- Email: [email protected]
Quantum bits generation using correlated photons: Fidelity and error corrections
Optik 121 (2010) 1976–1980
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Quantum bits generation using correlated photons: Fidelity and error corrections T. Tanengtang a,b, K. Paithoonwattanakij a, P.P. Yupapin b,, S. Suchat c a
Department of Electrical Engineering, 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 c Faculty of Science and Technology, Phranakhon Rajabhat University, Bangkok 10220, Thailand b
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 9 May 2009
We propose a new system of quantum bits generation using a soliton pulse within a micro ring resonator. A quantum gate can be formed using a polarization control unit incorporating into the system. The random signal and idler pairs can be formed within the photon correlation bandwidth, which can be generated and randomly formed the packet quantum bits, i.e. quantum packet codes. Each random code (logic) can be performed by combining the signal and idler of each entangled photon pair via the quantum gate. Results obtained have shown that the packet of quantum logic bits can be generated using the entangled photon pairs generated by the proposed system. The quantum bits transmission fidelity and error corrections are also described. & 2009 Elsevier GmbH. All rights reserved.
Keywords: Quantum bits Error corrections Photon correlation Fidelity Entanglement
1. Introduction Qubit is introduced in order to emphasize that the aim is to stabilize a complete quantum state, not just a chosen observable. Also, we are concerned with the properties of the quantum state, not with the physical system expressing it. For example, a single qubit may be expressed by a system whose Hilbert space has more than two dimensions. Among the possible changes such a system may undergo, some will affect the stored single qubit of quantum information, but others will not. Both quantum communication and quantum information processing have been shown to be fundamentally different from its classical counterpart. Examples where this difference is highlighted are secure key distribution for cryptography and the existence of fast algorithms for an idealized quantum computer. Quantum network has also been introduced and is becoming the promising technology that can be used to fulfill the perfect network security. Some research works have been reported in various forms of applications [1,2]. The use of quantum key distribution via optical network has been reported [3,4]. To date, quantum key distribution (QKD) is the only form of information that can provide the perfect communication security. The use of QKD has been proposed in many research works, whereas the applications in different forms – such as point to point link [5], optical wireless [6], satellite [7], long distance [8] and network [9] – have been reported. However, a more reliable system for network security is needed, which has both high capacity and is secure. The concept of continuous variable in the Corresponding author. Fax: + 66 2 3264354.
E-mail address: [email protected] (P.P. Yupapin). 0030-4026/$ - see front matter & 2009 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2009.05.032
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 available for a large demand. Some works have proposed the use of continuous variable QKD with quantum router and network [10,11]. However, the requirement of large bandwidth signal and dense wavelength multiplexing has become practical problems. Yupapin et al. [12] have also shown that the continuous wavelength can be generated using a soliton pulse in a micro ring resonator, which can be used to overcome such problems. In this paper, we show that the large bandwidth of soliton pulse is generated and filtered within the micro ring resonator system. The selected (tuned) pulse is coherently localized and stored within the ring device. In applications, we can use the tuned soliton pulses to perform the required applications such as dense wavelength division multiplexing (DWDM) based soliton communication, optical and quantum memory, and multi-soliton sources. We also propose a new concept of quantum gate where the packet of quantum of logic bits can be formed using a simple scheme. By using a soliton pulse traveling within nonlinear micro and ring resonators, large bandwidth signals occurred and dense wavelengths were generated, which were available for quantum dense coding, i.e. quantum packet switching application. The quantum bits transmission fidelity and error corrections are also analyzed.
2. Multi-spatial solitons An optical soliton is recognized as a powerful laser pulse that can be used to enlarge the optical bandwidth when propagating
T. Tanengtang et al. / Optik 121 (2010) 1976–1980
within the nonlinear micro ring resonator [12]. Moreover, the superposition of self-phase modulation (SPM) soliton pulses can keep the large output power. Initially, the optimum energy is coupled into the waveguide by a lager effective core area device, i.e. ring resonator. Then the smaller one is connected to form the stopping behavior. The filtering characteristic of the optical signal is presented within a ring resonator, where the suitable parameters can be controlled to obtain the required output energy. To maintain the soliton pulse propagating within the ring resonator, the suitable coupling power into the device is required, whereas the interference signal is a minor effect compared to the loss associated with the direct passing through. We are looking for a stationary soliton pulse, which is introduced into the multi-stage micro ring resonators as shown in Fig. 1, the input optical field (Ein) is given by T z ð1Þ exp io0 t ; Ein ¼ A sech T0 2LD where A and z are the optical field amplitude and propagation distance, respectively. T is a soliton pulse propagation time in a frame moving at the group velocity T= t b1z, where b1 and b2 are the coefficients of the linear and second order terms of Taylor expansion of the propagation constant, respectively. LD ¼ T02 =jb2 j is the dispersion length of the soliton pulse. T0 in equation is a soliton pulse propagation time at initial input, where t is the soliton phase shift time, and the frequency shift of the soliton is o0. This solution describes a pulse that keeps its temporal width invariance as it propagates, and thus is called a temporal soliton. When a soliton peak intensity ðjb2 =GT02 jÞ is given, then T0 is known. For the soliton pulse in the micro ring device, a balance should be achieved between the dispersion length (LD) and the nonlinear length (LNL = (1/GfNL), where G = n2k0 is the length scale over which dispersive or nonlinear effects make the beam becomes wider or narrower. For a soliton pulse, there is a balance between dispersion and nonlinear lengths; hence LD = LNL. When light propagates within the nonlinear material (medium), the refractive index (n) of light within the medium is given by n2 n ¼ n0 þn2 I ¼ n0 þ P; ð2Þ Aeff where n0 and n2 are the linear and nonlinear refractive indexes, respectively. I and P are the optical intensity and optical power, respectively. The effective mode core area of the device is given by Aeff. For the micro ring and nano ring resonators, the effective mode core areas range from 0.50 to 0.1 mm2. When a soliton pulse is input and propagated within a micro ring resonator as shown in Fig. 1, which consists of a series micro ring resonators, the resonant output is formed; thus, the normalized output of the light field is the ratio between the output and input fields (Eout(t) and Ein(t)) in each roundtrip, which can be
Fig. 1. A schematic of multi-soliton pulse generation system, where Rs: ring radii, ks: coupling coefficients.
