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Quantum key distribution via fiber optic, fidelity and error corrections
Optik 121 (2010) 1944–1947
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Quantum key distribution via fiber optic, fidelity and error corrections S. Suchat a, N. Haisirikul a, T. Thaengtang b, K. Paithoonwattanakij b, P.P. Yupapin c, a b c
Physics Division, Faculty of Science and Technology, Phranakhon Rajabhat University, Bangkok 10220, Thailand Faculty of Engineering, King Mongkut’s Institute of Technology Ladkrabang, Bangkok 10520, Thailand Advanced Research Center for Photonics, Faculty of Science, King Mongkut’s Institute of Technology Ladkrabang, Bangkok 10520, Thailand
a r t i c l e in f o
a b s t r a c t
Article history: Received 20 January 2009 Accepted 20 May 2009
We propose an interesting scheme on photon states generation using a fiber optic Mach Zehnder interferometer incorporating a fiber optic ring resonator without any optical pumping parts including in the system, which is available for long-distance link. In principle, the state of a quantum bit, it is known, unknown, or entangled to other systems. The desired quantum states are generated and transmitted in the link via a fiber optic. The transmission quality in terms of quantum fidelity is analyzed, where a high fidelity to the noiseless quantum channel is achieved by adding an ancillary photon after the signal photon within the correlation time of the fiber noise and by performing the quantum parity checking method. The error correction is also analyzed. For simplicity, feature and robustness against path-length mismatches among the nodes make our scheme suitable for multi-user quantum communication networks. & 2009 Elsevier GmbH. All rights reserved.
Keywords: Quantum bits Quantum key distribution Error corrections
Recently, Yupapin and Suchat [1] have demonstrated the use of an all-fiber optic scheme to generate an entangled photon pair, which is the technique similarly proposed by Brendel et al. [2]. However, the former have shown a scheme that is remarkably simple and operated without any optical pumping parts or components included in the system. This paper proposes the extension of the all-fiber optical arrangement setup to describe the system that can be used to generate more photon states. The use of two MZIs, which are connected in a series incorporating a fiber ring resonator, is analyzed. However, this has also been proposed in the way of using bulky optical components with the pumping parts in two papers [3,4]. Both techniques require the pumping parts to be included in the systems. This is difficult to implement in quantum communication where small-scale device is required to fabricate and use. To simplify the problem and make the process easy to understand, in this paper we propose that an all-fiber optic MZIs incorporating a nonlinear effect in a fiber optic ring resonator can be used to generate the pulsed polarization-entangled photon pairs; the nonlinear effects are introduced by using a weak input light pulse incorporating a four-wave mixing phenomenon in the fiber ring resonator [5], where the time-bin entangled photons can be performed. Quantum key distribution (QKD) is the only form of information that can provide the perfect communication security. The use
Corresponding author.
E-mail addresses: [email protected] (S. Suchat), [email protected] (P.P. Yupapin). 0030-4026/$ - see front matter & 2009 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2009.05.023
of QKD has been proposed in many research works, whereas the applications in different forms such as point-to-point link [6], optical wireless [7], satellite [8], long distance [9] and network [10] have been reported. However, a more reliable system for network security is needed. A system that has both high capacity and security is required. The concept of continuous variable in the form of dense wavelength multiplexing is introduced to overcome such a problem. By using the continuous variable concept, the continuous QKD can be formed and be available for a large demand. There are some works that proposed the use of continuous variable QKD with quantum router and network [11,12]. In this proposal, we introduce the idea of the scheme of an alloptical arrangement and then describe the method of using two photons generated in a spontaneous parametrical way, using the linear optical elements and the photon detectors. Suppose that we are given a photon in the unknown state a|HS+b|VS, where |HS and |VS represent the horizontal (H) and the vertical (V) polarization states, respectively, and |a|2+|b|2 ¼ 1. The photon in a time-bin follows the reference photon with a temporal delay Dt1, where the two-photon state can be written as jDS ðajHSDt1 þ bjVSDt1 Þ
ð1Þ
This represents the temporal delay from the front time-bin as shown in Fig. 1. The photons in the H and the V polarization states are transmitted through the second MZI on the channel CH and CV, respectively. Here, a length of an ordinary single-mode fiber can be employed to transmit the quantum channels. For simplicity, we
S. Suchat et al. / Optik 121 (2010) 1944–1947
1945
two photons arrived at the fiber coupler. In this case, the state just can be written as
ae2ifH jVSLDt2 jHSSDt1 þ be2ifV jHSLtþDt2 jVSSDt1 þt þei
Fig. 1. A schematic diagram set-up composed of photon states generation. PBC: Polarization Beam Combiner, MZIs: a fiber optic Mach Zehnder interferometer, Ds: detectors, PCs: polarization control fiber, L: long path, S: short path.
