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We report the first Sagnac quantum secret sharing (in three-and four-party implementations) over 1550 nm single mode fiber (SMF) networks, using a single qubit protocol with phase encoding. Our secret sharing experiment has been based on a single qubit protocol, which has opened the door to practical secret sharing implementation over fiber telecom channels and in free-space. The previous quantum secret sharing proposals were based on multiparticle entangled states, difficult in the practical implementation and not scalable. Our experimental data in the three-party implementation show stable (in regards to birefringence drift) quantum secret sharing transmissions at the total Sagnac transmission loop distances of 55-75 km with the quantum bit error rates (QBER) of 2.3-2.4% for the mean photon number μ = 0.1 and 1.7-2.1% for μ = 0.3. In the four-party case we have achieved quantum secret sharing transmissions at the total Sagnac transmission loop distances of 45-55 km with the quantum bit error rates (QBER) of 3.0-3.7% for the mean photon number μ = 0.1 and 1.8-3.0% for μ = 0.3. The stability of quantum transmission has been achieved thanks to our new concept for compensation of SMF birefringence effects in Sagnac, based on a polarization control system and a polarization insensitive phase modulator. The measurement results have showed feasibility of quantum secret sharing over telecom fiber networks in Sagnac configuration, using standard fiber telecom components.
©2009 Optical Society of America
1. Introduction
Splitting a secret message in the way that a single person is not able to reconstruct it is a common task in information processing and security applications. For instance, let us assume that withdrawing cash from a joint banking account is possible only when all account owners cooperate by generating a code required by an automated teller machine (ATM) or by a banker [1]. A solution for this problem and its generalization, including several variations, is provided by classical cryptography and is called secret sharing. It consists of a way of splitting the message using mathematical algorithms and the distribution of the resulting pieces to two or more legitimate users by classical communication. However, all ways of classical communication currently used are susceptible to eavesdropping attacks. As the usage of quantum resources can lead to unconditionally secure communication, a protocol introducing the quantum information scheme to secret sharing has been developed [2, 3]. This protocol provides information splitting and eavesdropping protection. However, the scheme is in practice non-scalable since it used multiphoton polarization entangled states that are difficult to generate and transmit. Furthermore, the use of polarization encoding is impractical for applications over commercial birefringent single mode fiber (SMF) networks. Nevertheless, three proof of principle experiments, using three [4, 5] and four [6] entangled polarized photons, have been carried out. A new protocol solving the above mentioned problems was proposed in 2005 [7]. The protocol requires only a single qubit for quantum information transmission, which allowed for its practical experimental realization and scalability.
This new innovative N-party quantum secret sharing protocol, with the parties called R 1,…,RN [7], is showed in Fig. 1. A qubit is prepared in an initial state |x⟩ = (|0⟩ + |1⟩)/ √2 by the party R 1 and is sent sequentially, from R 1 to RN, over the quantum channel, until it is measured by the last party RN. Each party Ri(i = 1, …,N - 1) modulates the photon with a randomly chosen phase ϕi equal to 0, π/2, π, or 3π/2 through the unitary phase operator
The parties R 1, R 2,…, and R N-1 modulating phase choices {0, π/2, π,3π/2} can be assigned into two bases {0,π} and {π/2,3π/2}. The RN party’s phase modulation choice ϕN are two phases only: 0 (which belongs to the basis {0, π}) and π/2 (which belongs to the basis {π/2,3π/2}). The probability that RN detects the state |±x⟩ = (|0⟩ ± |1⟩)/√2 is
Fig. 1. N-party single qubit secret sharing. A qubit (in our case a photon) is prepared in an initial state by the part R 1, using a single qubit source, and is sent sequentially, from R 1 to RN, over the quantum channel, until it is measured by the last part RN. Each party modulates the photon with a randomly chosen phase ϕi(i = 1,…,N - 1) of 0, π/2, π or 3π/2. The part RN, carrying out the measurement, sets measurement’s base with its phase modulator by choosing ϕN between 0 and π/2. In half of the cases the phases add up in such a way that the measurement results are deterministic. These cases can be used for the secret sharing.
In half of the cases the phases add up in such a way that the measurement results are deterministic indicating a constructive or destructive interference with the total correlation function given by
These cases can be used for the secret sharing. After the photon has been detected, the parties in a reverse order [8] (i.e. RN, R N-1,…, R 2, R 1) announce on a public channel their basis choices, but keep their particular modulating phases secret. From the shared (on the public channel) basis information, the parties can determine which runs led to deterministic measurements with cos(ϕ 1 + … + ϕN) = ±1. In these cases, any subset of N - 1 parties is able to infer the modulating phase of the remaining part if all the N - 1 parties of the subset collaborate and reveal among themselves their modulating phases. The way in which the parties collaborate and reveal modulating phases depends on the particular secret sharing application (and there are many of them). If the subset includes the part RN, it must reveal the measurement result (it has already revealed its basis choices on the public channel). In such a way the goal of secret sharing has been achieved.