1977
expressed as 2 Eout ðtÞ2 ¼ ð1 gÞ41 E ðtÞ in
3 ð1 ð1 gÞ=x2 Þk 5: pffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffi 2 pffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffi 2 ð1 x= 1 g 1 kÞ þ4x 1 g 1 k sin f2
ð3Þ The closed form of Eq. (3) indicates that a ring resonator in the particular case is very similar to a Fabry–Perot cavity, which has an input and output mirror with a field reflectivity (1 k) and a fully reflecting mirror. k is the coupling coefficient, and x= exp( aL/2) represents a roundtrip loss coefficient, f0 =kLn0 and fNL =kLn2jEinj2 are the linear and nonlinear phase shifts, respectively, and k= 2p/l is the wave propagation number in a vacuum, where L and a are a waveguide length and linear absorption coefficient, respectively. In this work, the iterative method is introduced to obtain the results as shown in Eq. (3), similar to when the output field is connected and input into the other ring resonators. After the large bandwidth signals are generated within the first micro ring device, then the filtering device is required to tune the required signals. In order to tune the signals from large bandwidth, we propose using the add/drop device using the appropriate parameters. This is given in details as follows. The optical circuit of the add/drop filters can be given by Eqs. (4) and (5), where b = kneff is the propagation constant, neff the effective refractive index of the waveguide and the circumference of the ring is L= 2pR, where R is the radius of the ring: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a 2 Et k2 e2L cosðkn LÞ þð1 k2 ÞeaL ¼ ð1 k1 Þ 2 1 k1 1 p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a ; E 1 þ ð1 k1 Þð1 k2 ÞeaL 2 1 k1 1 k2 e2L cosðkn LÞ in ð4Þ where Eqs. (5) and (4) from the add/drop equation as 2 a Ed k1 k2 e2L ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a ; E a L 1 þð1 k1 Þð1 k2 Þe 2 1 k1 1 k2 e2L cosðkn LÞ in ð5Þ where Et and Ed are the throughput and drop port signals, respectively, and f = bL is the phase constant. The required tunable signals can be managed using the specific parameters of the add/drop device. When the two polarized pumping pulses p1 and p2 are coupled into a micro ring resonator as shown in Fig. 2, whereas polarized photons are randomly coupled and converted between horizontal|H4 and vertical |V4 components, we can obtain the following polarization entangled state as in Eq. (6), where more details have been described by [6]: jFS ¼ j2; HSs j2; HSi þ exp½iðfs þ fi Þj2; VSs j2; VSi :
ð6Þ
In the expression jk; HS; k is the number of time slots (1 or 2), and the subscript identifies whether the state is the signal (s) or the idler (i) state. We assume that the response time of the Kerr effect is much less than the cavity roundtrip time. Because of the
Fig. 2. A schematic of an entangled photon pair generation via a nonlinear four wave mixing (FWM) type in a ring resonator. Bell’s state propagates to a rotatable polarizer and then is split by a beam splitter (PBS) flying to two detectors (D1 and D2).
1978
T. Tanengtang et al. / Optik 121 (2010) 1976–1980
Kerr nonlinearity of the optical device, the strong pulses acquire an intensity-dependent phase shift during propagation. The interference of light pulses at a coupler introduces the output beam, which is entangled. Because the polarization states of light pulses are changed and converted while circulating in the delay circuit, where the polarization entangled photon pairs can be generated, the entangled photons of the nonlinear ring resonator are separated to be the signal and idler photon probability. The polarization angle adjustment device is applied to investigate the orientation and optical output intensity.
3. Quantum bits transmission In operation, the large bandwidth signals can be generated within the micro ring device using a soliton pulse input into the nonlinear micro ring resonator. The schematic diagram of the proposed system is as shown in Fig. 1. A soliton pulse with 20 ns pulse width, peak power at 500 mW is input into the system. The suitable ring parameters are used, for instance, ring radii R1 = 10 mm, R2 = 5 mm, and Rd = 200 mm. In order to associate the system with the practical device [12], the selected parameters of the system are fixed to l0 =1.55 mm, n0 = 3.34 (InGaAsP/InP), Aeff =0.25 mm2 for all micro ring resonators, a =0.5 dB mm 1, g = 0.1. The coupling coefficient (kappa, k) of the micro ring resonator ranged from 0.50 to 0.975. The nonlinear refractive index is n2 = 2.2 10 17 m2/W. In this case, the wave guided loss used is 0.5 dB mm 1. The signal is chopped (sliced) into a smaller signal spread over the spectrum as shown in Fig. 3(b), which shows that the large bandwidth (continuous variable) signal is formed within the first ring device. The compress bandwidth with smaller group velocity is obtained within the ring R2, i.e. filtering signals. The localized soliton concept is formed when the resonant soliton pulses are seen as shown in Fig. 3(d). We have found that the tuned light pulse can be obtained by connecting the add/drop device into the system (i.e. R2). However, there are two types of soliton pulses, i.e. temporal (time) and spatial 9 (wavelength) solitons. In application, the quantum dense coding (different wavelengths) can be performed as shown in Figs. 2 and 3. For example, one of the selected wavelengths is the second harmonic one, which is selected using the specific ring parameters (R3) and detected by the different time slot at the ends (users).This means that the dense wavelengths generation allows the packet
Fig. 3. Results of the multi-soliton pulse generation, where (a) input soliton, (b) large bandwidth signals, (c) temporal soliton, (d) spatial soliton. The free spectrum range is 600 pm, and the FWHM of the pulse is 10 pm.