also use the polarization control fibers included in the experiment. In this case, the polarization rotations of the photons in each channel do not occur, but unknown phase shifts fH and fV are added to the photons in each channel independently due to the fluctuations of light in the optical path lengths. We assume that the interval Dt1 between the signal and the reference photons is much shorter than the correlation time of the fluctuations, so that the phase shifts are considered to be correlated such that fH(t) ¼ fH(t+Dt1) ¼ fH and fV(t) ¼ fV(t+Dt1) ¼ fV. The photons in both modes, long and short arms, are then mixed together, and the received state becomes 1 pffiffiffi ½ae2ifH jHSjHSDt1 þ be2ifV jVSt jVSDt1 þt 2 þ eiðfH þfV Þ ðajVSt jHSDt1 þ bjHSjVSDt1 þt Þ
ð2Þ
Here the optical path lengths of long and short may differ, which is indicated by the temporal delay t in the subscripts of the V polarization states. We can easily see that the state ajVSt jHSDt1 þ bjHSjVSDt1 þt is invariant under the phase shifts. The schematics of the proposed setup is shown in Fig. 1. Two photons in distinct modes are generated by fiber optic ring resonator (FORR). One photon in |HS passes through long path, thus adding a phase shift by a polarization control (PC1). The other photon in |HS passes through short path; these photons are mixed by a non-polarizing coupler. The temporal delay Dt1 between the signal and the reference photon is in nanosec (ns). The photons are split into H and V polarization modes by a polarization beam combiner (PBC1), which transmits the H-polarization photons and reflects the V polarization photons. These photons are then transmitted to Bob though CV and CH. At Bob’s location, these photons are mixed by PBC2 again. If the optical path lengths of CV and CH were precisely adjusted with high stability, the received state would be the same as the state prepared by Alice. The signal state from the received two-photon state can be passively performed in the following way. The received two photons are first split into fiber optic MZI pass long path (L) and short path (S) by a coupler, and then mixed by PBC3 again. The polarization controller (PC) is tilted by rotating the polarization of photons in the long path by 901. The polarization of the photon in mode X is projected onto the diagonal state. The difference between the lengths of L and S corresponds to a temporal delay Dt2, which is adjusted by the length L on the stress fiber optics. The successful events (i.e. input pulses) that are post-selected by is criminating the time delay between the arrival of photons at detectors DX and DY by using the time resolving coincidence detection as shown in Fig. 1. We consider the successful case where the signal photon passes through S and the reference photon passes through L. This happens with the probability when
ðfH þfV Þ
ðajHSLtþDt2 jHSSDt1 þ bjVSLDt2 jVSSDt1 þt Þ
ð3Þ
where the superscripts represent the spatial modes. Here fH and fV include the phase shifts added in the output interferometer. If one photon is emitted in each mode of X and Y, the output state just after the polarization beam combiner is ajHSXDt1 jHSYtþDt2 þbjVSXDt2 jVSYDt1 þt . This operation is referred to as quantum parity checking, which is also useful for other quantum information tasks. Let us consider the case where Dt1 ¼ Dt2. When detector 1 finds a photon, the state in mode X is projected onto the detector. At that time, the state in mode Y is projected onto the state ajHSYtþDt1 þ bjVSYDt1 þt ; implying that we faithfully obtain the signal state in mode Y. It is worthwhile to mention here that the delay t affects only the arrival time but not the fidelity of the output states as long as the correlation time of the fluctuations of the phase shifts added in the interferometer is much larger than t. The entangled photons probability is