Experimental demonstration of high-rate measurement-device-independent quantum key distribution over asymmetric channels
Hui Liu, Wenyuan Wang, Kejin Wei, Xiao-Tian Fang, Li Li, Nai-Le Liu, Hao Liang, Si-Jie Zhang, Weijun Zhang, Hao Li, Lixing You, Zhen Wang, Hoi-Kwong Lo, Teng-Yun Chen, Feihu Xu, Jian-Wei Pan
EExperimental demonstration of high-ratemeasurement-device-independent quantum key distribution overasymmetric channels
Hui Liu,
1, 2, ∗ Wenyuan Wang, ∗ Kejin Wei,
1, 2
Xiao-Tian Fang,
1, 2
Li Li,
1, 2
Nai-LeLiu,
1, 2
Hao Liang,
1, 2
Si-Jie Zhang,
1, 2
Weijun Zhang, Hao Li, Lixing You, ZhenWang, Hoi-Kwong Lo, Teng-Yun Chen,
1, 2
Feihu Xu,
1, 2 and Jian-Wei Pan
1, 21
Shanghai Branch, Hefei National Laboratory for PhysicalSciences at Microscale and Department of Modern Physics,University of Science and Technology of China, Shanghai, 201315, China CAS Center for Excellence and Synergetic InnovationCenter in Quantum Information and Quantum Physics,University of Science and Technology of China, Shanghai 201315, P. R. China Centre for Quantum Information and Quantum Control (CQIQC),Dept. of Electrical & Computer Engineering and Dept. of Physics,University of Toronto, Toronto, Ontario, M5S 3G4, Canada State Key Laboratory of Functional Materials for Informatics,Shanghai Institute of Microsystem and Information Technology,Chinese Academy of Sciences, Shanghai 200050, China a r X i v : . [ qu a n t - ph ] A p r bstract Measurement-device-independent quantum key distribution (MDI-QKD) can eliminate all de-tector side channels and it is practical with current technology. Previous implementations of MDI-QKD all use two symmetric channels with similar losses. However, the secret key rate is severelylimited when different channels have different losses. Here we report the results of the first high-rate MDI-QKD experiment over asymmetric channels. By using the recent 7-intensity optimizationapproach, we demonstrate >
10x higher key rate than previous best-known protocols for MDI-QKDin the situation of large channel asymmetry, and extend the secure transmission distance by morethan 20-50 km in standard telecom fiber. The results have moved MDI-QKD towards widespreadapplications in practical network settings, where the channel losses are asymmetric and user nodescould be dynamically added or deleted. ∗ These authors contributed equally to this work.
IG. 1. An illustration of a star-type MDI-QKD network providing six users with access to theuntrusted relay, Charlie. Inset: an example of the possible implementation by Charlie. quantum network infrastructure.In asymmetric MDI-QKD with two users, Alice and Bob have channel transmittances η A and η B ( η A (cid:54) = η B ). The main question is how to choose the optimal intensities of theweak coherent pulses for Alice and Bob, denoted by s A and s B , so as to maximize the keyrate [25]. A natural option is to choose the intensities to balance the channel losses, i.e., s A η A = s B η B . By doing so, a symmetry of photon flux can arrive at Charlie, and thusresulting in a good Hong-Ou-Mandel (HOM) dip [35]. The dependency of HOM visibilityversus balance of photon number flux can be seen in [34, 36]. However, such an option issub-optimal, and may even generate no key rate at all when the channel asymmetry is high.The fundamental reason is that MDI-QKD is related but different from HOM dip. That is,HOM dip affects only the errors in X basis (i.e., the phase error rate estimated with decoystate method), but has no effect to errors in Z basis (i.e., the bit error rate). Therefore, theoptimal method is to decouple the decoy state estimation in X basis from key generation in Z basis. This is the key idea of the 7-intensity optimization method proposed in [34]. Notethat Ref. [28] also mentioned on passing the possibility of using different intensities for Aliceand Bob, but no analysis on this important asymmetric case was performed there.In the 