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Physical Feasibility of QKD Based on Probabilistic Quantum Circuits
ISSN 1751-8644 doi: 0000000000 www.ietdl.org
A.Gueddana1 V.Lakshminarayanan2 1
Green, Smart Communication Systems Lab, Gres’Com, Engineering School of Communication of Tunis, Sup’Com, Ghazela Technopark, 2083, Ariana, Tunisia 2 Theoretical & Experimental Epistemology Lab, TEEL, School of Optometry and Vision Science, Department of Physics, Department of Electrical and Computer Engineering and Department of Systems Design Engineering , University of Waterloo 200 University Avenue West, Waterloo, Ontario N2l 3G1, Canada E-mail:
[email protected]
Abstract: We present a comparison between the basic BB84 Quantum Key Distribution (QKD) and its optimized version based on Quantum Dense Coding (QDC), in terms of theoretical modeling and feasibility of circuits implementations. These circuits are mainly composed by probabilistic quantum gates, single photon sources, Beam Splitters (BS), Polarizing BS (PBS) and single photon detectors. We assume that the photonic devices composing the circuits are not perfect. We assess the average success probabilities of the two protocols depending on the success probability of the photonic sources used and the distance between Alice and Bob. We demonstrate that typical real-world photonic devices permit the implementation of secure communication with 0.65 of success probability over 120 km, while it could reach more than 0.9 over 148 km if just deterministic sources are provided.
1
Introduction
QKD promises unconditional security for information communication [1]. Real implementation of efficient QKD protocol has been widely discussed in the literature [2–5]. Advanced optical fiber based QKD systems have already been commercialized to allow secret key generation over about hundred km of distance [6–9]. In Beijing and Shanghai, a network of fibers, 2000 km long, based on QKD is underway for deployment. The UK is actually realizing a quantum network that helps integration of quantum communication. Japan and the US have already started installing their own QKD networks [10]. Nonetheless, before this solution can be widely adopted, a number of significant challenges have to be taken into consideration. These include the key rate, the distance and the security of the implementation using the state-of-the-art telecom band devices. A multitude of works addressed the realization of QKD when using imperfect Single Photon Source (SPS) and Single Photon Detector (SPD) and associated to other photonic devices. In their work, D. Gottesman et al. [11] proved the security of the QKD-BB84 in the case where the errors of both source and detector are under the control of an adversary, their main study addressed the basicdependent detector efficiency and the basis dependent misalignment in the source or detector, and they proposed an estimation of the key generation rate when such errors are provided by different models of the sources and detectors. W.Y.Liang et al. [7] proposed an optimized version of QKD based on the single photon Bell state measurement. They experimentally demonstrated the feasibility of the protocol over 175 km using Attenuated Laser (AL) emitting at 1550 nm with weak coherent pulses. The optical devices used in the setup, including the BS, the intensity modulator and phase modulator were just referred in the setup without taking into consideration their imperfections. T. Gehring et al. [8] implemented a QKD based on Podolsky-Rosen entanglement continuous variable with composable security when facing a coherent attack. The SPS consisted of two continuous wave squeezed vacuum beams, generated by parametric down conversion at 1550 nm. The Decoy state method has been considered to compensate the imperfection of such SPS. Another interesting work is proposed by K. Takemoto et al. [9] who implemented a QKD-BB84 over 120 km with a low secure key rate of 0.307 bps, their work originally considered a Quantum Dot (QD)
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SPS when applying a very short and resonant excitation pulse (instead of attenuated laser) and a superconducting nanowire SPD. The QD-SPS used in this work is still non deterministic and provides g 2 (0) = 0.0051 which could influence the security of the links, where no discussion was aborted in this matter. All of the above mentioned works have in common that they suppose perfect BS, PBS, CNOT and other single and double qubits gates, which is experimentally impractical. Moreover, the success probability of the circuit realizing the QKD has to take into consideration probabilistic SPS and SPD. We consider in this work actual SPS technological realizations and exclude the AL from our study. To our knowledge, no work addressed the assessment of the success probability of the entire circuit that implements the QKD, when based on imperfect photonic devices and using probabilistic SPS for single photon emission. We already proposed a circuit realization of the BB84 protocol based on single and two qubits gates [12]. Then, we suggested a more secure version that is based on Quantum Dense Coding (QDC), we proposed the circuit that realizes the optimized BB84 version when using a specific probabilistic CNOT gate model 4 ) and optimized heralded SPS. (providing a success probability of 27 We showed that the realization is feasible with a very low success probability. In this paper, we extend our work to any SPS and show the advantages of using our version of the BB84 when using actual photonic devices [14–19]. The objective of this work is to use the serial and parallel decomposition of quantum circuits to model the average success probability of the quantum circuits that implements both of the classical and the optimized BB84 circuits. We show the efficiency of our solution when deterministic components are provided and we enumerate as well it’s drawbacks when using nowadays optical devices. This paper is composed of four sections, section 2 presents the BB84 protocol and corresponding optimized version based on QDC while considering possible circuit realization. In this section, we demonstrate how to model these circuits when using the serial and parallel decomposition into a set of one qubit and two qubits gates. Section 3 is dedicated to the simulation results, where we present the average of the success probability of correctly realizing the BB84 when using probabilistic devices, a discussion about physical realizability is also presented. We enclose this paper by a conclusion and different perspectives in the section 4.