shown in terms of the optical output intensity, which is generated using an all-fiber optic system. The entangled photon states are created by fourwave mixing of the directed pulse train and the polarized controlled outputs from FORR. It is possible to use such a system because the multi-entangled photon states are also possible by using a series of more MZIs. We present a scheme to generate an entangled four-photon state from two pairs of entangled twophoton states using linear optical elements. This entangled fourphoton state is equivalent to two maximally entangled polarizations. The violation of the Bell inequality has shown this kind of multi-photon states. We assume that two pairs of maximally entangled two-photon states have been prepared: 1 1 pffiffiffi ðjHS1 jVS2 þ jVS1 jHS2 Þ and pffiffiffi ðjHS3 jVS4 þ jVS3 jHS4 Þ 2 2
ð4Þ
We have presented a technique that could be used to generate pulsed photon states polarization-entangled photon pairs using the orthogonal pulse polarization delay circuit. The system used is based on the system called time-bin quantum entanglement, incorporating a fiber optic ring resonator. The entangled photon pairs were formed by the interference of randomly delayed orthogonal polarized light pulses while circulating in the fiber ring resonator. Consider the scheme of Fig. 1. Alice has a single-photon qubit in an arbitrary unknown state |cS ¼ a|HS+b|VS, where the kets denote the ‘horizontal’ and ‘vertical’ polarization modes of the photon, respectively, and |aS2+|bS2 ¼ 1. She wants to transmit |cS to Bob over a noisy channel in a manner that enables him to reject any qubit errors that may have occurred and keep only the uncorrupted states. To this end, she possesses an unbalanced polarization interferometer, based on two polarizing beam combiner (PBC), and transmits |HS photons (so that they propagate through the short path, S) and reflects |VS photons (so that they propagate through the long path, L). If Alice’s photon is in the form of an ultrashort wave-packet typical of pulse, and the time-of-flight difference of the unbalanced interferometer is on the order of a few nanoseconds, her qubit transforms as |cS ¼ a|HSS+b|VSL. Alice activates only when the L-path component is present, effecting the transformation |VSL+|HSL. Hence, the final encoded state launched into the noisy channel is of the form a|HSS+b|HSL.
1946
S. Suchat et al. / Optik 121 (2010) 1944–1947
Suppose that the noisy channel is a long-distance optical fiber, where random birefringence induces unknown transformations of the polarization state. However, because Alice has assigned the components of her single-photon qubit to two different time-bins, a few nanoseconds apart, the above causes of birefringence are virtually in a steady state during this temporal interval. In other words, with this encoding technique, whatever unknown unitary operator, U, acts on the early component also acts on the later component. The operator U can be expressed by the transformation |HS-eif cos y|HS+eiw sin y|VS, |VS-eiw sin y|HS+eif cos y|VS, which describes a general qubit transformation (excluding a global phase which is of no physical significance in this context). However, because of Alice’s encoding, the polarization state in both time-bins is |HS and the action of U over the long-distance channel can be described by |HSj-eif cos y|HSj+eiw sin y|VSj, where the subscript j ¼ S, L denotes the time-bin. The evolution of |cS from Alice to Bob can be written as jcSj
Interferometer
!
U
ajHSS þ bjVSL !aðeif cos yjHSS þ eiw sin yjVSS Þ
þbðeif cos yjHSL þ eiw sin yjVSL Þ ¼ jcSU
ð5Þ
Bob is equipped with a same polarization interferometer as Alice. Bob activates when the S-path components of the qubit are present, effecting the transformation |HSS2|VSS. The interferometer on the received state |cSU is given by jcSU
Interferometer
!