7-intensity optimization method [34], Alice and Bob each selects a set of 4 in-tensities, namely signal state { s A , s B } in the Z basis, and decoy states { µ A , ν A , ω } and { µ B , ν B , ω } in the X basis, respectively. The parameters that Alice and Bob choose include7 different intensities in total, as well as the proportions to send them. The secret key isgenerated only from the Z basis, while the data in the X basis are all used to perform thedecoy state analysis. The decoy state intensities are chosen to compensate for asymmetry4nd ensure good HOM visibility in the X basis (and roughly satisfy µ A µ B = ν A ν B ≈ η B η A , whichmaintains symmetry of photon flux arriving at Charlie). On the other hand, the signalstate is decoupled from the decoy states, and can be freely adjusted to maximize key ratein the Z basis (and generally s A s B (cid:54) = η B η A ). Overall, Alice and Bob optimize 12 implementa-tion parameters: [ s A , µ A , ν A , p s A , p µ A , p µ A , s B , µ B , ν B , p s B , p µ B , p µ B ] . To efficiently choose theoptimal parameters, we use a local search algorithm and follow the optimization techniquein [34], which converts the 12 parameters into polar coordinate and searches them whilelocking the decoy state intensities at: µ A ν A = µ B ν B [37]. The optimization technique is highlyefficient, and takes less than 0.1s for each run of full optimization on a common desktop PC(with a quad-core Intel i7-4790k processor, using parallelization with 8 threads). FIG. 2. MDI-QKD setup. Alice’s (Bob’s) signal laser pulses are modulated into signal and decoyintensities by three amplitude modulators (AM1-AM3). Key bits are encoded by a Mach-Zehnderinterferometer, AM4, and a phase modulator (PM). In Charlie, the polarization stabilization systemin each link includes an electric polarization controller (EPC), a polarization beam splitter (PBS)and a superconducting nanowire single-photon detector (SNSPD); the Bell state measurement(BSM) system includes a 50/50 beam splitter (BS), SNSPD1 and SNSPD2. Abbreviations ofother components: DWDM, dense wavelength division multiplexer; ConSys, control system; ATT,attenuator; PSL, phase-stabilization laser; Circ, circulator; PC, polarization controller; PS, phaseshifter; SPAPD, single-photon avalanche photodiode.
To implement MDI-QKD over two asymmetric channels, we construct a time-bin-phaseencoding MDI-QKD setup in Fig. 2. Alice and Bob each possesses an internally modulatedlaser which emits phase-randomized laser pulses at a clock rate of 75 MHz. The gain-switchedlaser diode can naturally generate optical pulses with random phases. AM1 (amplitude5
ABLE I. List of parameters characterized from experiment: detector dark count rate Y , detectorsystem efficiency η d , optical misalignment e Xd , e Zd in the X and Z bases, fiber loss coefficient α indB/km, error-correction efficiency f , security parameter (cid:15) , and the total number of laser pulses N sent by Alice/Bob. Y η d e Zd e Xd α f (cid:15) N . × −
46% 0.5% 4% 0.19 1.16 10 − modulator) is used to tailor the pulse shape by cutting off the overshoot rising edge of laserpulses. AM2 and AM3 are employed to randomly modulate the intensities of signal stateand weak decoy states. The time-bin encoding is implemented by utilizing a combinationof a Mach-Zehnder interferometer (MZI), AM4 and a phase modulator (PM). For Z basis,the key bit is encoded in time bin | (cid:105) or | (cid:105) by AM4, while for the X basis, it is encodedin the relative phase 0 or π by the PM. Alice and Bob send their laser pulses through twostandard fiber spools, L A and L B , to Charlie, who performs Bell state measurement (BSM).The BSM includes a 50/50 beam splitter (BS) and two superconducting nanowire single-photon detectors (SNSPD1 and SNSPD2). The main system parameters characterized inthe experiment are shown in Table I.To compensate for the relative phase drift and establish a common