Fig. 4. The chaotic signals generated by the micro ring resonators: (a) a continuous spectrum, (b, c) the filtering signals.
quantum key to be performed. One pair of the second harmonic signals, i.e. signal and idler, is shown in Fig. 4, where the output entangled photon pair can be formed by the polarization control unit. The initial quantum key is formed by Alice at time t1. The required codes are retrieved by Bob when the secret codes are given by Alice at different time (t2, t3, t4, y, tn). In general, the quantum dense coding can be provided using the packet of entangled photon generation, where each pair of the entangled photon is detected, which is recognized by the time difference, i.e. different time slot. The key advantage of the proposed system is the multi-soliton pulse generation, which is available for highcapacity and secure switching called quantum packet switching. Since a soliton communication has been recognized as a good candidate for long-distance communication, the increase in more soliton channels (wavelengths) is interesting. The increase in communication capacity is obtained by using more available channels and large bandwidth. After this multi-soliton pulses are generated and then transmitted into the optical gate, i.e. PBS. The multi-entangled photon pairs, i.e. correlated photons, are introduced and the entangled photons formed by the increase in soliton pulses (li), which is formed by spatial soliton. Finally, the high-capacity and secured signals via quantum packet switching can be transmitted and retrieved by the required users in the quantum network. The required signals can be filtered and retrieved, and the quantum codes can be formed between Alice and Bob. To understand the application of these ideas to the quantum regime, it is best to start with a simple example. Thus, suppose we have a collection of photons, each of which is subject independently to random ‘states’ jHS or jVS, but which otherwise is stable. Whenever such a change occurs, the relevant two-state system may become entangled with its environment. In order to stabilize a single qubit, in the general state A|H4 + B|V4, we express it by means of three two-state systems, with the ‘encoding’ j0S ¼ jHHHS; j1S ¼ jVVVS. Thus the total initial state of the three photons is AjHHHS þ BjVVV S. After a period of time, during which random flips may occur, the three-photon system is measured twice. The first measurement is a projection onto the two-state basis: jHHHS þ jVVVS þjHHVS þ jVVHS; jVVHS þ jHVHS þ jVHVS þjVHHS:
T. Tanengtang et al. / Optik 121 (2010) 1976–1980
Fig. 5. Schematic of a large bandwidth signal generation system (a), and (b) a schematic of a quantum router and network system, where Rs: ring radii and ks, ks1, ks2 are the coupling coefficients.
The second measurement is a projection onto the two-state basis: jHHHS þjVVV Sþ jHVHS þjVHVS; jVVHSþ jVVHS þ jVHHS þjHVVS: Each measurement has two possible results, which we will call 0 and 1. Depending on which results R are obtained, an appropriate action is carried out: if R=00, do nothing; if R= 01, convert the right photon; if R= 10, convert the middle photon; if R= 11, convert the left photon. If, during the time interval when the system was left to evolve freely, no more than one photon coupled, then this procedure will return the three-photon state to AjHHHSþ BjVVVS. It is remarkable that this can be done without gaining information about the values of A and B and thus disturbing the stored quantum information. During the correction procedure, the entanglement between the system and its environment is transferred to an entanglement between the measuring apparatus and the environment. The qubit is actively isolated from its environment by means of this carefully controlled entanglement transfer. In general, the quantum dense coding/packet switching of the photons within the correlation bandwidth can be performed in a similar manner.
4. Fidelity Generally, the criterion for quantum network performance is the channel capacity, which is restricted by the quantum fidelity, especially when the large bandwidth is employed in the system (i.e. continuous variable). In this proposed system, we form the continuous wavelength that is available for continuous variable QKD; therefore, the fidelity causes the network performance degradation. In principle, fidelity occurred when the channel spacing is narrower than the signal bandwidth. From Fig. 5, the network fidelity of the continuous wavelength router and quantum network is well described in Ref. [13]. Fidelity is a popular measure of distance between density operators. It is not a metric, but has some useful properties. There are various ways of introducing a notion of distance between two quantum states. In general, a distance measure quantifies the extent to which two
1979
quantum states behave in the same way. While these distance measures are usually given by certain mathematical expressions, they often possess a simple operational meaning, i.e. they are related to the problem of distinguishing two systems. This is why the distance measures in quantum information are sometimes referred to as distinguishing ability measures. A density matrix, or density operator, is used in quantum theory to describe the statistical state of a quantum system. It is the quantummechanical analogue to a phase-space density (probability distribution of position and momentum) in classical statistical mechanics. The need for a statistical description via density matrices arises because it is not possible to describe a quantum-mechanical system that undergoes general quantum operation such as measurement, using exclusive states represented by key vectors. In general a system is said to be in a mixed state, except in the case where the state is not reducible to a convex combination of other statistical states. In that case it is said to be in a pure state. Fidelity as a distance measure between pure states used to be called ‘transition probability’. For two states given by unit vectors j, c it is j/f; cSj. For a pure state (vector c) and a mixed state (density matrix r) this generalizes to /c; rcS, and for two density matrices r, s it is generalized as the largest fidelity between any two purifications of the given states. According to a theorem, this leads to the expression as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 pffiffiffiffiffiffiffipffiffiffiffi Fðr; sÞ ¼ tr rs r ;
ð7Þ
where the term fidelity appears to have been used first. However, one can also start from j/f; cSj, leading to the alternative form as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffipffiffiffiffi rs r: Fðr; sÞ ¼ tr
ð8Þ
pffiffiffi This second quantity is sometimes denoted as F and called square root fidelity. It has no interpretation as a probability, but appears in some estimates in a simpler way. This is equivalent to the density matrix being a one-dimensional projector, i.e. of the form r ¼ jcS/cj for some unit vector c. In this case the formula for the expectation value of an operator A in the state simplifies as trðrAÞ ¼ /c; AcS. Equivalently, a state r is pure if tr(r2) =1. If r ¼ jcS/cj is pure, then Fðr; sÞ ¼ /cjsjcS and if both states are pure i.e. r ¼ jcS/cj and s ¼ jfS/fj, then Fðr; sÞ ¼ j/cjfSj2 . Fidelity measures the difference between two quantum bit vectors. Because of quantum entanglement, each of the 2n combinations of bits in a vector of size n is physically separated states. For a given problem, one particular vector is considered a reference state that other vectors are compared against. For example, if we start with a bit vector of zeros [0000], and we send the bits through a noisy channel in which bit 3 is flipped with probability p, we would end up with a probabilistic vector of ((1 p) [0000] + p[0010]). The fidelity of the final state in relation to the starting (‘error free’) state is just (1 p). So, in the case of an operational state vs. a reference ‘correct’ state, the fidelity describes the amount of error introduced by the system on the operational state. Fidelity of 1 indicates that the system is definitely in the reference state, where fidelity of 0 indicates that the system has no overlap with the reference state. We characterize errors by calculating the fidelity of qubits traversing the various quantum channels and gates necessary to route and move bits around a communication network. We will combine models of the individual communication components so that we get an overall fidelity of communication as a function of distance and architecture [14,15].