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2 Circuit realization of the Quantum Key Distribution Protocols 2.1
Description
The classical BB84 protocol, denoted in this paper by BB84-QKD, is based on four quantum states that build two different bases: Horizontal-Vertical basis, denoted , and diagonal-anti diagonal basis, denoted . In the basis, the logical 0 and 1, are encoded by the Horizontally and Vertically polarized single photon, denoted by |←→i and |li, respectively. In the basis, the notations |%i and |-i stand for single photon being polarized at 45° and 135° and represent the logical 0 and 1, respectively. In this circuit, we consider the presence of an eavesdropper capable of powering an Intercept and Resend Attack (IRA) and a Beam Splitter Attack (BSA) [12]. For security purposes of the BB84, we assume that Eve has enough technological power and we integrate a realistic optimized attack strategy that combines both IRA and BSA (stage 2 of figure 3). We consider a quantum Channel (QC) between Alice and Bob as illustrated by figure 3. The set of the SPSs, QC, BSs, PBSs and SPDs do introduce some extra errors to the transmitted qubits, which will affect the sifted key, even in the case where Alice and Bob make identical choices of basis. Since it is hardly complicated to estimate the success probability of the most general attack technique, we consider the optimized attack strategy where Eve is permitted to attack by any manner allowed by quantum mechanics [12]. On the other hand, we consider an implementation of the Optimized BB84-QKD using QDC, denoted BB84-QDC, proposed in [12], and illustrated by level 1 of figure 4. The measuring process uses the same SPDs, denoted SP D1B , SP D2B , SP D3B and SP D4B , necessary to detect the states |-i, |%i, |←→i and |li, respectively. Moreover, two PBSs denoted P BS1B and P BS2B are used for the 1st and the 2nd photon, and working in the and basis, respectively. Errors of measurements due to decoherence arising from interaction with environment are not considered here in. Also, the problem of entanglement distillation is not considered and we suppose that all errors are due to the channel noise and eventually, Eve’s interference. We estimate errors due to the physical parameters and exclusively in the case where we have a right choices of basis measurement. 2.2
Circuit modeling
The basic optical devices composing a quantum circuits are the quantum gates. These quantum gates can perform their functions whether in a deterministic or in a probabilistic way. In this paper, five deterministic single qubit gates are addressed, I2 , X, Z , Y , and H, realizing the functions of the identity, the negation, the phase shift, the negation-phase shift and the Hadamard transform, respectively. These single qubit gates have the following matrix transforms: I2 = |0i h0| + |1i h1| =
1
0 0 UX = |1i h0| + |0i h1| = 1 0 UY = |0i h1| − |1i h0| = 1 1 UZ = |0i h0| − |1i h1| = 0 1 1 UH = √1 1 −1 2
0 1 1 0 −1 0 0 −1
For the second category, we denote α = {α1 , α2 }, for 0 ≤ αi, i={1,2} ≤ 1, as the set of probability values realizing correctly the functions of these devices. Their transfer matrices are given as follows: √ α1 √0 α USuccess = (3) 0 α2 Also, we denote by β = {β1 , β2 , β3 , β4 }, for 0 ≤ βj, j={1,2,3,4} ≤ 1, the probability values describing the probability of success of a device in the 4 × 4 system states as given by the following: √ β = WSuccess
β1 0 0 0
√0 β2 0 0
0 √0 β3 0
0 0 0 √ β4
(4)
The final transfer matrices corresponding to circuits of figure 3 and QDC figure 4, are respectively denoted by U BB84 Circuit and U Circuit , and are defined as follows: BB84 BB84 BB84 BB84 = UBob .UEve .UAlice UCircuit BB84 UAlice ,