phase reference, Aliceemploys a phase-stabilization laser (PSL) and Bob employs a phase shifter (PS) in one of thearms of his MZI and a single-photon avalanche photodiode. To properly interfere the twopulses at Charlie, we develop a real-time polarization feedback control system, an automatictime calibration system and a temperature feedback control system [37]. Thanks to thefeedback control systems, the observed visibility of the two photon interference is about46% and the system has a long-term stability over tens of hours. This stability enablesus to collect a large number of signal detections, thus properly considering the finite-keyeffect [26].We implement the 7-intensity method over different choices of channel lengths [37]. First,we fix the distance between Alice and Charlie at 10 km, i.e., L A = 10 km, while the distancebetween Bob and Charlie L B varies from 40 km to 90 km. At each channel setting, weuse the system parameters listed in Table I to perform a numerical optimization on theimplementation parameters, based on three optimization strategies: (i) 4-intensity method,where the same intensities and proportions for Alice and Bob are selected and optimized6 IG. 3. Simulation (curve) and experiment results (data points) for secret rate (bit/pulse) vs thetotal distance L AB in standard telecom fiber. (a) L A is fixed at 10 km, while L B is selected at 40, 60,80, 90 km. (b) L A is fixed at 0 km, while L B is selected at 40, 60, 80, 100 km. The points (curves)in the figure indicate the experimental (simulation) results for (i) 4-intensity method shown in bluediamond points (blue dashed line), where the same intensities and proportions for Alice and Bobare selected and optimized in the 4-intensity protocol [21, 28]; (ii) 4-intensity+fiber method [14]shown in black circle points (black dot-dash line);(iii) 7-intensity method [34], shown in red squarepoints (red solid line). As can be seen, for the 4-intensity methods, adding fibers improves thekey rate in long distances, but it does not in short distances. In contrast, the 7-intensity methodalways achieves substantially higher key rate than any of the other two methods, especially whenchannel asymmetry is high. in the 4-intensity protocol [21, 28]; (ii) 4-intensity+fiber method, where the asymmetry ofchannels is first compensated by adding additional losses [14] and then the same intensitiesand proportions for Alice and Bob are selected; (iii) 7-intensity method. The results areshown in Fig. 3(a). 7-intensity method can substantially increase the key rate and maximumdistance of MDI-QKD in the case of high channel asymmetry: at L B = 60 km, the 7-intensity method generates a secret key rate of over an order of magnitude higher thanthe 4-intensity+fiber method, and extends the maximum distance for approximately 20kmcompared to 4-intensity+fiber method, and 40km compared to 4-intensity method alone.Next, we demonstrate for the first time a “single-arm” MDI-QKD, as shown in the insetfigure in Fig. 3(b), where we place Alice and Charlie at the same location, i.e., L A = 0 km. L B varies from 40 km to 100 km. The results are shown in Fig. 3(b). Such a single-arm setuponly uses one public channel, and could be highly useful in free-space QKD, where Aliceand Bob typically have a single free-space channel, in the middle of which adding a relay is7 ABLE II. Example implementation parameters and experimental results for L A =10 km and L B =60 km. s Z is the estimated yield of single photons in the Z basis and e X is the estimatedphase-flip error rate of single photons in the X basis. Q Zss and E Zss are the observed gain and QBERfor signal states. R is the secret key rate (bit/s). Ratio is the key rate advantage of the 7-intensitymethod over the given method. Parameters 7-intensity 4-intensity 4-intensity+fiber s A s B µ A µ B ν A ν B p s A p s B p µ A p µ B p ν A p ν B s Z . × − . × − . × − e X Q Zss . × − . × − . × − E Zss .