The reported work includes the first, to the best of our knowledge, secret sharing experiment over fiber networks in Sagnac interferometric configuration (in three- and four-party implementations), using a single qubit protocol with phase encoding. The experiment has mainly been focused on proving feasibility of Sagnac quantum secret sharing in fiber, using single photons. The protocol [7], which we based our experiment on, constitutes the main transmission layer. More layers could be added on it in order to provide means for a particular quantum transmission application. The achieved stable (in regards to birefringence drift) secure quantum transmission distances were of 55-75 km in the three-party and 25 – 50 km in the four-party implementation. The stability has been achieved thanks to our new concept for compensation of SMF birefringence effects in Sagnac (see Sec. 2.1) based on a polarization control system and a polarization insensitive phase modulator (PM) scheme, discussed in Sec. 2.2.
2. Experimental setup
2.1. System configuration
Figure 2 shows the configuration of our Sagnac QKD setup with phase encoding. Sagnac interferometer’s main feature is its optic path’s loop (a ”circular network” in the case of its fiber implementation) connected to a coherent light source through a coupler. Thus, the light propagates over the interferometer in opposite directions (clockwise and counterclockwise).
The experimental setup could be divided into three main parts: Alice’s station; SMF quantum transmission channels in Sagnac configuration; and party stations (Bob’s, Charlie’s, and David’s), consisting mainly of polarization insensitive phase modulators (PM), see Sec. 2.2.
Fig. 2. Four party Sagnac secret sharing. All the measurements were carried out at μ = 0.1 and μ = 0.3. The laser pulse repetition rate in the burst mode was set to 2 MHz with the burst duty cycle of 20 %. The setup does not show a control electronics and wavelength-division multiplexing (WDM) layer used to send synchronization and trigger from Alice’s station to Bob’s, Charlie’s, and David’s stations. For this purpose a second pulsed laser with 1538 nm wavelength was used.
Alice’s station contains a weak coherent pulsed 1550 nm laser from id Quantique generating 1 mW, 500 ps wide pulses at 2 MHz repetition rate (the rate has been limited by the single photon detectors’ afterpulsing effect); a phase modulator (PMA); a 50/50 coupler (C 3) dividing the weak laser pulses into clockwise and counterclockwise parts; two interferometers (INT 1 and INT 2); and two single photon detectors (SPD 1 and (SPD 2) connected to the C 3 coupler’s outputs. The detectors (PGA600 from Princeton Lightwave Inc.) implement InGaAs avalanche photodiodes. They provide quantum efficiency of 20% and 10−5 dark count probability per 1 ns gating pulse.
The interferometer INT 2 contains a polarizing beam splitter (PBS 2) and a 50/50 coupler (C 2). The interferometer INT 1 has additionally a fiber stretcher (from General Photonics Inc.), controlled by the NI6602, a digital acquisition card (DAQ) from National Instrument, and a delay line, which length is matched to stretcher’s fiber length of 18 m. The interferometers are used for removing the different SMF birefringence effects on the clockwise and counterclockwise pulses by converting their polarization (into the horizontal one) after they have propagated over the Sagnac loop and arrived back to Alice’s station. The conversion is necessary since the birefringent SMF channel differently changes polarization of the clockwise and counterclockwise pulses, which would decrease interference visibility in the coupler C 3. Stretcher’s main function is to minimize the amount of energy losses [9, 10, 11, 12] in the couplers C 1 and C 2 arms that are not connected into the coupler C 3. This energy loss is monitored by an additional single photon detector ((SPD 3) connected to the coupler C 1. The detector’s output is read, once per second, by the NI6602 DAQ card, programmed in LabView. After the reading process, a simple proportional–integral–derivative (PID) controller in LabView, working in one second long time frames, adjusts stretcher’s length (thus the phase of the signal in stretcher’s arm) in order to minimize the interference losses in the couplers C 1 and C 2.
All Alice station’s components are polarization maintaining since the phase modulator (a standard telecom component) is polarization sensitive: its attenuation for the vertical polarization (fast axis component) is very high while it is low (3–3.5 dB) for the horizontal one (slow axis). Also outputs of the polarizing beam splitters PBS 1 and PBS 2 are aligned to the horizontal axis as well as the interconnecting fiber cords.