aðeif cos yjVSSL þ eiw sin yjHSSS Þ
þ bðeif cos yjHSLS þ eiw sin yjVSLL Þ
ð6Þ
From the last line of expression (6) we see that the first term indicates that the original qubit state (free of errors) emerges at a definite time-of-arrival, corresponding to the delay of propagation once through path S and once through path L (SL or LS). The second term indicates that the |HS and |VS components of the qubit are temporally separated and arrive too late (LL) or too early (SS), respectively. Therefore, Bob’s time gate can readily discard all events that correspond to the transmitted qubit having been projected onto an error-state. With this encoding/decoding technique, the parameters f and w cannot induce errors and the uncorrupted qubit state is obtained with a probability equal to cos2 y. This property is desirable since it means that for small values of y the probability is close to 1. Allowing y varying over its entire range during transmissions (indicating strong environmental influence on the channel) only means that the probability of obtaining the uncorrupted state tends to 0.5. The alternative schemes only work for bit-flip errors while our method rejects any qubit error that may occur due to the action of an unknown unitary operator over the noisy channel. Because the alternative schemes rely on parity checks within multi-photon entangled states, a bit-flip on two qubits renders the method ineffective. This constrains the bit-flip probability to values much less than 1 so that fatal double-errors become negligible. In our case there is only a single-photon qubit involved in the process and therefore the ‘double-error’ possibility does not exist, there is no need of multi-photon entanglement, and the variation of the error-inducing parameter, y, does not affect the error-rejecting capability of the scheme. Furthermore, because of the linear optical encoding of the unknown single-photon qubit within a multi-photon entangled state, the alternative schemes are inherently probabilistic and require post-selection in the coincidence basis in order to determine whether proper encoding was achieved. The scheme of Fig. 1 is deterministic because every
encoding attempt on the unknown qubit state (by Alice) is valid and every decoding attempt (by Bob) succeeds in revealing an error on the transmitted qubit state. To implement such a scheme incorporating in an optical communication networks, the effects on fiber optic properties in network such as spectrum noise, signal dispersion, fiber losses, and timing walk-off on the entangled states are required to make the system validate. When light propagates in a fiber ring resonator, where the ring radius could be ranged from micrometer to few kilometers, the problem is the signal spacing among the propagating signals. In case of multi photons, the signal spacing regions known as the signal free spectral ranges (FSR) are required in order to design to meet the specific purpose. This parameter can cause the intermodulation noises of the signals in the ring resonator and network link. Further, the problem of time delay of each photon pair in the networks caused by the polarization mode dispersion (PMD) of polarized light in fiber optic needs to addressed; otherwise, this effect can cause the problem of timing walk-off the photon entangled states, especially in case of multientangled photons. To compensate this problem the polarization mode dispersion shift fiber, i.e. dispersion shift fiber (DSF fiber), is required to implement into the system. In general case, the optical signals in long-distance communication are degraded due to the fiber optic property, which is usually addressed as a common problem that can be recovered by an amplification system. However, the problem of noisy signals could also be amplified along with the used signals; therefore, the technique that can take care of noisy signal is needed to be implemented into the system. In the optical/quantum networks the entangled states are seen whenever the projection devices are applied, which means the small signals, i.e. signal and idler pairs, can be presented in terms of the detection probability, which is enough to perform the entangled states. In conclusion, we have shown that the advantage of this technique is the remarkably simple optical arrangement, which does not require any optical pumping parts or bulky components, and which has also shown the potential of making device fabrication. Long haul quantum communication via fiber optic link can also be realized by using polarization-entangled photon pairs to encrypt the signals, which can then be recovered by decrypting the signals at the far end of the optical line. However, the problem of qubit missing in the link may cause the information error; however, the error corrections can overcome the problem. This means that top security using fiber optic link based on quantum cryptography by light is plausible. This scheme also has the potential to be used for quantum dense coding for quantum communication and networks that seem suitable for multi-user quantum communication networks.
References [1] P.P. Yupapin, S. Suchat, Entangle photon generation using fiber optic MachZehnder interferometer incorporating nonlinear effect in a fiber ring resonator, J. Nanophoton. 1 (2007) 13504. [2] J. Brendel, N. Gisin, W. Tittel, H. Zbinden, Pulsed energy-time entangled twinphoton source for quantum communication, Phys. Rev. Lett. 82 (12) (1999) 4656–4659. [3] Chunhui Li, Four-photon, five-dimensional entanglement for quantum communication, Phys. Lett. A 313 (5) (2003) 389–392. [4] T.B. Pittman, On the use of double entanglement in four-photon experiments, Phys. Lett. A 204 (3) (1995) 193–197. [5] J.E. Heebner, V. Wong, A. Schweinsberg, R.W. Boyd, D.J. Jackson, Optical transmission characteristics of fiber ring resonators, IEEE J. Quantum Electron. 40 (6) (2004) 726–730. [6] T.S. Manderbach, H. Weier, M. Furst, R. Ursin, F. Tiefenbacher, T. Scheidl, J. Perdigues, Z. Sodnik, C. Kurtsiefer, J.G. Rarity, A. Zeilinger, H. Weinfurter, Experimental demonstration of free-space decoy-state quantum key distribution over 144 km, Phys. Rev. Lett. 98 (2007) 010504.