11 25 . L A = 10km and L B = 60 km in Table II. Note that the parameters in 7-intensity method are quitedifferent from those two types of 4-intensity methods. We obtain a secret key rate of 343bits/s with 7-intensity method, which is 13.5 times higher than that of 4-intensity+fibermethod. By using the joint-bound analysis [28], the key rate can be further improved to 645bits/s. Moreover, the 7-intensity optimization method can greatly extend the transmissiondistance by about 50 km fiber. Furthermore, we also tested an extreme case where L A =0kmand L B =100km. 7-intensity produces a secret key rate of 0.049 bit/s. In contrast, no keybits can be extracted with either strategy of using 4-intensity method with/without fiber.8he method of asymmetric intensities and decoupled bases we demonstrated can be ap-plied to general quantum information protocols. First, the asymmetric method is importantto the future implementation of free-space MDI-QKD with a moving relay such as satellite.For instance, the channel transmittances in satellite-based quantum communication are con-stantly changing with up to 20-dB channel mismatch [38]. Second, the asymmetric methodcan be readily applied to MDI quantum digital signature (QDS) [39, 40] and twin-field (TF)QKD [29]. The key generation formula of MDI-QDS is similar to that of MDI-QKD, wherethe proposed method can be directly implemented [37]. TF-QKD relies on single-photoninterference, where the intensity-asymmetry affects both the interference visibility and thesingle-photon gain [37]. Our methods of asymmetric choice of intensities and optimizationof parameters can be implemented to improve the key rate for asymmetric TF-QKD [41].However, we note that the two encoding bases are symmetric in TF-QKD, thus the methodof decoupled bases might not be applicable [37]. Finally, other protocols that rely on single-photon or two-photon interference, such as comparison of coherent states [42] and quantumfingerprinting [43–45], can also be benefited from our methods when they are working in anasymmetric setting.In conclusion, by using the recent 7-intensity method, we demonstrate an order of mag-nitude higher key rate and an extension of 20-50 km distance over previous best-knownMDI-QKD protocols. While previous methods of adding fibers inconveniently require themodification of every existing node with the addition/deletion of a new node, our 7-intensitymethod implements the optimization in software only and provides much better scalability.Overall, our results have moved MDI-QKD towards a more practical network setting, wherethe channel losses can be asymmetric and nodes can be dynamically added or deleted. ACKNOWLEDGMENTS
This work has been supported by the National Key R&D Program of China, National Nat-ural Science Foundation of China, Anhui Initiative in Quantum Information Technologies,Fundamental Research Funds for the Central Universities, Shanghai Science and TechnologyDevelopment Funds. F. X. acknowledges the support from Thousand Young Talent programof China. W. W. and H.-K. L. were supported by NSERC, U.S. Office of Naval Research,CFI, ORF, and Huawei Canada. 9 ppendix A: Optimization Algorithm
Here we briefly describe the optimization algorithm, as proposed in Ref. [34], for theintensities and the probabilities of sending them in the 7-intensity optimization method.As described in the main text, there are a total of 12 parameters that need to be optimized:[ s A , µ A , ν A , p s A , p µ A , p µ A , s B , µ B , ν B , p s B , p µ B , p µ B ] (A1)To navigate such a large parameter space, a local search algorithm is necessary. However,the key rate function versus these parameters is in fact non-smooth, hence local search doesnot work well in this case. To address this, we need to convert the parameters to polarcoordinate: [ r s , θ s , r µ , r ν , θ µν , p s A , p µ A , p µ A , p s B , p µ B , p µ B ] (A2)where the conversion follows that: r s = (cid:112) s A + s B θ s = arctan ( s A s B ) (A3)and the same applies for r µ , θ µν and r ν , θ µν . Note that here we have locked the polar angleof µ and ν to be always equal. This is because, the optimal values of the decoy intensitiesfor Alice and Bob always satisfy [34] µ A ν A = µ B ν B (A4)for the 7-intensity optimization method. The intensity probabilities are not involved innon-smoothness and therefore do not need to be changed. With the 11 parameters now,we can perform a local search algorithm, such as coordinate descent [27, 34], to efficientlyfind the set of optimal intensities. Coordinate descent algorithm alternatively optimizeseach variable at a time while keeping others constant. When all variables are searched, thealgorithm starts over again with the first variable. With enough iterations, this algorithmcan reach a local maximum point. For asymmetric MDI-QKD, the algorithm can find theglobal maximum point (which is the only local maximum).10 ppendix B: The Detail of MDI-QKD Systems Alice and Bob each possess an internally modulated laser which emits phase-randomizedlaser pulses at a clock rate of 75 MHz. The pulse width is about 3.4 ns, and the centerwavelength is at 1550.12 nm with a full width at half maximum (FWHM) of about 15 pm.AM4, is used to modulate the vacuum state, where the extinction ratio between the signalstate and the vacuum state is larger than 23 dB. The MZI divides each incoming pulseinto two time-bins with 6.4 ns time interval. The superconducting nanowire single-photondetectors (SNSPD1 and SNSPD2) have detection efficiency 70% and dark count rate 30cps. Due to an extra insertion loss 1.2 dB in Charlie and 15% non-overlap between laserpulse and detection time window, the total system detection efficiency is 46%.To get a high visibility of two-photon interference , we adopt several feedback systems [18]to calibrate the polarization, time and spectrum modes of the signal pulses generated bytwo independent laser sources, as shown in Fig. 2 in main text.First, to make sure that the polarization of two pulses are indistinguishable, we plug aEPC and a PBS before the BS in the BSM. Intensities of the reflection port of the PBS aremonitored by a SNSPD, which outputs a feedback signal to control the EPC to minimizethe intensities.Besides, to precisely overlap timing mode of the two signal pulses, there are two calibra-tion processes in the experiments. First, Charlie generates two synchronization lasers(Synls,1570nm) pulses which are synchronized by a crystal oscillator circuit (COC). The pulsesare respectively sent to Alice and Bob, who detected it by a PD. The output signals of thePDs are used to synchronize lasers in Alice and Bob. Second, Alice and Bob respectivelysend their signal laser pulses to Charlie, who use the SNSPD to measure the arriving timeof the pulses. Then, by using a programmable delay chip, Charlie adjusts the time delaybetween the two SynL according to the arriving time difference. The total timing calibrationprecision is below 10ps and the arriving time jitter of SNSPD is below 50ps, both them arefar smaller that the pulse width of 2.4 ns.For the spectrum mode, we first use an optical spectrum analyzer (OSA,YOKOGAWAAQ6370B) to calibrate Alice’s signal pulses and phase feedback pulses wavelength at1550.12nm. Then we observe the Hong-Ou-Mandel (HOM) interference of two pulsesin Charlie and adjust the operating temperature of the laser in Bob to the value where the11OM dip are found.Alice and Bob need to have a shared phase reference frame which fluctuates with tem-perature and stress. Using a PSL, Alice sends laser pulses from her AMZI to Bob’s AMZIvia an additional link between Alice and Bob. Bob monitors the power at one of outputs ofhis interferometer with a SPAPD and then minimizes the counts of SPAPD by using a PSin Bob’s AMZI.