The laser pulse enters the interferometer through the circulator and, as already mentioned, gets equally split in the coupler C 3 into clockwise and counterclockwise parts. The pulses are sent by Alice as a high-speed burst by burst-pulsing the laser driver with the aid of a programmable pulse generator (PG9528 from Quantum Composers Inc.). Alice waits with a new burst sending until all the burst pulses have been transmitted over the Sagnac loop. The duty cycle (20% in our setup with 400 active and 1600 inactive pulses) of the burst should be chosen in such a way that only the clockwise pulses are phase modulated in Charlie’s and David’s stations while the counterclockwise ones are modulated in Bob’s and Alice’s station (see Fig. 2 showing, with the arrows, directions of the phase modulation). The duty cycle of 20% gives the system’s effective pulsing rate of 400 kHz.
The main reason for using the burst mode was the limited analog bandwidth (5 MHz) of the PG9528 PM driver. The bandwidth limits the minimum width of the modulating pulses to 100 ns. Using such wide modulating pulses, as in our case, makes it difficult to avoid unwanted modulation of the clockwise and counterclockwise pulses assigned by setup’s configuration for the modulation at other stations (as already mentioned, the clockwise pulses are assigned for the modulation at Charlie’s and David’s stations while the counterclockwise at the remaining stations). The limitation does not allow to fully utilize the PM bandwidth of 0.5 GHz.
Alice’s PMA and Bob’s PMB have to be kept inactive during the clockwise pulse’s passage through Sagnac loop. First at the output of Charlie’s station the pulses are phase modulated and their energy is set to a single photon level [14] by the digitally controlled optical attenuator from OZ Optics Inc. These attenuated (”faint”) laser pulses are Poisson distributed, which gives the probability that a non-empty pulse contains more than one photon: P(n > 1) ≃ μ/2, where μ is the mean photon number. By adjusting the pulse attenuation it is possible to limit the probability that a non-empty weak pulse contains more than one photon to an arbitrary small number. Assuming the attenuation giving one single photon for ten laser pulses (μ = 0.1) we are getting P(n > 1) ≃ μ/2 = 0.05, a low number. It should be pointed out that to the total attenuation of the pulses contribute not only the fiber spools and digital attenuator, but also other setup components such as Alice’s PM, PBS, and circulator as well as the other party stations (Bob’s, Charlie’s, and David’s).
2.2. Polarization insensitive phase modulator
The clockwise and counterclockwise pulses leaving the interferometers INT 1 and INT 2 (see Fig. 2) are generally elliptically polarized. However, their polarization changes along the setup’s transmission path since the SMF is birefringent. It differently changes polarization of the clockwise and counterclockwise pulses so they would arrive back to the coupler C 3, (after their propagation over the Sagnac loop) at different polarization states, which would decrease the interference visibility. In order to avoid it the interferometers INT 1 and INT 2 are used for the polarization back conversion into the horizontal one, which is necessary since all Alice’s station components, including its phase modulator (PMA), are polarization maintaining and aligned horizontally. As mentioned above, the standard telecom phase modulator (PMA) is used at Alice’s station. However, at the party stations there is a need of polarization insensitive phase modulators since the SMF channel’s birefringence would cause significant, slowly varying, transmittance changes of the standard telecom phase modulators (usually based on a LiNbO 3 crystal).
Fig. 3. Polarization insensitive phase modulator. The R output of the PBS denotes the reflected component (vertical), while the T output denotes the transmitted component (horizontal).
Another difficulty, facing a designer of any ”faint-pulse” based QKD system, using telecom phase modulators, is the need of precise attenuation of the laser pulses to a single photon level, in accordance with the ”faint pulse” approach [14, 15]. Since the telecom phase modulators are polarization sensitive and the SMF’s birefringence causes slow, random polarization changes of the laser pulses then there is no way to guarantee a stable attenuation of the entire system. In our Sagnac configuration, it means that it is not possible to guarantee that the ”faint” pulses leaving Charlie’s station are on an assumed mean photon number μ.
In order to resolve the above mentioned issues there is a need to provide a polarization insensitive phase modulator for Bob’s, Charlie’s, and David’s stations. Unfortunately, there are no, to the best of our knowledge, commercially available polarization insensitive phase modulators on the market so we have realized a polarization insensitive device, based on commercially available optical fiber components.
Figure 3 shows the block diagram of our scheme, which implements two 1550nm, 500 MHz bandwidth phase modulators from JDSU Inc. The polarization maintaining pigtails of the device are aligned to the slow (horizontal) axis, which requires the same alignment of the polarizing beam splitters PBSA and PBSB. The design’s working principle is simple: it splits horizontal and vertical polarization components into two separately controlled phase modulators.
In detail, let us consider an input light pulse horizontally, diagonally, circularly, or generally elliptically polarized arriving into PBSA. The pulse’s horizontal polarization component will be transmitted into the port T, while the vertical one will be ”reflected” into the port R and rotated 90° to the horizontal polarization. Thus, both horizontally aligned components can be transmitted (and modulated) by the phase modulators PM 1 and PM 2. The outputs of the modulators are connected to the PBSB, which recreate the original light pulse polarization. The design is bidirectional, which makes it possible to implement it in our setup. Both phase modulators, with a Vπ around 3 Volts, are controlled by the same pulse generator (PG9528). The design guarantees a stable, polarization insensitive optical insertion loss of 5.0 dB.