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[7] S. Suchat, W. Khannam, P.P. Yupapin, Quantum key distribution via an optical wireless communication link for telephone networks, Opt. Eng. 46 (2007) 100502. [8] M. Pfennigbauer, M. Aspelmeyer, W. Leeb, G. Baister, T. Dreischer, T. Jennewein, G. Neckamm, J. Perdigues, H. Weinfurter, A. Zeilinger, Satellitebased quantum communication terminal employing state-of-the-art technology, J. Opt. Netw. 4 (2005) 549. [9] H. Takesue, S.W. Nam, Q. Zhang, R.H. Hadfield, T. Honjo, K. Tamaki, Y. Yamamoto, Quantum key distribution over 40 dB channel loss using superconducting single photon detectors, Nat. Photon. 1 (2007) 343.
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[10] P.P. Yupapin, P. Phiphithirankarn, S. Suchat, A quantum CODEC design via an optical add/drop multiplexer in a fiber optic network, Far East J. Electron. Commun. 1 (2007) 259. [11] Z.L. Yuan, A.J. Shields, Continuous operation of a one-way quantum key distribution system over installed telecom fibre, Opt. Exp. 13 (2005) 660. [12] B. Qi, Single-photon continuous-variable quantum key distribution based on the energy–time uncertainty relation, Opt. Lett. 31 (2006) 2795.
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Quantum key distribution via fiber optic, fidelity and error corrections S. Suchat a, N. Haisirikul a, T. Thaengtang b, K. Paithoonwattanakij b, P.P. Yupapin c, a b c
Physics Division, Faculty of Science and Technology, Phranakhon Rajabhat University, Bangkok 10220, Thailand Faculty of Engineering, King Mongkut’s Institute of Technology Ladkrabang, Bangkok 10520, Thailand Advanced Research Center for Photonics, Faculty of Science, King Mongkut’s Institute of Technology Ladkrabang, Bangkok 10520, Thailand
a r t i c l e in f o
a b s t r a c t
Article history: Received 20 January 2009 Accepted 20 May 2009
We propose an interesting scheme on photon states generation using a fiber optic Mach Zehnder interferometer incorporating a fiber optic ring resonator without any optical pumping parts including in the system, which is available for long-distance link. In principle, the state of a quantum bit, it is known, unknown, or entangled to other systems. The desired quantum states are generated and transmitted in the link via a fiber optic. The transmission quality in terms of quantum fidelity is analyzed, where a high fidelity to the noiseless quantum channel is achieved by adding an ancillary photon after the signal photon within the correlation time of the fiber noise and by performing the quantum parity checking method. The error correction is also analyzed. For simplicity, feature and robustness against path-length mismatches among the nodes make our scheme suitable for multi-user quantum communication networks. & 2009 Elsevier GmbH. All rights reserved.
Keywords: Quantum bits Quantum key distribution Error corrections
Recently, Yupapin and Suchat [1] have demonstrated the use of an all-fiber optic scheme to generate an entangled photon pair, which is the technique similarly proposed by Brendel et al. [2]. However, the former have shown a scheme that is remarkably simple and operated without any optical pumping parts or components included in the system. This paper proposes the extension of the all-fiber optical arrangement setup to describe the system that can be used to generate more photon states. The use of two MZIs, which are connected in a series incorporating a fiber ring resonator, is analyzed. However, this has also been proposed in the way of using bulky optical components with the pumping parts in two papers [3,4]. Both techniques require the pumping parts to be included in the systems. This is difficult to implement in quantum communication where small-scale device is required to fabricate and use. To simplify the problem and make the process easy to understand, in this paper we propose that an all-fiber optic MZIs incorporating a nonlinear effect in a fiber optic ring resonator can be used to generate the pulsed polarization-entangled photon pairs; the nonlinear effects are introduced by using a weak input light pulse incorporating a four-wave mixing phenomenon in the fiber ring resonator [5], where the time-bin entangled photons can be performed. Quantum key distribution (QKD) is the only form of information that can provide the perfect communication security. The use
Corresponding author.