BB84 UEve
(5)
BB84 , UBob
are respectively the transfer and where matrices representing Alice, Eve and Bob stages, and: QDC QDC QDC QDC UCircuit = UBob .ULink .UAlice
(6)
QDC QDC QDC where UAlice , ULink and UBob , are respectively the transfer matrices representing Alice, the link and Bob stages. In the following, we detail how we modeled the transfer matrices of equations 5 and 6 while considering realistic errors [13]. BS1A and BS1B have reflectivity coefficients ηBS A = 1 and 1 ηBS B = 21 , respectively. By taking into consideration the er1 rors ξ1 and ξ2 that could affect these reflectivity coefficients, ξ1 ξ2 we denote by UBS A and UBS B the respective transfer matri1 1 ces. − ξ1 ) , (1 − ξ1) , (1 − ξ1 ) , (1 − ξ1 )} and η2 = 1 For η1 = {(1 1 1 1 − ξ , − ξ , − ξ , − ξ , the transfer matri2 2 2 2 2 2 2 2 ces are directly obtained from equation 4 as follows: ξ1 UBS A
η1 ξ2 η2 WSuccess ; UBS B = WSuccess
=
1
(7)
1
For P BS1A , P BS2A , P BS1B and P BS2B in stage 1 and stage 3 of figure 3, we denote by ς1 , ς2 , ς3 and ς4 the probability of errors that alter their functioning, The resulting transfer matrices, denoted by UPς1BS A . UPς2BS A , UPς3BS B and UPς4BS B , respectively, are obtained 1 2 1 2 according to equation 2 as follows: ς1 ς2 UPς1BS A = UErr ; UPς2BS A = UErr ;
(1)
1
2
1
2
ς3 ς4 UPς3BS B = UErr ; UPς4BS B = UErr
Concerning the probabilistic devices used in figure 3 and figure 4 (lower parts of the figures), they are classified into two categories: the first one being the devices that introduce errors between the basis states, i.e. the PBSs and the link. The second category being the lossy components that provide a certain probability of being only in one state of the basis, i.e. the SPSs and the SPDs. For the first category, Perr denotes the probability of errors, where 0 ≤ Perr ≤ 1. Each of these components can be seen as a one qubit quantum gate acting
2
Perr , and given on the 2 × 2 quantum system by a transfer matrix UErr as follows: √ √ 1√− Perr √ Perr Perr UErr = (2) Perr 1 − Perr
(8)
The optimized intrusion of Eve is illustrated by stage 2 of figure 3. To go unnoticed, Eve needs to optimize her intrusion rate when performing IRA, therefore, the induced error rate is denoted by EveErr and is obtained as EveErr = 0.25 × IntruRate. For this error rate, the matrix representing the errors induced by Eve is BB84 denoted as UIntrusion and is given equal to: BB84 EveErr EveErr UIntrusion = UErr ⊗ UErr
(9)
Also, this stage includes the two links Alice-Eve (dAE ) and EveBob (dEB ). Let us consider L (km)=(dAE + dEB ) and Af (dB),
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representing the length and the losses of the link, and At (dB) is the approximated total losses due to the BSs, PBSs and other optical devices (not shown in figure 3), composing QC (i.e connectors, filters, etc...). The efficiency of the link, depending on these parameters, is L denoted Tlink and is given as: L Tlink = 10
(Af L+At) 10
dAE
P
dAE
UFdAE
=
ErrLink ErrLink ⊗ UErr UErr
UFdEB
=
ErrLink ErrLink ⊗ UErr UErr
P
dEB
P
p1 ε1 ε 2 ε3 p1 ε1 ε 2 ε3
p, UCN OT
=
p, UICN OT
=
(10)
L Since Tlink of equation 10 represents the efficiency, the correspondL ing probability of errors, depending on L, is denoted by PErrLink , L and is approximated by PErrLink = T L1 . This latter allows us to link compute the transfer matrices of the two links having distances dAE and dEB , as follows: P
are expressed as follows [13]: ε4 p2 ε5 ε6
ε7 ε8 ε9 p3
ε4 ε5 ε6 p2
ε7 ε8 p3 ε9
ε10 ε11 p4 ε12 ε10 p4 ε11 ε12
(17)
(18)
where p = (pi )1≤i≤4 are the probability amplitude of realiz ing correctly the CNOT function, and ε = εj 1≤j≤12 are the probability amplitude of ending in an erroneous output state.
dEB
(11)
For BB84-QDC circuit, we define the transfer matrix of the optical link transmitting the first photon of the pair, depending on the length dAB , as follows: P
dAB
ErrLink UFdAB = UErr
where
dAB PErrLink
=
1 d
AB Tlink
(12)
.
(a) CNOT gate
(b) ICNOT
Fig. 1: CNOT and Inverted CNOT in a 2-qubits system.
Regarding the SPSs and the SPDs used by o the BB84n 1 1 1 1 1 QKD, and for s11 = PSP S1 , PSP S2 , PSP S3 , PSP S4 and d1 = n o PSP DB , PSP DB , PSP DB , PSP DB , their transfer matrices 1 2 3 4 are given as:
We consider the three gates CN OT1A , CN OT1B and CN OT2B , used by the circuit of the BB84-QDC. CN OT1A and CN OT1B have p,ε p,ε transfer matrices denoted by UCN andUCN , and obtained OT A OT B 1
s11
d11
BB84 BB84 = WSuccess ; Udet = WSuccess USource
(13)
On the other hand, in BB84-QDC circuit, the 2 SPSs and the 4 SPDs used by Alice and Bob are modeled as follows: QDC USource
=
QDC Udet
=
BB84 Ualice
γ1 γ2 USuccess ⊗ USuccess γ4 γ3 ⊗ USuccess USuccess
(14)
BB84 Ueve BB84 Ubob
o n 1 1 1 1 where γ1 = PSP S1 , PSP S1 , γ2 = PSP S2 , PSP S2 ,γ3 = n o n o PSP DB , PSP DB and γ4 = PSP DB , PSP DB . 1 2 3 4 Let us denote the success probability of detecting a photon by an SPD as PSP D and it is obtained by: o
n
e PSP D = 1 − PSP D
(15)
e where PSP D is the probability of having an erroneous detection, given by the probability of having a dark count over the total probability of detection, which is obtained as [1, 12]:
e PSP D =
c DSP D ∆TSP D µSP S ηSP D
(16)
c where DSP D , ∆TSP D and ηSP D are the dark count rate, the detection time window and the quantum efficiency of the SPD and µSP S is the mean photon number of photon per pulse emitted by the SPS. Another two qubits probabilistic gate that is required for modeling the BB84-QDC is the photonic two qubits CNOT gate. In the 2-qubits system spanned by the basis of equation 4, the abstract probabilistic CNOT (figure 1a) and Inverted CNOT (figure 1b), have p,ε p, transfer matrices denoted UCN OT and UICN OT , respectively, and
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1
according to equation 17, while the transfer matrix of CN OT2B is p,ε denoted UICN , which is obtained according to equation 18. OT2B According to the above considerations, the transfer matrices of Alice, Eve and Bob for BB84-QKD (figure 3) , are simply:
=
ξ1 BB84 ς2 ς1 .USource UBS A . UP BS A ⊗ UP BS A
=
UFdEB .UIntrusion .UFdAE
=
2
1
1
(19)
ξ2 BB84 Udet . UPς3BS B ⊗ UPς4BS B .UBS B 1
2
1
On the other hand, stage 1, 2 and 3 of the BB84-QDC (figure 4) are computed as follows: QDC Ualice
=
QDC ULink
=
QDC Ubob
=
p,ε QDC UQDC ⊗ I2 .UCN . (UH ⊗ I2 ) .USource OT A 1
UFdAB ⊗ I2 (20) QDC p,ε Udet . UPς3BS B ⊗ UPς4BS B .UICN .U p,ε OT B CN OT B 1
2
2
1
BB84 The transfer matrix UCircuit of equation 5 traduces the probability amplitude of ending in one of the |-i, |%i, |←→i and |li states, according to the input state (only one SPS among the four SPSs is QDC active at a given time). The transfer matrix UCircuit of equation 6 permits to predict the probability amplitude of measuring one of the |00i, |01i, |10i and |11i states. By considering each element squared QDC BB84 of the two matrices UCircuit and UCircuit , we obtain the success QDC BB84 probability matrices denoted PCircuit and PCircuit , respectively. By considering these matrices, we are able to assess the average of the success probabilities of correctly realizing the BB84-QKD and the BB84-QDC, the next section discusses this issue.