Appendix C: Application to asymmetric quantum digital signature
In MDI quantum digital signature (QDS) [39, 40, 46], there exists a sufficiently largesignature length which makes the protocol secure, if the condition Q , Z + Q , Z [1 − h ( e , X )] − h ( E Z ) > Q i,iZ is the lower bound on the count rate when Alice and Bob sent Z -basispulses containing i photons, e , X is the upper bound for the single-photon phase error rateand E Z is the quantum bit error rate (QBER) in Z basis.Apparently, Eq. (C1) is the same as the key rate formula of MDI-QKD [13] except forthat MDI-QDS omits the inefficient factor of error correction. Hence, we can directly applythe proposed asymmetric method to improve the performance of MDI-QDS over asymmetricchannels. Appendix D: Application to asymmetric twin-field QKD
Unlike MDI-QKD, in the case of TF-QKD [29], the key bits are generated from singles | (cid:105) + | (cid:105) rather than coincidences | (cid:105) . Alice and Bob send signals whose phases areannounced and post-selected to be in the same ”phase-slice”, and the signals are receivedby a third party, Eve, who announces the detector events at detectors C and D . The keyrate can be written as [29], R = dM { Q [1 − h ( e )] − f Q µ h ( E µ ) } (D1)where M is the number of phase slices, d is a phase slice post-selection factor, Q , e are12he estimated gain and phase-error rate of single photons, Q µ , E µ are the observed gain andQBER for the signal states, and f is the error correction efficiency.Conceptually, for single photons, the asymmetry between transmittances in the two chan-nels decreases the visibility of single-photon interference, resulting in higher QBER. In re-ality, Alice and Bob both use WCP sources. Consider Alice and Bob sending intensities µ A and µ B . Let us define γ A = √ µ A η A η d γ B = √ µ B η B η d (D2)where η A , η B are the channel transmittances between Alice (Bob) and Charlie, and η d is thedetector efficiency. Following [25], the received intensities D C , D D at the detectors (here forsimplicity we ignore the dark counts) are: D C = ( γ A + γ B − γ A γ B cosφ ) / D D = ( γ A + γ B + 2 γ A γ B cosφ ) / φ is the relative phase between Alice’s and Bob’s signals. Consider the detector C ,the visibility can be written as: v = I max − I min I max + I min = 2 γ A γ B γ A + γ B = 2 k + k (D4)where k = (cid:113) µ A η A µ B η B is the ratio between arriving intensities at Eve’s beam splitter. v is afunction that reaches maximum 1 when k = 1, and monotonically decreases with k when k deviates from 1. In fact, we can compare this with the two-photon interference visibilityfrom two WCP sources as used in MDI-QKD: v = 1 − P coin P C P D = 2 µ A η A µ B η B ( µ A η A + µ B η B ) = 22 + k + k (D5)where P C , P D , P coin are respectively the count probability of detector C , detector D , and thecoincidence count probability.We have plotted both visibilities versus ratio of arriving intensities in Fig. 4. We cansee that just like for MDI-QKD, the single-photon interference visibility in TF-QKD heavily13epends on the balance of arriving intensities. The visibility of the interference betweenthe two pulses directly affects the observed QBER for both the signal and decoy states.Therefore, we can also apply our method to compensate for channel asymmetry (such asusing η A µ A = η B µ B ) and obtain higher single-photon interference visibility and lower QBER.Note that the single-photon gain Q also depends on the intensity values [41]. Hence anasymmetric choice of intensities, i.e., η A µ A = η B µ B , may not be the optimal method. Acareful optimization of the parameters, following the approach in [34], are likely required toproduce the optimal key rate in asymmetric TF-QKD. Note also that the decoupling of basismight not work in TF-QKD. The reason is that, unlike MDI-QKD where the two bases areinherently asymmetric (Charles only measures in Z basis [34]), for TF-QKD the two basesare symmetric, and both will depend on the visibility of interference. Some extensions of theinitial TF-QKD protocol use only one basis X for encoding and another basis Z for decoy-state analysis, where decoupling of bases and using different intensity choices potentiallymight work, but a rigorous study of this will be the subject of future studies. FIG. 4. The visibility for a single-photon interference (used in TF-QKD) and for a two-photoninterference (used in MDI-QKD) versus the ratio of intensities used by Alice and Bob, usingWCP sources. Here we consider two channels with 50km difference in standard fiber (i.e. with η A /η B = 0 . µ A η A = µ B η B , and the visibility drops as ratio of arriving intensities at Charles becomesimbalanced. Note that a two-photon interference with WCP sources can only reach a maximum of50% visibility while single-photon interference can reach 100%. Therefore, just like for MDI-QKD,the visibility of single-photon interference in TF-QKD (and as a result its QBER) heavily dependson the balance of intensities.