E-mail addresses: [email protected] (S. Suchat), [email protected] (P.P. Yupapin). 0030-4026/$ - see front matter & 2009 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2009.05.023
of QKD has been proposed in many research works, whereas the applications in different forms such as point-to-point link [6], optical wireless [7], satellite [8], long distance [9] and network [10] have been reported. However, a more reliable system for network security is needed. A system that has both high capacity and security is required. The concept of continuous variable in the form of dense wavelength multiplexing is introduced to overcome such a problem. By using the continuous variable concept, the continuous QKD can be formed and be available for a large demand. There are some works that proposed the use of continuous variable QKD with quantum router and network [11,12]. In this proposal, we introduce the idea of the scheme of an alloptical arrangement and then describe the method of using two photons generated in a spontaneous parametrical way, using the linear optical elements and the photon detectors. Suppose that we are given a photon in the unknown state a|HS+b|VS, where |HS and |VS represent the horizontal (H) and the vertical (V) polarization states, respectively, and |a|2+|b|2 ¼ 1. The photon in a time-bin follows the reference photon with a temporal delay Dt1, where the two-photon state can be written as jDS ðajHSDt1 þ bjVSDt1 Þ
ð1Þ
This represents the temporal delay from the front time-bin as shown in Fig. 1. The photons in the H and the V polarization states are transmitted through the second MZI on the channel CH and CV, respectively. Here, a length of an ordinary single-mode fiber can be employed to transmit the quantum channels. For simplicity, we
S. Suchat et al. / Optik 121 (2010) 1944–1947
1945
two photons arrived at the fiber coupler. In this case, the state just can be written as
ae2ifH jVSLDt2 jHSSDt1 þ be2ifV jHSLtþDt2 jVSSDt1 þt þei
Fig. 1. A schematic diagram set-up composed of photon states generation. PBC: Polarization Beam Combiner, MZIs: a fiber optic Mach Zehnder interferometer, Ds: detectors, PCs: polarization control fiber, L: long path, S: short path.
also use the polarization control fibers included in the experiment. In this case, the polarization rotations of the photons in each channel do not occur, but unknown phase shifts fH and fV are added to the photons in each channel independently due to the fluctuations of light in the optical path lengths. We assume that the interval Dt1 between the signal and the reference photons is much shorter than the correlation time of the fluctuations, so that the phase shifts are considered to be correlated such that fH(t) ¼ fH(t+Dt1) ¼ fH and fV(t) ¼ fV(t+Dt1) ¼ fV. The photons in both modes, long and short arms, are then mixed together, and the received state becomes 1 pffiffiffi ½ae2ifH jHSjHSDt1 þ be2ifV jVSt jVSDt1 þt 2 þ eiðfH þfV Þ ðajVSt jHSDt1 þ bjHSjVSDt1 þt Þ
ð2Þ
Here the optical path lengths of long and short may differ, which is indicated by the temporal delay t in the subscripts of the V polarization states. We can easily see that the state ajVSt jHSDt1 þ bjHSjVSDt1 þt is invariant under the phase shifts. The schematics of the proposed setup is shown in Fig. 1. Two photons in distinct modes are generated by fiber optic ring resonator (FORR). One photon in |HS passes through long path, thus adding a phase shift by a polarization control (PC1). The other photon in |HS passes through short path; these photons are mixed by a non-polarizing coupler. The temporal delay Dt1 between the signal and the reference photon is in nanosec (ns). The photons are split into H and V polarization modes by a polarization beam combiner (PBC1), which transmits the H-polarization photons and reflects the V polarization photons. These photons are then transmitted to Bob though CV and CH. At Bob’s location, these photons are mixed by PBC2 again. If the optical path lengths of CV and CH were precisely adjusted with high stability, the received state would be the same as the state prepared by Alice. The signal state from the received two-photon state can be passively performed in the following way. The received two photons are first split into fiber optic MZI pass long path (L) and short path (S) by a coupler, and then mixed by PBC3 again. The polarization controller (PC) is tilted by rotating the polarization of photons in the long path by 901. The polarization of the photon in mode X is projected onto the diagonal state. The difference between the lengths of L and S corresponds to a temporal delay Dt2, which is adjusted by the length L on the stress fiber optics. The successful events (i.e. input pulses) that are post-selected by is criminating the time delay between the arrival of photons at detectors DX and DY by using the time resolving coincidence detection as shown in Fig. 1. We consider the successful case where the signal photon passes through S and the reference photon passes through L. This happens with the probability when