1980
T. Tanengtang et al. / Optik 121 (2010) 1976–1980
5. Error corrections When photons are sent through a fiber as part of a quantum communication protocol, the error that is most difficult to correct is photon loss. Here, we propose and analyze a 2–4 qubit encoding scheme, which can recover the loss of one qubit in the transmission. When single photons are presented in all four channels, we can skip the correction process. Once the absence of the photon is found in one channel, a single photon is added in that channel. Whatever the quantum state of the single photon that is added, the correction process then will fix the error, since the quantum error-correction (QEC) [16,17] is for an arbitrary error. Including the single-photon quantum nondemolition (QND), the quantum circuit for the correction process is depicted in Fig. 5. Let us consider a two-qubit input state, for example, jcin S ¼ j01S. The code word is then jcen S ¼ pffiffi2ðj0110S1þ j1001SÞ. Suppose now that the last qubit is lost. The state of the system is given by the density operator r1 ¼ 12ðj011S/011jþ j100S/100jÞ, which is obtained from the initial state jcen S by tracing out the last qubit. For the sake of simplicity, let us consider the mixed state r1 represented as a probability distribution over the pure states, instead of a density matrix. Thus the mixed state after the photon loss can be written as
1 1 : ð9Þ r1 ¼ j011S; ; j100S; 2 2 The quantum nondemolition device that signals the loss of the last qubit is followed by a qubit state preparation device that substitutes the missing qubit with a new qubit in the ground state j0S the new density operator is then
1 1 : ð10Þ r2 ¼ j0110S; ; j1000S; 2 2 Including the two ancilla bits, the total system is in the mixed state
r3 ¼ r2 j00S/00j ¼
1 1 ; j1000Sj00S; : j0110Sj00S; 2 2
ð11Þ
After applying the transform on the ancilla bits, this becomes
1 1 r4 ¼ j00S þ j01S þ j10Sþ j11S; ; j1000S½j00Sþ j01S 2 2 1 g: ð12Þ þ j10S þ j11S; 2 This is followed by the transform on the ancilla bits, then yields the mixed state as
1 1 ðj0110Sþ j1001SÞj00S þðj0110S j1001SÞj10S; r5 ¼ 2 2 1 1 ½ðj1000Sþ j0111SÞj01Sþ ðj1000S j0111SÞj11S; : 2 2 ð13Þ
The measurement outcome of the two ancilla now determines the error-correcting operator on the last qubit. Note that after the measurement of the ancilla photons, the result is always a pure state. Furthermore, all the results are equally likely, so this process does not reveal any information about the original encoded state. Our scheme, on the other hand, utilizes quantum error correction to relay an unknown quantum state with high fidelity down a
quantum channel, and can be named as a ‘quantum transponder’ or a ‘quantum relay’. For quantum key distribution schemes such as BB84, only a transponder is required for long-distance key transfer. Furthermore, if the fidelity of the transponder is sufficiently high, one can also use it to distribute entanglement by relaying, say, one half of an entangled pair. The entangled photons can perform the qubits, where the error corrections can be formed similar to the case as discussed in Section 3.
6. Conclusion We have proposed the very interesting concept where the packet quantum key distribution can be performed using a remarkably simple system. The system consists of a series of nonlinear micro ring devices, where the key advantage of the system is that the quantum dense coding, i.e. quantum packet switching, can be formed within the single system. The optical power in the system is generated using a soliton pulse within the nonlinear Kerr type micro ring devices. Therefore, the remaining optical power is able to perform the link due to the soliton behavior. The quantum bits transmission failure in the quantum network known as quantum fidelity and error corrections are also analyzed and discussed.
References [1] T. Zhang, X.F. Mo, Z.F. Han, G.C. Guo, Extensible router for a quantum key distribution network, Phys. Lett. A 372 (2008) 3957. [2] B.S. Ham, A novel method of all-optical switching: quantum router, ETRI J. 23 (2001) 106. [3] T. Zhang, Z.F. Han, X.F. Mo, G.C. Guo, Extensible router for multi-user quantum key distribution network, quant-ph/0608238vl, 2006. [4] P.P. Yupapin, S. Mitatha, Multi-users quantum key distribution via wavelength routers in an optical network, Recent Pat. Comput. Sci. 2 (1) (2009) 14. [5] 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. [6] S. Suchat, W. Khannam, P.P. Yupapin, Quantum key distribution via an optical wireless communication link for telephone networks, Opt. Eng. 46 (2007) 100502-1. [7] 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. [8] 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. Photonics 1 (2007) 343. [9] 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. [10] 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. [11] B. Qi, Single-photon continuous-variable quantum key distribution based on the energy–time uncertainty relation, Opt. Lett. 31 (2006) 2795. [12] P.P. Yupapin, N. Pornsuwanchroen, S. Chaiyasoonthorn, Attosecond pulse generation using nonlinear micro ring resonators, Microwave Opt. Technol. Lett. 50 (2008) 3108. [13] L. Molnar, Fidelity preserving map on density operators, arXiv: math / 0108060v2, 2002. [14] H. Hubel1, M.R. Vanner, T. Lederer, B. Blauensteiner, T. Lorunser, A. Poppe1, A. Zeilinger, High-fidelity transmission of polarization encoded qubits from an entangled source over 100 km of fiber, Opt. Exp. 15 (2007) 7853. [15] K. Tsujino, S. Takeuchi, K. Sasaki, Detailed analysis of the fidelity of quantum teleportation using photons: considering real experimental parameters, Phys. Rev. A 66 (2002) 042314. [16] D. Kalamidas, Single-photon quantum error rejection and correction with linear optics, Phys. Lett. A 343 (2005) 331. [17] R.M. Gingrich, P. Kok, H. Lee, F. Vatan, J.P. Dowling, An all linear optical quantum memory based on quantum error correction, Phys. Rev. Lett. 91 (2003) 217901.