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This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication in an issue of the journal. To cite the paper please use the doi provided on the Digital Library page. Table 1 Physical parameters of the photonic devices used in the simulation process. Physical Value Parameters Link [1] Fiber Losses Af = 0.2 db/km Bob’s setup losses At = −2 dB SPD [7] Detection Time Window
∆TSP D1
Dark Count Rate
Quantum Efficiency Gates [13, 14, 16, 17]
CNOT [21]
Eve [12]
=
∆TSP D2
= ∆TSP D3 = ∆TSP D4 = 0.16 ns c c DSP D = DSP D2 c 1 c = DSP D3 = DSP D4 −8 = 5 × 10 ηSP D1 =ηSP D2 =ηSP D3 = ηSP D4 = 0.153 %
Errors of the BS
ξ1 = 10−3.2 , ξ2 = 10−3.4
Errors of the PBS
ς1 = 10−3.1 , ς2 = 10−2.9 ς3 = 10−2.5 , ς4 = 10−2.2 p1 ,ε1 p1 ,ε1 UCN = UCN and OT A OT B
Transfer Matrices of the CNOTs
Optimal Intrusion Rate
1
1
p1 ,ε1 UICN OT2B for p1 = {pi |pi = 0.969 ∀i = [1..4]} and ε1 = εj |εj = 0.141 ∀j = [1..12] 10 %
3 Numerical Simulation and Physical Realizability 3.1
Numerical Simulation
In this section, we assess the average success probabilities of the basic BB84-QKD and the improved BB84-QDC, while considering common physical parameters for each of the two circuits of figures 3 and 4. We consider non deterministic devices allowed by actual technologies and providing a realistic range of the errors. For our simulation, we consider physical features of photonic devices that have already been used for experimental realization, i.e. the superconducting nanowire SPD [7] and the deterministic CNOT in a double sided optical micro-cavity [16]. These features are summarized and referred in table 1. For simplicity considerations, we suppose that the gates gQDC , H and I2 perform their functions in the deterministic case. We suppose also that Eve is performing an optimized intrusion with 10 % of intrusion rate, and being exactly in the middle between Alice and Bob (dAE = dEB = dAB 2 ). For BB84-QKD, we compute the average of the success probabilities of the elements at the diagoBB84 nal of PCircuit , and then we multiply this value by 2 since BS1B (stage 3 of figure 3) allows us to choose in 1/2 of the cases the correct basis, which are discarded after the basis announcement on the public channel. This average success probability of the BB84-QKD is denoted PBB84−QKD . For the BB84-QDC, we consider in our simulation the specific |01i input state (the result is the same if we consider another input state) and therefore, we use the UX transform (equation 1) to model the transfer matrix of the gate gQDC [12]. In QDC QDC this case, PCircuit (2, 1) and PCircuit (4, 1) allows us to compute the average success probability denoted PBB84−QDC . Two parameters are under study in our simulation, they are the success probability of the SPSs, and the distance (dAB ) over which the protocols are providing secure key transfer. We consider the same technological realization for all the six SPSs of figure 3 (SP S1 , SP S2 , SP S3 and SP S4 ) and 4 (SP S1 and SP S2 ), we assume they all provide a single photon with common success probability value denoted PSP S , and to be
4
Fig. 2: Simulation of the average success probability of correctly realizing the circuits of BB84-QKD and BB84-QDC
closer to implementation setup, we generate randomly for every SPS an approximated error at the order of ±10−3 , denoted {ιk }k=[1..6] . Otherwise, in equations 13 and 14, we consider s11 = {PSP S + ι1 , PSP S + ι2 , PSP S + ι3 , PSP S + ι4 }, γ1 = {PSP S + ι5 , PSP S + ι5 } and γ2 = {PSP S + ι6 , PSP S + ι6 }. Let us recall that PSP S stands for the probability of having exactly one photon per pulse emitted by the SPS, while the mean photon number ( µSP S in equation 16) stands for the average of having a single photon per pulse. ALs are very used in the QKD systems [20] with µSP S ranging in [0.01..0.2], but since this work addresses only the SPS with high PSP S values (PSP S ≥ 0.5), we use in our