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P a r a m e t e r s L A = k m L B = k m L B = k m L B = k m L B = k m - i n t - i n t - i n t + fib e r - i n t - i n t - i n t + fib e r - i n t - i n t + fib e r - i n t s A . . . . . . . . . s B . . . . . . . . . µ A . . . . . . . . . µ B . . . . . . . . . ν A . . . . . . . . . ν B . . . . . . . . . p s A . . . . . . . . . p s B . . . . . . . . . p µ A . . . . . . . . . p µ B . . . . . . . . . p ν A . . . . . . . . . p ν B . . . . . . . . . s Z . E - . E - . E - . E - . E - . E - . E - . E - . E - e X . % . % . % . % . % . % . % . % . % Q Z s A s B . E - . E - . E - . E - . E - . E - . E - . E - . E - E Z s A s B . % . % . % . % . % . % . % . % . % R . E - . E - . E - . E - . E - . E - . E - . E - . E - L A = k m L B = k m L B = k m L B = k m L B = k m s A . . . . . . . . . s B . . . . . . . . . µ A . . . . . . . . . µ B . . . . . . . . . ν A . . . . . . . . . ν B . . . . . . . . . p s A . . . . . . . . . p s B . . . . . . . . . p µ A . . . . . . . . . p µ B . . . . . . . . . p ν A . . . . . . . . . p ν B . . . . . . . . . s Z . E - . E - . E - . E - . E - . E - . E - . E - . E - e X . % . % . % . % . % . % . % . % . % Q Z s A s B . E - . E - . E - . E - . E - . E - . E - . E - . E - E Z s A s B . % . % . % . % . % . % . % . % . % R . E - . E - . E - . E - . E - . E - . E - . E - . E - A B L E I V . L i s t o f t h e t o t a l ga i n s a nd e rr o r ga i n s o f b e ll s t a t e ψ − i n t h ec a s e s o f L A = k m . T h e fi r s t r o w i nd i c a t e s t h ec h a nn e l d i s t a n ce l e n g t h a nd a tt e nu a t i o n f r o m B o b t o C h a r li e . T h e n o t a t i o n α β s h o w n i n t h e s ec o nd c o l u m nd e n o t e s t h e pu l s e p a i r f r o m A li ce s o u r ce α a nd B o b s o u r ce β , r e s p ec t i v e l y . L B ( A tt) k m ( . d B ) k m ( . d B ) k m ( . d B ) k m ( . d B ) M e t h o d - i n t - i n t - i n t + fib e r - i n t - i n t - i n t + fib e r - i n t - i n t + fib e r - i n t T o t a l G a i n s ss µµ νν µ ν ν µ µ o o µ ν o o ν oo E rr o r G a i n s ss µµ νν µ ν ν µ µ o o µ ν o o ν oo A B L E V . L i s t o f t h e t o t a l ga i n s a nd e rr o r ga i n s o f b e ll s t a t e ψ − i n t h ec a s e s o f L A = k m . T h e fi r s t r o w i nd i c a t e s t h ec h a nn e l d i s t a n ce l e n g t h a nd a tt e nu a t i o n f r o m B o b t o C h a r li e . T h e n o t a t i o n α β s h o w n i n t h e s ec o nd c o l u m nd e n o t e s t h e pu l s e p a i r f r o m A li ce s o u r ce α a nd B o b s o u r ce β , r e s p ec t i v e l y . L B ( A tt) k m ( . d B ) k m ( . d B ) k m ( . d B ) k m ( . d B ) M e t h o d - i n t - i n t - i n t + fib e r - i n t - i n t - i n t + fib e r - i n t - i n t + fib e r - i n t T o t a l G a i n s ss µµ νν µ ν ν µ µ ω ω µ ν ω ω ν ωω E rr o r G a i n s ss µµ νν µ ν ν µ µ ω ω µ ν ω ω ν ωω31721457301