉 in collective rotation should zero. Fig. 2 also shows the outcomes of measuring |ψ ⁺〉 and |ψ ^-〉 in the basis Z ⊗ Z, X ⊗ X, and Y ⊗ Y respectively. From the results in Fig. 2, we can calculate the flipping rate t _ψ^-→ψ⁺ and t _ψ⁺→ψ^- to be about 3.2% and 3.1%. Considering the average fidelity of the entangled states |ψ ⁺〉 and |ψ ^-〉 were about 0.97, it means that the observed flipping rate mainly results from the imperfection of our entangled photon states.

 

Fig. 3. Bit-flip with two qubit states in the subspace {|HV〉, |VH〉} and polarization encoding single photon states.

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The protocol measures the error rate under collective polarization rotation for subspace S = {|HV〉, |VH〉}. we assume that the channel noise are mainly from the collective unitary noise. For the qubit states |HV〉 and |VH〉, bit-flip only comes from the collective rotation of polarization. To do the measurement, we can put a half wave plate to check the bit-flip as the collective rotation. As shown in Fig. 3, Alice sends a state |HV〉 in the subspace S, Bob measures the state in the basis Z ⊗ Z. Bob rejects all the outcomes of the states |HH〉 and |VV〉, which are outside the subspace S, and keeps the outcomes of the states |HV〉 and |VH〉. We estimated the bit-flip rate by changing the polarization rotation angle. Figure 3 shows the error rate r_b of the state |HV〉, which fits the prediction by Eq. (1) very well.

We compared our protocol with the single photon QKD encoded in the polarization states (exactly as same as BB84 protocol) by examining the error rate. We used one arm of SPDC as single photon source and use the other arm for coincidence measurement in order to remove the noise coming from background. We measured the bit-flip error rate of the single photon QKD by putting a half wave plate in QKD channel and rotating the polarization. The results are also shown in Fig. 3. The error rate of the single photon QKD is theoretically given as follows. If Alice sends a state |H〉, the probability for Bob’s measurement to get the state |H〉 is cos² θ and the state |V〉 is sin² θ, so the error rate of QKD is sin² θ as shown in Fig. 3. The error rate of the single photon QKD was clearly higher than that of the proposed protocol using the qubit states |HV〉 and |VH〉. From our results, the error rate was about (2.4 ± 0.1)% for two-qubit states, but (13.0 ± 0.5)% for single qubit state, when polarization was rotated by 20°.

We also compared our protocol with polarization entangled states QKD by using decoherence free state |ψ ^-〉. Two photons are separated in space, so there is no any collective rotation for the two photons. Alice and Bob do measurement at the basis Z ⊗ Z and X ⊗ X randomly [13, 14]. The relative rotation angle of polarization θ can result from misalignment of the axis between two parities, or from random polarization rotation due to the environment. It is not difficult to calculate that the error rate of keys is sin² θ, which is exactly same as the error of single photon polarization encoding QKD. Therefore, our protocol yields a lower error rate than the polarization entangled states QKD.

 

Fig. 4. Error rate measurement for |ψ ^-〉 under decoherence environment with half wave plate and quarter wave plate rotation.

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For the states |ψ ^-〉 and |ψ ⁺〉, experiments in Ref. [20] has also shown that |ψ ^-〉 is decoherence free, so |ψ ^-〉 will keep unchanged under collective noise. Any collective operation will not change the outcomes of measurement in the basis Z ⊗ Z, X ⊗ X, and Y ⊗ Y. The state |ψ ⁺〉 is not decoherence free state, and it will be changed under collective rotation. In our experiment, we need to examine the bit flipping from |ψ ^-〉 to |ψ ⁺〉 or from |ψ ⁺〉 to |ψ ^-〉. We used a half wave plate or a quarter wave plate to realize collective operation in the environment as shown in Fig. 1. We measured the error rate in the basis Z ⊗ Z, X ⊗ X, and Y ⊗ Y by rotating the angle of the half or quarter wave plate. From Eq. (2), we estimated the net flipping rate from |ψ ^-〉 to |ψ ⁺〉. As shown in Fig.4a, b, the total flipping for |ψ ^-〉 is very low (less than 1%) due to decoherence free nature of the state. We also estimated the error rate for the state |ψ ⁺〉 with Eq. (3). The result is shown in Fig. 5a, b. Although the flipping rate from |ψ ⁺〉 to |ψ ^-〉 is zero in theoretical analyst, there have peak error rate when rotation angle is close to 22.5° with half wave plate or 45° with quarter wave plate. We will find that ε_z is close to unity for both conditions, so that it is difficult to calculate the error rate from Eq. (3). For both wave plates, Fig. 5 shows that the net flipping rate is insensitive when the rotation angle of the wave plates is less than 20°. For example, when the rotation angle is 15° with half wave plate, which refers to the polarization rotation by 30°, the error rate is 6.0% for the state |ψ ⁺〉 as shown in Fig. 5. The error rate is (25.0 ± 1.0)% for single photon polarization encoding QKD as shown in Fig. 3. This result also means that error tolerance for our protocol is better than BB84. In our experiment, accident coincidence rate is about 1.5counts/second. All our results include accident coincidences.

For practical realization of our QKD protocol, Alice sends 3 (1 + δ)n two-qubit states from {|HV〉, |VH〉} and (1 + δ)n two-qubit states from {|ψ ^±〉} to Bob as mentioned in Ref. [16]. Two photons can be combined in space and separated in time of several nanoseconds. The time separation should be small to satisfy the condition for the collective unitary random noise, as long as realizable with the current techniques. Bob’s measurement can be synchronized to distinguished the two photons.

 

Fig. 5. Error rate for |ψ ⁺〉 under decoherence environment introduced with half wave plate and quarter wave plate rotation.

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4. Conclusion

In conclusion, we have demonstrated a fault tolerant quantum key distribution using four two-qubit polarization encoding states. The error rates were estimated under the collective polarization and phase noise. Our results show that the quantum key distribution is robust to collective random unitary noise. The experiment has shown that the error rate is insensitive to collective polarization rotation when the polarization rotation is less than 25°. The proposed QKD protocol would be suitable for a ground-to-satellite transmission link because of its immunity against the collective random noise.