simulation the approximation PSP S ' µSP S . According to the above considerations, PBB84−QKD and PBB84−QDC , depending on PSP S and dAB are illustrated by figure 2. In figure 2, the dashed curve depicts PBB84−QKD and the flat curve depicts PBB84−QDC . Small fluctuations on the surfaces of both curves are due to the errors {ιk }k=[1..6] . PSP S varies in [0.5..1] while dAB varies in [1..200 km]. For the specific range of values PSP S =[0.5..0.92] and dAB =[1..90 km], the BB84-QKD offers better performances in term of success probability. This is basically due to the number of probabilistic SPSs used at the emission. However, while BB84-QKD uses only one SPS to encode one qubit at a given time, our BB84-QDC uses simultaneously two SPSs to encode one of the four basis states (|00i, |01i, |10i or |11i). In the other hand, for PSP S > 0.92, our proposal of the BB84-QDC offers a higher success probability (more than 90%). It is worth to notice that for deterministic SPS (PSP S ≈ 1), it is possible to reach a distance of 148 km with PBB84−QKD ≥ 0.9. 3.2
Discussion and Physical Realizability
Our simulation results clearly highlight the importance of having deterministic SPS or, probabilistic SPS with success probability approaching unity. In the literature, the best SPSs that we have come across are related to the work of McCuster, K.T. et al. [19], where the experimental setup deterministically creates photon when using spontaneous down conversion and providing PSP S ≈ 0.7. Another interesting SPS is related to the quantum dot embedded in microcavity proposed by Peter, E. et al. [18], offering around PSP S ≈ 0.73 of having exactly a single photon per pulse, if we consider this source for our simulation, we find that the BB84-QKD offers a better success probability. For example, over a distance of 120 km, PBB84−QKD = 0.65 whereas PBB84−QDC = 0.5. This result clearly highlights again the advantage of having SPS with success probability near 100% to benefit from our improved BB84-QDC version, which is more secure and permits to overlap the key rate problem encountered by the BB84-QKD.
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In the CNOT circuit shown in Figure 4, we have added the physical features proposed by Wang, H.F. et al.[21]. These investigators showed theoretically that in the strong coupling regime, a success probability of 94% can be attained with these CNOT circuits. It is possible for the proposed gate to reach a 100% success rate only under certain conditions. However, these gates are not easy to implement in practice. On the other hand, recall that the objective of this work is to assess the probability of success of the circuit realizing both BB84-QKD and BB84-QDC when an optimized attack is considered. Our attack strategy combined both the BSA and the IRA with an optimized intrusion rate [12]. We believe that other possible attack strategies are possible when imperfect devices are used, e.g., Gottesman, D. et al. [11] who showed that tagging, basis dependent detector efficiency, basis dependent misalignment in the SPS and SPD, multiple photons emission by the SPS [18, 19], mismatch of trajectories in Alice’s setup, and different light trajectories from the SPSs to BS1 (stage 1 of figure 3), could end in photon counting mismatch or different bandwidths. These facts may be used by Eve to pinpoint the source leading to knowledge on the light state emitted, which could improve her information rate about the final key without introducing extra errors. It is possible also that these attacks will slightly affect the success probability of our proposed solution. Considering all of the above possible attack strategies together, and by introducing physical features of a specific SPS realization [18, 23], will allow computing the success probability of both BB84QKD and BB84-QDC with high efficiency and will permit deeper study of Eve’s information rate about the final sifted key.