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Quantum bits generation using correlated photons: Fidelity and error corrections T. Tanengtang a,b, K. Paithoonwattanakij a, P.P. Yupapin b,, S. Suchat c a
Department of Electrical Engineering, 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 c Faculty of Science and Technology, Phranakhon Rajabhat University, Bangkok 10220, Thailand b
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 9 May 2009
We propose a new system of quantum bits generation using a soliton pulse within a micro ring resonator. A quantum gate can be formed using a polarization control unit incorporating into the system. The random signal and idler pairs can be formed within the photon correlation bandwidth, which can be generated and randomly formed the packet quantum bits, i.e. quantum packet codes. Each random code (logic) can be performed by combining the signal and idler of each entangled photon pair via the quantum gate. Results obtained have shown that the packet of quantum logic bits can be generated using the entangled photon pairs generated by the proposed system. The quantum bits transmission fidelity and error corrections are also described. & 2009 Elsevier GmbH. All rights reserved.
Keywords: Quantum bits Error corrections Photon correlation Fidelity Entanglement
1. Introduction Qubit is introduced in order to emphasize that the aim is to stabilize a complete quantum state, not just a chosen observable. Also, we are concerned with the properties of the quantum state, not with the physical system expressing it. For example, a single qubit may be expressed by a system whose Hilbert space has more than two dimensions. Among the possible changes such a system may undergo, some will affect the stored single qubit of quantum information, but others will not. Both quantum communication and quantum information processing have been shown to be fundamentally different from its classical counterpart. Examples where this difference is highlighted are secure key distribution for cryptography and the existence of fast algorithms for an idealized quantum computer. Quantum network has also been introduced and is becoming the promising technology that can be used to fulfill the perfect network security. Some research works have been reported in various forms of applications [1,2]. The use of quantum key distribution via optical network has been reported [3,4]. To date, quantum key distribution (QKD) is the only form of information that can provide the perfect communication security. The use of QKD has been proposed in many research works, whereas the applications in different forms – such as point to point link [5], optical wireless [6], satellite [7], long distance [8] and network [9] – have been reported. However, a more reliable system for network security is needed, which has both high capacity and is secure. The concept of continuous variable in the Corresponding author. Fax: + 66 2 3264354.
E-mail address: [email protected] (P.P. Yupapin). 0030-4026/$ - see front matter & 2009 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2009.05.032
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 available for a large demand. Some works have proposed the use of continuous variable QKD with quantum router and network [10,11]. However, the requirement of large bandwidth signal and dense wavelength multiplexing has become practical problems. Yupapin et al. [12] have also shown that the continuous wavelength can be generated using a soliton pulse in a micro ring resonator, which can be used to overcome such problems. In this paper, we show that the large bandwidth of soliton pulse is generated and filtered within the micro ring resonator system. The selected (tuned) pulse is coherently localized and stored within the ring device. In applications, we can use the tuned soliton pulses to perform the required applications such as dense wavelength division multiplexing (DWDM) based soliton communication, optical and quantum memory, and multi-soliton sources. We also propose a new concept of quantum gate where the packet of quantum of logic bits can be formed using a simple scheme. By using a soliton pulse traveling within nonlinear micro and ring resonators, large bandwidth signals occurred and dense wavelengths were generated, which were available for quantum dense coding, i.e. quantum packet switching application. The quantum bits transmission fidelity and error corrections are also analyzed.
2. Multi-spatial solitons An optical soliton is recognized as a powerful laser pulse that can be used to enlarge the optical bandwidth when propagating
T. Tanengtang et al. / Optik 121 (2010) 1976–1980
within the nonlinear micro ring resonator [12]. Moreover, the superposition of self-phase modulation (SPM) soliton pulses can keep the large output power. Initially, the optimum energy is coupled into the waveguide by a lager effective core area device, i.e. ring resonator. Then the smaller one is connected to form the stopping behavior. The filtering characteristic of the optical signal is presented within a ring resonator, where the suitable parameters can be controlled to obtain the required output energy. To maintain the soliton pulse propagating within the ring resonator, the suitable coupling power into the device is required, whereas the interference signal is a minor effect compared to the loss associated with the direct passing through. We are looking for a stationary soliton pulse, which is introduced into the multi-stage micro ring resonators as shown in Fig. 1, the input optical field (Ein) is given by T z ð1Þ exp io0 t ; Ein ¼ A sech T0 2LD where A and z are the optical field amplitude and propagation distance, respectively. T is a soliton pulse propagation time in a frame moving at the group velocity T= t b1z, where b1 and b2 are the coefficients of the linear and second order terms of Taylor expansion of the propagation constant, respectively. LD ¼ T02 =jb2 j is the dispersion length of the soliton pulse. T0 in equation is a soliton pulse propagation time at initial input, where t is the soliton phase shift time, and the frequency shift of the soliton is o0. This solution describes a pulse that keeps its temporal width invariance as it propagates, and thus is called a temporal soliton. When a soliton peak intensity ðjb2 =GT02 jÞ is given, then T0 is known. For the soliton pulse in the micro ring device, a balance should be achieved between the dispersion length (LD) and the nonlinear length (LNL = (1/GfNL), where G = n2k0 is the length scale over which dispersive or nonlinear effects make the beam becomes wider or narrower. For a soliton pulse, there is a balance between dispersion and nonlinear lengths; hence LD = LNL. When light propagates within the nonlinear material (medium), the refractive index (n) of light within the medium is given by n2 n ¼ n0 þn2 I ¼ n0 þ P; ð2Þ Aeff where n0 and n2 are the linear and nonlinear refractive indexes, respectively. I and P are the optical intensity and optical power, respectively. The effective mode core area of the device is given by Aeff. For the micro ring and nano ring resonators, the effective mode core areas range from 0.50 to 0.1 mm2. When a soliton pulse is input and propagated within a micro ring resonator as shown in Fig. 1, which consists of a series micro ring resonators, the resonant output is formed; thus, the normalized output of the light field is the ratio between the output and input fields (Eout(t) and Ein(t)) in each roundtrip, which can be
Fig. 1. A schematic of multi-soliton pulse generation system, where Rs: ring radii, ks: coupling coefficients.