ðfH þfV Þ
ðajHSLtþDt2 jHSSDt1 þ bjVSLDt2 jVSSDt1 þt Þ
ð3Þ
where the superscripts represent the spatial modes. Here fH and fV include the phase shifts added in the output interferometer. If one photon is emitted in each mode of X and Y, the output state just after the polarization beam combiner is ajHSXDt1 jHSYtþDt2 þbjVSXDt2 jVSYDt1 þt . This operation is referred to as quantum parity checking, which is also useful for other quantum information tasks. Let us consider the case where Dt1 ¼ Dt2. When detector 1 finds a photon, the state in mode X is projected onto the detector. At that time, the state in mode Y is projected onto the state ajHSYtþDt1 þ bjVSYDt1 þt ; implying that we faithfully obtain the signal state in mode Y. It is worthwhile to mention here that the delay t affects only the arrival time but not the fidelity of the output states as long as the correlation time of the fluctuations of the phase shifts added in the interferometer is much larger than t. The entangled photons probability is shown in terms of the optical output intensity, which is generated using an all-fiber optic system. The entangled photon states are created by fourwave mixing of the directed pulse train and the polarized controlled outputs from FORR. It is possible to use such a system because the multi-entangled photon states are also possible by using a series of more MZIs. We present a scheme to generate an entangled four-photon state from two pairs of entangled twophoton states using linear optical elements. This entangled fourphoton state is equivalent to two maximally entangled polarizations. The violation of the Bell inequality has shown this kind of multi-photon states. We assume that two pairs of maximally entangled two-photon states have been prepared: 1 1 pffiffiffi ðjHS1 jVS2 þ jVS1 jHS2 Þ and pffiffiffi ðjHS3 jVS4 þ jVS3 jHS4 Þ 2 2
ð4Þ
We have presented a technique that could be used to generate pulsed photon states polarization-entangled photon pairs using the orthogonal pulse polarization delay circuit. The system used is based on the system called time-bin quantum entanglement, incorporating a fiber optic ring resonator. The entangled photon pairs were formed by the interference of randomly delayed orthogonal polarized light pulses while circulating in the fiber ring resonator. Consider the scheme of Fig. 1. Alice has a single-photon qubit in an arbitrary unknown state |cS ¼ a|HS+b|VS, where the kets denote the ‘horizontal’ and ‘vertical’ polarization modes of the photon, respectively, and |aS2+|bS2 ¼ 1. She wants to transmit |cS to Bob over a noisy channel in a manner that enables him to reject any qubit errors that may have occurred and keep only the uncorrupted states. To this end, she possesses an unbalanced polarization interferometer, based on two polarizing beam combiner (PBC), and transmits |HS photons (so that they propagate through the short path, S) and reflects |VS photons (so that they propagate through the long path, L). If Alice’s photon is in the form of an ultrashort wave-packet typical of pulse, and the time-of-flight difference of the unbalanced interferometer is on the order of a few nanoseconds, her qubit transforms as |cS ¼ a|HSS+b|VSL. Alice activates only when the L-path component is present, effecting the transformation |VSL+|HSL. Hence, the final encoded state launched into the noisy channel is of the form a|HSS+b|HSL.
1946
S. Suchat et al. / Optik 121 (2010) 1944–1947
Suppose that the noisy channel is a long-distance optical fiber, where random birefringence induces unknown transformations of the polarization state. However, because Alice has assigned the components of her single-photon qubit to two different time-bins, a few nanoseconds apart, the above causes of birefringence are virtually in a steady state during this temporal interval. In other words, with this encoding technique, whatever unknown unitary operator, U, acts on the early component also acts on the later component. The operator U can be expressed by the transformation |HS-eif cos y|HS+eiw sin y|VS, |VS-eiw sin y|HS+eif cos y|VS, which describes a general qubit transformation (excluding a global phase which is of no physical significance in this context). However, because of Alice’s encoding, the polarization state in both time-bins is |HS and the action of U over the long-distance channel can be described by |HSj-eif cos y|HSj+eiw sin y|VSj, where the subscript j ¼ S, L denotes the time-bin. The evolution of |cS from Alice to Bob can be written as jcSj
Interferometer
!
