1. Introduction

Secret sharing [1] is an important branch of cryptography, its objective being to distribute a secret message among certain clients, such that all of them can share the secret if they all agree to collaborate, while if a subset tries to sidestep the others, no-one can obtain the message. Classical secret sharing is vulnerable to eavesdropping attacks, but the emergence of quantum cryptography promises a new path since it allows the unconditionally secure distribution of information between all the clients, based on the laws of quantum mechanics. Since Hillery et al. first presented a quantum secret sharing (QSS) scheme in 1999 using three-particle and four-particle Greenberger-Horne-Zeilinger (GHZ) entangled states [2], many other QSS schemes [3–8] have been proposed in both theoretical and experimental aspects. In these schemes, multiparticle polarization entangled states are widely used and so are rather impractical since entangled states are difficult to generate and transmit. Some of the protocols that propose using a single photon qubit [9–11] have not been demonstrated experimentally. Moreover, since commercial single-mode fiber has birefringence, polarization encoding is impractical in applications over commercial networks. Recently, Bogdanski et al [12, 13] reported using a single qubit protocol [14] with phase encoding in a 1550 nm fiber network, thus solving the above mentioned problems. However, the stability of their systems was based on a complicated polarization control system [15] with a polarization insensitive phase modulator, and the bit generation efficiency was just 50%.

In this letter we present a robust single-photon QSS experiment for three parties over a single-mode fiber network in a Mach-Zehnder interferometric setup using the single qubit protocol proposed by Deng et al [16] with phase encoding. This protocol just requires a single qubit for information transmission, which has allowed its practical experimental realization and scalability, and its efficiency can reach nearly 100%. Moreover, our scheme provides autocompensation of birefringence with just a commercial polarization-sensitive phase modulator connected to a Faraday mirror, which has been shown to provide full compensation of the birefringence [17]. Thanks to this design, our setup can also automatically provide completely polarization-insensitive phase modulation; to the best of our knowledge, there is no commercially available polarization-insensitive phase modulator, and our method is simpler than other previously reported ways to realize polarization-insensitive phase modulation [12, 18].

The scheme of this protocol for the three parties is shown in Fig. 1.

 

Fig. 1 Scheme of the circular secret sharing protocol.

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Using two sets of conjugate measuring bases, Alice randomly prepares her qubits which are polarized single photons into one of the following four states:

Bob and Charlie can perform randomly one of the two unitary operations, U₀ and U₁ which represent the bits 0 and 1 respectively, on the single photons that they receive:

The identity operator U0 does nothing on the single photon. The nice feature of the U1 operation is that it flips or negates the states in both measuring bases. i. e.

To create the private key U_A, Alice sends a quantum signal to Bob, who can chose the encoding mode or the control mode randomly. In the encoding mode, bob then encodes the photon with the two unitary operations U₀ and U₁ chosen randomly, marked by U_B, and sends it to Charlie. Charlie performs a similar operation in the same way as Bob, and returns the photon to Alice after his operation marked as U_C. For each photon that she receives, Alice performs a single-photon measurement with the same basis as the one she originally used to prepare it. As the two unitary operations U₀ and U₁ do not change the measuring bases, Alice obtains a deterministic outcome for almost all the photons returned, e. g., U_A = U_B ⊗ U_C. In order to prevent any eavesdropper from getting information about U_A, Bob and Charlie can chose the control mode, in the control mode, they measure a few photons he received with one of the two measuring bases randomly. Alice also can measure a few photons in this way. In the eavesdropping check Bob, Charlie or Alice will publish the result of measurements and negotiate with the parties who sent these photons to him. In essence, the security of this QSS protocol is ensured by the analysis of the error rates in a similar way to the BB84 and LM05 protocol [19, 20].

In the current work only the encoding mode of the Deng’s protocol is experimentally realized. So, as a consequence of what said above, we cannot claim the security of the scheme against a number of specific attacks, like, for instance, the Trojan-horse. However, the fact is that the Trojan-horse attack is a serious threat for all the known QSS protocols and quantum key distribution [21]. The current partial implementation of the Deng’s protocol represents a meaningful result, as it is directly connected to the final secure rate achievable with a complete implementation.

2. Experimental setup

Figure 2 shows the configuration of our QSS setup, which can be divided into two main parts:the sender Alice and the two other parties Bob and Charlie who can only reconstruct the secret code together. Alice’s station contains a pulsed 1310 nm laser diode LD of pulsewidth 20ps, an attenuator A, a phase-sensitive modulator PM_A connected to a Faraday mirror FM₁, two single-photon detectors D1 and D2, a 50 / 50 coupler, and two polarization beamsplitters PBS₁ and PBS₂ which are connected to the other parties. All Alice station’s components are polarization maintaining and aligned to the “horizontal” axis.

 

Fig. 2 Experimental setup. LD: laser diode; A: attenuator; Cir₁, Cir₂: circulators; PM_A, PM_B, PM_C: phase-sensitive modulators; FM₁, FM₂, FM₃: Faraday mirror; D₁, D₂: single photon detector; C: a 50 / 50 coupler; PBS₁, PBS₂, polarization beam splitter; PM_A, PM_B, PM_C: phase modulator; SW₁, SW₂: fiber switch; CM₁, CM₂: the configuration of control mode; QC₁; QC₂: quantum channel.