1977
expressed as 2 Eout ðtÞ2 ¼ ð1 gÞ41 E ðtÞ in
3 ð1 ð1 gÞ=x2 Þk 5: pffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffi 2 pffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffi 2 ð1 x= 1 g 1 kÞ þ4x 1 g 1 k sin f2
ð3Þ The closed form of Eq. (3) indicates that a ring resonator in the particular case is very similar to a Fabry–Perot cavity, which has an input and output mirror with a field reflectivity (1 k) and a fully reflecting mirror. k is the coupling coefficient, and x= exp( aL/2) represents a roundtrip loss coefficient, f0 =kLn0 and fNL =kLn2jEinj2 are the linear and nonlinear phase shifts, respectively, and k= 2p/l is the wave propagation number in a vacuum, where L and a are a waveguide length and linear absorption coefficient, respectively. In this work, the iterative method is introduced to obtain the results as shown in Eq. (3), similar to when the output field is connected and input into the other ring resonators. After the large bandwidth signals are generated within the first micro ring device, then the filtering device is required to tune the required signals. In order to tune the signals from large bandwidth, we propose using the add/drop device using the appropriate parameters. This is given in details as follows. The optical circuit of the add/drop filters can be given by Eqs. (4) and (5), where b = kneff is the propagation constant, neff the effective refractive index of the waveguide and the circumference of the ring is L= 2pR, where R is the radius of the ring: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a 2 Et k2 e2L cosðkn LÞ þð1 k2 ÞeaL ¼ ð1 k1 Þ 2 1 k1 1 p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a ; E 1 þ ð1 k1 Þð1 k2 ÞeaL 2 1 k1 1 k2 e2L cosðkn LÞ in ð4Þ where Eqs. (5) and (4) from the add/drop equation as 2 a Ed k1 k2 e2L ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a ; E a L 1 þð1 k1 Þð1 k2 Þe 2 1 k1 1 k2 e2L cosðkn LÞ in ð5Þ where Et and Ed are the throughput and drop port signals, respectively, and f = bL is the phase constant. The required tunable signals can be managed using the specific parameters of the add/drop device. When the two polarized pumping pulses p1 and p2 are coupled into a micro ring resonator as shown in Fig. 2, whereas polarized photons are randomly coupled and converted between horizontal|H4 and vertical |V4 components, we can obtain the following polarization entangled state as in Eq. (6), where more details have been described by [6]: jFS ¼ j2; HSs j2; HSi þ exp½iðfs þ fi Þj2; VSs j2; VSi :
ð6Þ
In the expression jk; HS; k is the number of time slots (1 or 2), and the subscript identifies whether the state is the signal (s) or the idler (i) state. We assume that the response time of the Kerr effect is much less than the cavity roundtrip time. Because of the
Fig. 2. A schematic of an entangled photon pair generation via a nonlinear four wave mixing (FWM) type in a ring resonator. Bell’s state propagates to a rotatable polarizer and then is split by a beam splitter (PBS) flying to two detectors (D1 and D2).
1978
T. Tanengtang et al. / Optik 121 (2010) 1976–1980
Kerr nonlinearity of the optical device, the strong pulses acquire an intensity-dependent phase shift during propagation. The interference of light pulses at a coupler introduces the output beam, which is entangled. Because the polarization states of light pulses are changed and converted while circulating in the delay circuit, where the polarization entangled photon pairs can be generated, the entangled photons of the nonlinear ring resonator are separated to be the signal and idler photon probability. The polarization angle adjustment device is applied to investigate the orientation and optical output intensity.
3. Quantum bits transmission In operation, the large bandwidth signals can be generated within the micro ring device using a soliton pulse input into the nonlinear micro ring resonator. The schematic diagram of the proposed system is as shown in Fig. 1. A soliton pulse with 20 ns pulse width, peak power at 500 mW is input into the system. The suitable ring parameters are used, for instance, ring radii R1 = 10 mm, R2 = 5 mm, and Rd = 200 mm. In order to associate the system with the practical device [12], the selected parameters of the system are fixed to l0 =1.55 mm, n0 = 3.34 (InGaAsP/InP), Aeff =0.25 mm2 for all micro ring resonators, a =0.5 dB mm 1, g = 0.1. The coupling coefficient (kappa, k) of the micro ring resonator ranged from 0.50 to 0.975. The nonlinear refractive index is n2 = 2.2 10 17 m2/W. In this case, the wave guided loss used is 0.5 dB mm 1. The signal is chopped (sliced) into a smaller signal spread over the spectrum as shown in Fig. 3(b), which shows that the large bandwidth (continuous variable) signal is formed within the first ring device. The compress bandwidth with smaller group velocity is obtained within the ring R2, i.e. filtering signals. The localized soliton concept is formed when the resonant soliton pulses are seen as shown in Fig. 3(d). We have found that the tuned light pulse can be obtained by connecting the add/drop device into the system (i.e. R2). However, there are two types of soliton pulses, i.e. temporal (time) and spatial 9 (wavelength) solitons. In application, the quantum dense coding (different wavelengths) can be performed as shown in Figs. 2 and 3. For example, one of the selected wavelengths is the second harmonic one, which is selected using the specific ring parameters (R3) and detected by the different time slot at the ends (users).This means that the dense wavelengths generation allows the packet
Fig. 3. Results of the multi-soliton pulse generation, where (a) input soliton, (b) large bandwidth signals, (c) temporal soliton, (d) spatial soliton. The free spectrum range is 600 pm, and the FWHM of the pulse is 10 pm.