2.3. Transmission and error rates
The raw rate in the protocol is defined as Rateraw = qμfηdetηlink, where q is a setup dependent coefficient, μ is the mean photon number per pulse, f is the laser pulsing frequency, ηdet is the probability of the photon being detected, and ηlink is the transfer efficiency of the Sagnac loop. The factor q = 0.5 in our setup since only in 50% of all the measurement cases the measurement basis are coincidental.
The QBER for the faint laser pulse can be written as a sum of two main contributing factors: QBER = QBERopt + QBERdet = Popt + Pnoise/Pphoton = Popt + Pnoise/μηdetμlink, where popt is the probability of a photon going to the wrong detector, and pnoise is the probability of getting a noise-count (mainly dark counts) per gating pulse window [13, 14, 15]. For the setup using phase modulation Popt = (1 − V)/2, where V is the interference visibility.
3. Experiment data
Fig. 4. Three-party Sagnac secret sharing. All the measurements were carried out at μ = 0.1 and μ = 0.3. The laser pulse repetition rate in the burst mode was set to 2 MHz with the burst duty cycle of 20 %
All measurements were carried out at a conservatively chosen [14] mean photon numbers per pulse μ = 0.1 and μ = 0.3. The laser pulse repetition rate was 2 MHz (the rate has been limited by the single photon detectors’ afterpulsing effect). The laser pulsing duty cycle of 20% gives the system’s effective rate of 400 kHz.
Figures 4 and 5 show measurement results for μ = 0.1 and μ = 0.3 in the three-party and four-party configurations, respectively. In the three-party implementation we have achieved stable (in regards to birefringence drift) quantum secret sharing transmissions at the total Sagnac transmission loop distances of 55-75 km with the quantum bit error rates (QBER) of 2.3-2.4% for the mean photon number μ = 0.1 and 1.7-2.1% for μ = 0.3. The achieved raw transmission rates were of 1275–1954 Hz for μ = 0.1 and 3684–6488 Hz for μ = 0.3. In the four-party configuration we have achieved the total Sagnac transmission loop distances of 45-55 km with QBER of 3.0-3.7% for μ = 0.1 and 1.8-3.0% for μ = 0.3. The achieved raw transmission rates were of 844–504 Hz for μ = 0.1 and 2794–1537 Hz for μ = 0.3.
Fig. 5. Four-party Sagnac secret sharing. All the measurements were carried out at μ = 0.1 and μ = 0.3. The laser pulse repetition rate in the burst mode was set to 2 MHz with the burst duty cycle of 20 %.
4. Conclusion
We have carried out the first Sagnac quantum secret sharing (in three- and four-party implementations) over 1550 nm single mode fiber (SMF) networks, with phase encoding. Our experiment has been based on a single qubit protocol, which has opened the door to practical secret sharing implementation over fiber telecom channels and in free-space. The protocol constitutes the main transmission layer (more layers could be added on it in order to provide means for a particular quantum secret sharing transmission application). The previous quantum secret sharing proposals were based on multiparticle entangled states, difficult in the practical implementation and not scalable.
Our experimental data in the three-party implementation show stable (in regards to birefringence drift) quantum secret sharing transmissions at the total Sagnac transmission loop distances of 55-75 km with the quantum bit error rates (QBER) of 2.3-2.4% for the mean photon number μ = 0.1 and 1.7-2.1% for μ = 0.3. The achieved raw transmission rates were of 1275–1954 Hz for μ = 0.1. For μ = 0.3 the rates were of 3684–6488 Hz. In the four-party configuration we have achieved the total Sagnac transmission loop distances of 45-55 km with QBER of 3.0-3.7% for μ = 0.1 and 1.8-3.0% for μ = 0.3. The achieved raw transmission rates were of 844–504 Hz for μ = 0.1 and 2794–1537 Hz for μ = 0.3.
The achieved rates could be much higher if instead of our standard InGaAs avalanche photodiode based single photon detectors the newly available superconducting devices were used in the experiment [16]. The stable, in regards to birefringence drift, measurement results have been made possible thanks to our novel concept for compensation of SMF birefringence effects in Sagnac, based on the polarization control system and the polarization insensitive phase modulator. These designs open the door not only to commercial application of Sagnac quantum secret sharing and quantum key distribution over telecom fiber networks, but also to other Sagnac-based applications over SMF links such as precise optical sensing, dispersion characteristics of optical fibers, acoustic and strain sensing, and generally sensing of any time varying phenomenon [17].
Acknowledgment
This work was supported by Swedish Defence Material Administration (FMV) and Swedish Research Council (VR).
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