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Gueddana, A., Attia, M., Chatta, R.: ’Optimized QKD BB84 protocol using quantum dense coding and CNOT gates: Feasibility based on probabilistic optical devices’,Proc SPIE Photonic Europe, Brussels, Belgium, 2014, pp. 913627-12 Chatta, R. , Gueddana, A., Attia, M. : ’Realizability Assessment of Probabilistic Quantum CNOT gates based on Experimental Implementation’, International Journal of Quantum Information, 2013, 11, (3), 1350032 Nemoto, K. , Munro, W.J. : ’Nearly Deterministic Linear Optical Controlled-NOT Gate’, Phys. Rev. Lett., 2004, 93: 250502 He, H. , Wu, J. , Zhu, X.: ’An Introduction to All-Optical Quantum ControlledNOT Gates’, Advanced Computer Architecture. ACA 2016. Communications in Computer and Information Science, Springer, Singapore, 2016, 626 Wang, H.F., Wen, J.J. , Zhu, A.D. , Zhang, S., Yeon, K.H.: ’Deterministic CNOT gate and entanglement swapping for photonic qubits using a quantum-dot spin in a double-sided optical microcavity’, Phys. Lett. A., 2013, 377: 2870-2876 Fiorentino, M., Wong, F. N. C.: ’Deterministic controlled-NOT gate for twoqubit single- photon quantum logic’, Conference on Lasers and ElectroOptics/International Quantum Electronics Conference and Photonic Applications Systems Technologies, Optical Society of America, 2004, IMC1 Peter, E., Senellart, P. , Martrou, D. et al.: ’Exciton-Photon Strong-Coupling Regime for a Single Quantum Dot Embedded in a Microcavity’, Phys. Rev. Lett, American Physical Society, 2005, 95, (067401) McCusker, K. T., Peters, N. A., VanDevender, A.P., Kwiat, P. G.: ’A Deterministic Single-Photon Source’, Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference and Photonic Applications Systems Technologies, Optical Society of America, 2008, JTuA117 Pearson, D. S. , Elliott, C. : ’On the optimal mean photon number for quantum cryptography’, arXiv:quant-ph/0403065v2, 2004 Wang, H.F., Wen, J.J., Zhu, A.D., Zhang, S., Yeon, K.H.:’ Deterministic CNOT gate and entanglement swapping for photonic qubits using a quantum-dot spin in a double-sided optical microcavity’, Phys. Lett. A., 2013, 377, pp. 2870-2876 Salewski ,M. , Poltavtsev, S. V., Kapitonov, Y. V., et al.: ’Photon echoes from (In , Ga) As quantum dots embedded in a Tamm-plasmon microcavity’, Phys. Rev. B, 2017, 95: 035312 Zhang, X., Xu, C, Ren, Z.: ’High fidelity heralded single photon source using cavity quantum electrodynamics’, Nature, Scientific Report, 2018, 8:3140
Conclusion
We proposed a gate-based decomposition method for modeling the average success probability of the circuits that realizes the classical BB84 protocol as well as it’s optimized version, denoted by BB84-QKD and BB84-QDC. We took into consideration real-world photonic device performance used in earlier experimental setups. A realistic set of the errors that could occur at the implementation level was also considered. We demonstrated via numerical simulation that a success probability of more than 0.9 can be achieved over 148 km if deterministic SPS are provided. Our BB84-QDC can still be improved by using quantum phase encoding techniques which will open new vistas for QKD with high key rate generation over long distances. To this end, decomposition into serial and parallel set of gates has to be based on the evolution of density matrix for probability matrices extraction. These probability amplitude values should be derived from a Hamiltonian that considers physical features of the photonic devices used, in order to obtain a high value prediction. Specific SPS technological realization is then required and multiple attack strategies in addition to our optimized version are to be considered for security proof. Our future work addresses this issue.