Fig. 4. The chaotic signals generated by the micro ring resonators: (a) a continuous spectrum, (b, c) the filtering signals.
quantum key to be performed. One pair of the second harmonic signals, i.e. signal and idler, is shown in Fig. 4, where the output entangled photon pair can be formed by the polarization control unit. The initial quantum key is formed by Alice at time t1. The required codes are retrieved by Bob when the secret codes are given by Alice at different time (t2, t3, t4, y, tn). In general, the quantum dense coding can be provided using the packet of entangled photon generation, where each pair of the entangled photon is detected, which is recognized by the time difference, i.e. different time slot. The key advantage of the proposed system is the multi-soliton pulse generation, which is available for highcapacity and secure switching called quantum packet switching. Since a soliton communication has been recognized as a good candidate for long-distance communication, the increase in more soliton channels (wavelengths) is interesting. The increase in communication capacity is obtained by using more available channels and large bandwidth. After this multi-soliton pulses are generated and then transmitted into the optical gate, i.e. PBS. The multi-entangled photon pairs, i.e. correlated photons, are introduced and the entangled photons formed by the increase in soliton pulses (li), which is formed by spatial soliton. Finally, the high-capacity and secured signals via quantum packet switching can be transmitted and retrieved by the required users in the quantum network. The required signals can be filtered and retrieved, and the quantum codes can be formed between Alice and Bob. To understand the application of these ideas to the quantum regime, it is best to start with a simple example. Thus, suppose we have a collection of photons, each of which is subject independently to random ‘states’ jHS or jVS, but which otherwise is stable. Whenever such a change occurs, the relevant two-state system may become entangled with its environment. In order to stabilize a single qubit, in the general state A|H4 + B|V4, we express it by means of three two-state systems, with the ‘encoding’ j0S ¼ jHHHS; j1S ¼ jVVVS. Thus the total initial state of the three photons is AjHHHS þ BjVVV S. After a period of time, during which random flips may occur, the three-photon system is measured twice. The first measurement is a projection onto the two-state basis: jHHHS þ jVVVS þjHHVS þ jVVHS; jVVHS þ jHVHS þ jVHVS þjVHHS:
T. Tanengtang et al. / Optik 121 (2010) 1976–1980
Fig. 5. Schematic of a large bandwidth signal generation system (a), and (b) a schematic of a quantum router and network system, where Rs: ring radii and ks, ks1, ks2 are the coupling coefficients.
The second measurement is a projection onto the two-state basis: jHHHS þjVVV Sþ jHVHS þjVHVS; jVVHSþ jVVHS þ jVHHS þjHVVS: Each measurement has two possible results, which we will call 0 and 1. Depending on which results R are obtained, an appropriate action is carried out: if R=00, do nothing; if R= 01, convert the right photon; if R= 10, convert the middle photon; if R= 11, convert the left photon. If, during the time interval when the system was left to evolve freely, no more than one photon coupled, then this procedure will return the three-photon state to AjHHHSþ BjVVVS. It is remarkable that this can be done without gaining information about the values of A and B and thus disturbing the stored quantum information. During the correction procedure, the entanglement between the system and its environment is transferred to an entanglement between the measuring apparatus and the environment. The qubit is actively isolated from its environment by means of this carefully controlled entanglement transfer. In general, the quantum dense coding/packet switching of the photons within the correlation bandwidth can be performed in a similar manner.
4. Fidelity Generally, the criterion for quantum network performance is the channel capacity, which is restricted by the quantum fidelity, especially when the large bandwidth is employed in the system (i.e. continuous variable). In this proposed system, we form the continuous wavelength that is available for continuous variable QKD; therefore, the fidelity causes the network performance degradation. In principle, fidelity occurred when the channel spacing is narrower than the signal bandwidth. From Fig. 5, the network fidelity of the continuous wavelength router and quantum network is well described in Ref. [13]. Fidelity is a popular measure of distance between density operators. It is not a metric, but has some useful properties. There are various ways of introducing a notion of distance between two quantum states. In general, a distance measure quantifies the extent to which two
1979
quantum states behave in the same way. While these distance measures are usually given by certain mathematical expressions, they often possess a simple operational meaning, i.e. they are related to the problem of distinguishing two systems. This is why the distance measures in quantum information are sometimes referred to as distinguishing ability measures. A density matrix, or density operator, is used in quantum theory to describe the statistical state of a quantum system. It is the quantummechanical analogue to a phase-space density (probability distribution of position and momentum) in classical statistical mechanics. The need for a statistical description via density matrices arises because it is not possible to describe a quantum-mechanical system that undergoes general quantum operation such as measurement, using exclusive states represented by key vectors. In general a system is said to be in a mixed state, except in the case where the state is not reducible to a convex combination of other statistical states. In that case it is said to be in a pure state. Fidelity as a distance measure between pure states used to be called ‘transition probability’. For two states given by unit vectors j, c it is j/f; cSj. For a pure state (vector c) and a mixed state (density matrix r) this generalizes to /c; rcS, and for two density matrices r, s it is generalized as the largest fidelity between any two purifications of the given states. According to a theorem, this leads to the expression as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 pffiffiffiffiffiffiffipffiffiffiffi Fðr; sÞ ¼ tr rs r ;
ð7Þ
where the term fidelity appears to have been used first. However, one can also start from j/f; cSj, leading to the alternative form as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffipffiffiffiffi rs r: Fðr; sÞ ¼ tr
ð8Þ
pffiffiffi This second quantity is sometimes denoted as F and called square root fidelity. It has no interpretation as a probability, but appears in some estimates in a simpler way. This is equivalent to the density matrix being a one-dimensional projector, i.e. of the form r ¼ jcS/cj for some unit vector c. In this case the formula for the expectation value of an operator A in the state simplifies as trðrAÞ ¼ /c; AcS. Equivalently, a state r is pure if tr(r2) =1. If r ¼ jcS/cj is pure, then Fðr; sÞ ¼ /cjsjcS and if both states are pure i.e. r ¼ jcS/cj and s ¼ jfS/fj, then Fðr; sÞ ¼ j/cjfSj2 . Fidelity measures the difference between two quantum bit vectors. Because of quantum entanglement, each of the 2n combinations of bits in a vector of size n is physically separated states. For a given problem, one particular vector is considered a reference state that other vectors are compared against. For example, if we start with a bit vector of zeros [0000], and we send the bits through a noisy channel in which bit 3 is flipped with probability p, we would end up with a probabilistic vector of ((1 p) [0000] + p[0010]). The fidelity of the final state in relation to the starting (‘error free’) state is just (1 p). So, in the case of an operational state vs. a reference ‘correct’ state, the fidelity describes the amount of error introduced by the system on the operational state. Fidelity of 1 indicates that the system is definitely in the reference state, where fidelity of 0 indicates that the system has no overlap with the reference state. We characterize errors by calculating the fidelity of qubits traversing the various quantum channels and gates necessary to route and move bits around a communication network. We will combine models of the individual communication components so that we get an overall fidelity of communication as a function of distance and architecture [14,15].