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Bob and Charlie have identical components, including a phase sensitive modulator terminated by a Faraday mirror (PM_B, FM₂ and PM_C, FM₃, respectively), and a quantum channel. They also have a control mode box and a fiber switch use to choose the control mode.

The pulsed laser emits vertically polarized pulses at a repetition rate of 1 MHZ which are attenuated at A to the level of single photon, pass through the circulator Cir₁, and are randomly modulated by the polarization-sensitive phase modulator PM_A to have a phase ϕ_A of 0, π/2, π or 3π/2. After reflection by the Faraday mirror FM₁ the polarization is rotated by 90° to the horizontal direction, and so the phase is not modulated in the return pass through PM_A. The pulses then proceed through Cir₂ and an ordinary 50/50 coupler C where they are divided along two paths and are transmitted through the two polarizing beamsplitters PBS₁ and PBS₂ to Bob and Charlie, respectively. The upper pulses, denoted by L₁ in Fig. 2, enter SW₁ in Bob’s port and passes through a quantum channel QC₁ to the phase modulator PM_B and Faraday mirror FM₂. On the first pass through PM_B there is no phase modulation since the photons are horizontally polarized, but after reflection at FM₂ they have undergone a polarization rotation of 90° so on the return pass they can acquire a phase ϕ_B of 0 or π On returning to Alice’s station this vertically polarized pulse is reflected by PBS₂ and PBS₁ and reaches Charlie’s port. Charlie has the same setup as Bob, and also modulates the phase by ϕ_C of 0 or π; the pulse is reflected by FM₃ back to PBS₁ and arrive at the lower arm of coupler C. The lower pulse, denoted by L₂ in Fig. 2, transmit over the opposite path so it arrives at the upper arm of coupler C at the same time of the upper pulse, but none of parties modulate phase in the lower pulse when it through the PM_B and PM_C. Thus interference occurs at C; a photon is detected at D1 when ϕ_B + ϕ_C = 0 or 2π, and at D2 if ϕ_B + ϕ_C = π. It should be particularly noted that our setup is based on phase encoding, so all three parties must modulate the phase of their photons, as mentioned above. After interference in the coupler C the photons arrive at detectors D1 or D2, depending on their phases shift. The detection probabilities of D1 and D2 for different phase values chosen by Alice, Bob and Charile are shown in Table 1. As Alice’s phase didn’t influence the detection probabilities of D1 and D2, the table just shown the result ϕ_A = 0 and ϕ_A = π. Thus Alice have an identical bit string according to their assumption such as: D1 click represents “0”, and D2 click represents “1”. Bob and Charlie must share its phase with each other when they want to know Alice’s sharing. The fiber switch is to switch between encoding mode and control mode, i.e. When Bob (Charlie) want to measure the photon he received, he can choose the SW₁ (SW₂) to CM₁ (CM₂) and completed the control mode presented in Deng’s paper. As said, the control mode is not currently implemented in our apparatus; however, its effect is simulated through the fiber switch since the counts are directed into CM₁ or CM₂ depicted in the figure. Compare to some paper using a beam splitter to realize a control mode [22, 23], the fiber switch provide a convenient way and the user can choose a mode alternatively according to the need of protocol.

Table 1. Detection probabilities for different phase values of ϕ_A, ϕ_B and ϕ_C with the Deng’s protocol

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3. Experimental data

The stability of this scheme was tested over a total fiber length of 50 km. Figure 3 shows the count rates in D₁ and D₂ versus the voltage V_PMB of Bob’s phase modulator, with the voltage of the other parties’ modulators fixed at 0. We can see that the there is good interference of the photons, and at the half-wave voltage (~4.2V), the counts in D₁ and D₂ reach their minimum and maximum, respectively, the photon pulses can contain the information all parties load in it and the “polarization insensitive phase modulator” modulator works well.

 

Fig. 3 Counts of D1、D2 vs modulation voltage of PMB.

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We measured the interference visibility when none of parties change their voltage of PM. The pluses sent from Alice’s port are attenuated at A to the level of single photon. At this, the visibility is equal to $\frac{C_1-C_2}{C_1+C_2}$, C₁ and C₂ mean the counts of D₁ and D₂ respectively.

Figure 4 shows the visibility measurement results tested over a total of 3 hours. The average visibility is 99.3% with a fluctuation of about 0.4%. The setup was found to be extremely stable and this visibility could be repeated without any adjustment to any part every time we started up the system.

 

Fig. 4 Visibility for the total fiber length of 50km during 3 hours.

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4. Conclusion

In summary, a robust single-photon circular QSS scheme with phase encoding has been presented. The average visibility is 99.3% with a fluctuation of about 0.4% over a total transmission distance of 50 km during a 3 hour test period. In effect, the visibility of our system can be maintained perpetually without any adjustment every time we switch on the system, demonstrating its stability and perfect birefringence compensation. Our experiment not only can provide perfect autocompensation for birefringence, but complete polarization insensitive phase modulators. Our method is simpler than other previously reported ways to realize polarization-insensitive phase modulation. The control mode of the protocol is simulated by using a fiber switch which is a novel way compared to previous paper.