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References Gisin, N., Ribordy, G., Tittel, W., Zbinden, H. : ’Quantum cryptography’, Rev. Mod. Phys., 2002, 74, (1), pp. 145-195 Brassard, G. , Lutkenhaus, N., Mor ,T. , Sanders, B. C.: ’Limitations on Practical Quantum Cryptography’, Phys. Rev. Lett., 2000, 85, (1330) Xiongfeng, Ma., Fung, C.H., Lu, H.K.: ’Quantum key distribution with entangled photon sources’, Phys. Rev. A., 2007, 76, 012307 Gisin, N. , Ribordy, G., Zbinden, H. , Stucki, D., Brunner, N. , Scarani V.: ’Towards practical and fast quantum cryptography’, arXiv:quant-ph/0411022, 2004 Shor, P.W., Preskill, J.: ’Simple proof of the security of the BB84 quantum key distribution protocol’, Phys.Rev.Lett., 2000, 85, pp. 441-444 Soujaeff, A. , Nishiola, T., Hasegaw, T. et al.: ’Quantum key distribution at 1550 nm using a pulse heralded single photon source’, Optics Express 726, 2007, 15, (2) Liang , W.Y. , Li ,M. , Yin , Z.Q., et al.: ’A simple implementation of quantum key distribution based on single-photon Bell state measurement’, quant-ph. arXiv:1505.00897v1, 2015. Gehring , T., Händchen, V. , Duhme, J., et al.: ’Implementation of continuousvariable quantum key distribution with composable and one-sided-deviceindependent security against coherent attacks’, Nature Communications, 2015, 6: 8795 Takemoto, K. , Nambu, Y. , Miyazawa, T. , et al.:’Quantum key distribution over 120 km using ultrahigh purity single-photon source and superconducting singlephoton detectors’, Nature, Scientific Reports, 2015, 5: 14383 Diamanti, E., Lo, H.K. , Qi, B., Yuan, Z.: ’Practical challenges in quantum key distribution’, Nature. npj Quantum Information, 2016, 2: 16025 Gottesman, D., Lo, H.K., Lütkenhaus, N., Preskill, J.: ’Security of quantum key distribution with imperfect devices’, arXiv:quant-ph/0212066v3, 2004
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Fig. 3: Alice’s set-up (stage 1) contains four SPS, denoted SP S1 , SP S2 , SP S3 and SP S4 , necessary to generate single photons being in the states |-i, |%i, |←→i and |li, respectively. Two Polarizing Beam Splitters (PBSs) are used. P BS1A (circular PBS), reflects the state |-i and transmit the state |%i, and a classical PBS, P BS2A reflects the state |li and transmits the state |←→i. BS1A is a Beam Splitter with reflectivity coefficient ηBS A = 1. dAB represents the length of the link between Alice and Bob. dAE and dEB state for the distances between Alice and 1
Eve, Eve and Alice, respectively. Bob’s set-up (stage 3) is composed by BS1B , with reflectivity coefficient ηBS B = P BS1B
basis) and two PBSs denoted and |%i ,|-i, |←→i and |li, respectively.
P BS2B .
Four SPDs, denoted
SP D1B ,
SP D2B ,
SP D3B
1
and
SP D4B ,
1 2
(random choice of the
are used to detect the states
Fig. 4: Alice set-up (stage 1), contains SP S1 and SP S2 that generate two single photons being horizontally polarized, denoted |0A i and |0B i, respectively. The circuit contained in the dashed circle in stage 1 permits generation of the Bell state Φ+ . The outputs of CN OT1A are the entangled pair of photons denoted |qA i and |qB i. |qB i is the photon taken by Bob without being applied to any transformation and |qA i is the photon undergoing the transform UQDC , where UQDC = {I2 ⊗ I2 , UX ⊗ I2 , UY ⊗ I2 , UZ ⊗ I2 , } depending on b = {00, 01, 10, 11} .|qA i is then transmitted through the link over a distance dAB , and applied to the control of CN OT1B and then to the target of CN OT2B at Bob setup (stage 3).
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