1980
T. Tanengtang et al. / Optik 121 (2010) 1976–1980
5. Error corrections When photons are sent through a fiber as part of a quantum communication protocol, the error that is most difficult to correct is photon loss. Here, we propose and analyze a 2–4 qubit encoding scheme, which can recover the loss of one qubit in the transmission. When single photons are presented in all four channels, we can skip the correction process. Once the absence of the photon is found in one channel, a single photon is added in that channel. Whatever the quantum state of the single photon that is added, the correction process then will fix the error, since the quantum error-correction (QEC) [16,17] is for an arbitrary error. Including the single-photon quantum nondemolition (QND), the quantum circuit for the correction process is depicted in Fig. 5. Let us consider a two-qubit input state, for example, jcin S ¼ j01S. The code word is then jcen S ¼ pffiffi2ðj0110S1þ j1001SÞ. Suppose now that the last qubit is lost. The state of the system is given by the density operator r1 ¼ 12ðj011S/011jþ j100S/100jÞ, which is obtained from the initial state jcen S by tracing out the last qubit. For the sake of simplicity, let us consider the mixed state r1 represented as a probability distribution over the pure states, instead of a density matrix. Thus the mixed state after the photon loss can be written as
1 1 : ð9Þ r1 ¼ j011S; ; j100S; 2 2 The quantum nondemolition device that signals the loss of the last qubit is followed by a qubit state preparation device that substitutes the missing qubit with a new qubit in the ground state j0S the new density operator is then
1 1 : ð10Þ r2 ¼ j0110S; ; j1000S; 2 2 Including the two ancilla bits, the total system is in the mixed state
r3 ¼ r2 j00S/00j ¼
1 1 ; j1000Sj00S; : j0110Sj00S; 2 2
ð11Þ
After applying the transform on the ancilla bits, this becomes
1 1 r4 ¼ j00S þ j01S þ j10Sþ j11S; ; j1000S½j00Sþ j01S 2 2 1 g: ð12Þ þ j10S þ j11S; 2 This is followed by the transform on the ancilla bits, then yields the mixed state as
1 1 ðj0110Sþ j1001SÞj00S þðj0110S j1001SÞj10S; r5 ¼ 2 2 1 1 ½ðj1000Sþ j0111SÞj01Sþ ðj1000S j0111SÞj11S; : 2 2 ð13Þ
The measurement outcome of the two ancilla now determines the error-correcting operator on the last qubit. Note that after the measurement of the ancilla photons, the result is always a pure state. Furthermore, all the results are equally likely, so this process does not reveal any information about the original encoded state. Our scheme, on the other hand, utilizes quantum error correction to relay an unknown quantum state with high fidelity down a
quantum channel, and can be named as a ‘quantum transponder’ or a ‘quantum relay’. For quantum key distribution schemes such as BB84, only a transponder is required for long-distance key transfer. Furthermore, if the fidelity of the transponder is sufficiently high, one can also use it to distribute entanglement by relaying, say, one half of an entangled pair. The entangled photons can perform the qubits, where the error corrections can be formed similar to the case as discussed in Section 3.
6. Conclusion We have proposed the very interesting concept where the packet quantum key distribution can be performed using a remarkably simple system. The system consists of a series of nonlinear micro ring devices, where the key advantage of the system is that the quantum dense coding, i.e. quantum packet switching, can be formed within the single system. The optical power in the system is generated using a soliton pulse within the nonlinear Kerr type micro ring devices. Therefore, the remaining optical power is able to perform the link due to the soliton behavior. The quantum bits transmission failure in the quantum network known as quantum fidelity and error corrections are also analyzed and discussed.
References [1] T. Zhang, X.F. Mo, Z.F. Han, G.C. Guo, Extensible router for a quantum key distribution network, Phys. Lett. A 372 (2008) 3957. [2] B.S. Ham, A novel method of all-optical switching: quantum router, ETRI J. 23 (2001) 106. [3] T. Zhang, Z.F. Han, X.F. Mo, G.C. Guo, Extensible router for multi-user quantum key distribution network, quant-ph/0608238vl, 2006. [4] P.P. Yupapin, S. Mitatha, Multi-users quantum key distribution via wavelength routers in an optical network, Recent Pat. Comput. Sci. 2 (1) (2009) 14. [5] 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. [6] S. Suchat, W. Khannam, P.P. Yupapin, Quantum key distribution via an optical wireless communication link for telephone networks, Opt. Eng. 46 (2007) 100502-1. [7] 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. [8] 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. Photonics 1 (2007) 343. [9] 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. [10] 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. [11] B. Qi, Single-photon continuous-variable quantum key distribution based on the energy–time uncertainty relation, Opt. Lett. 31 (2006) 2795. [12] P.P. Yupapin, N. Pornsuwanchroen, S. Chaiyasoonthorn, Attosecond pulse generation using nonlinear micro ring resonators, Microwave Opt. Technol. Lett. 50 (2008) 3108. [13] L. Molnar, Fidelity preserving map on density operators, arXiv: math / 0108060v2, 2002. [14] H. Hubel1, M.R. Vanner, T. Lederer, B. Blauensteiner, T. Lorunser, A. Poppe1, A. Zeilinger, High-fidelity transmission of polarization encoded qubits from an entangled source over 100 km of fiber, Opt. Exp. 15 (2007) 7853. [15] K. Tsujino, S. Takeuchi, K. Sasaki, Detailed analysis of the fidelity of quantum teleportation using photons: considering real experimental parameters, Phys. Rev. A 66 (2002) 042314. [16] D. Kalamidas, Single-photon quantum error rejection and correction with linear optics, Phys. Lett. A 343 (2005) 331. [17] R.M. Gingrich, P. Kok, H. Lee, F. Vatan, J.P. Dowling, An all linear optical quantum memory based on quantum error correction, Phys. Rev. Lett. 91 (2003) 217901.