U
ajHSS þ bjVSL !aðeif cos yjHSS þ eiw sin yjVSS Þ
þbðeif cos yjHSL þ eiw sin yjVSL Þ ¼ jcSU
ð5Þ
Bob is equipped with a same polarization interferometer as Alice. Bob activates when the S-path components of the qubit are present, effecting the transformation |HSS2|VSS. The interferometer on the received state |cSU is given by jcSU
Interferometer
!
aðeif cos yjVSSL þ eiw sin yjHSSS Þ
þ bðeif cos yjHSLS þ eiw sin yjVSLL Þ
ð6Þ
From the last line of expression (6) we see that the first term indicates that the original qubit state (free of errors) emerges at a definite time-of-arrival, corresponding to the delay of propagation once through path S and once through path L (SL or LS). The second term indicates that the |HS and |VS components of the qubit are temporally separated and arrive too late (LL) or too early (SS), respectively. Therefore, Bob’s time gate can readily discard all events that correspond to the transmitted qubit having been projected onto an error-state. With this encoding/decoding technique, the parameters f and w cannot induce errors and the uncorrupted qubit state is obtained with a probability equal to cos2 y. This property is desirable since it means that for small values of y the probability is close to 1. Allowing y varying over its entire range during transmissions (indicating strong environmental influence on the channel) only means that the probability of obtaining the uncorrupted state tends to 0.5. The alternative schemes only work for bit-flip errors while our method rejects any qubit error that may occur due to the action of an unknown unitary operator over the noisy channel. Because the alternative schemes rely on parity checks within multi-photon entangled states, a bit-flip on two qubits renders the method ineffective. This constrains the bit-flip probability to values much less than 1 so that fatal double-errors become negligible. In our case there is only a single-photon qubit involved in the process and therefore the ‘double-error’ possibility does not exist, there is no need of multi-photon entanglement, and the variation of the error-inducing parameter, y, does not affect the error-rejecting capability of the scheme. Furthermore, because of the linear optical encoding of the unknown single-photon qubit within a multi-photon entangled state, the alternative schemes are inherently probabilistic and require post-selection in the coincidence basis in order to determine whether proper encoding was achieved. The scheme of Fig. 1 is deterministic because every
encoding attempt on the unknown qubit state (by Alice) is valid and every decoding attempt (by Bob) succeeds in revealing an error on the transmitted qubit state. To implement such a scheme incorporating in an optical communication networks, the effects on fiber optic properties in network such as spectrum noise, signal dispersion, fiber losses, and timing walk-off on the entangled states are required to make the system validate. When light propagates in a fiber ring resonator, where the ring radius could be ranged from micrometer to few kilometers, the problem is the signal spacing among the propagating signals. In case of multi photons, the signal spacing regions known as the signal free spectral ranges (FSR) are required in order to design to meet the specific purpose. This parameter can cause the intermodulation noises of the signals in the ring resonator and network link. Further, the problem of time delay of each photon pair in the networks caused by the polarization mode dispersion (PMD) of polarized light in fiber optic needs to addressed; otherwise, this effect can cause the problem of timing walk-off the photon entangled states, especially in case of multientangled photons. To compensate this problem the polarization mode dispersion shift fiber, i.e. dispersion shift fiber (DSF fiber), is required to implement into the system. In general case, the optical signals in long-distance communication are degraded due to the fiber optic property, which is usually addressed as a common problem that can be recovered by an amplification system. However, the problem of noisy signals could also be amplified along with the used signals; therefore, the technique that can take care of noisy signal is needed to be implemented into the system. In the optical/quantum networks the entangled states are seen whenever the projection devices are applied, which means the small signals, i.e. signal and idler pairs, can be presented in terms of the detection probability, which is enough to perform the entangled states. In conclusion, we have shown that the advantage of this technique is the remarkably simple optical arrangement, which does not require any optical pumping parts or bulky components, and which has also shown the potential of making device fabrication. Long haul quantum communication via fiber optic link can also be realized by using polarization-entangled photon pairs to encrypt the signals, which can then be recovered by decrypting the signals at the far end of the optical line. However, the problem of qubit missing in the link may cause the information error; however, the error corrections can overcome the problem. This means that top security using fiber optic link based on quantum cryptography by light is plausible. This scheme also has the potential to be used for quantum dense coding for quantum communication and networks that seem suitable for multi-user quantum communication networks.
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