# Passive Decoy-State Quantum Key Distribution with Coherent Light

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Passive Decoy-State BB84 Transmitter

_{1}, both prepared in +45° linear polarization and with arbitrary phase relationship, $|\sqrt{2\mu}{e}^{i{\theta}_{1}}{\rangle}_{{a}_{0},+45\xb0}$ and $|\sqrt{2\mu}{e}^{i{\theta}_{2}}{\rangle}_{{b}_{0},+45\xb0}$, at a 50 : 50 BS. The output states in modes a

_{1}and b

_{1}(see Figure 1) are given by

_{1}and d

_{1}have the form

_{2}, $|\sqrt{\mu}{e}^{i{\theta}_{3}}{\rangle}_{{a}_{2},+45\xb0}$ and $|\sqrt{\mu}{e}^{i{\theta}_{4}}{\rangle}_{{b}_{2},+45\xb0}$, in a nonlinear medium using the SFG process, the resulting output states at frequency w

_{3}= w

_{1}+ w

_{2}, after the polarization rotation R, can be written as (see Appendix A)

_{3}(see Figure 1) is a coherent state of the form

_{2}− θ

_{1}, ϕ = π + θ

_{1}+ θ

_{3}+ arg (1 + e

^{iθ}), and the Fock states |n

_{ψ}〉 are given by

_{4}− θ

_{3}. Finally, Alice sends the quantum state given by Equation (4) through a BS of transmittance t ≪ 1. Then, the output states in modes c

_{3}and d

_{3}are given by

_{i}, with i ∈ {1,…, 4}, is randomized and inaccessible to the eavesdropper is now straightforward. It can be solved by just integrating the signals $|\sqrt{t\zeta (\theta )}{e}^{i\varphi}{\rangle}_{\psi ,{c}_{3}}$ and $|\sqrt{(1-t)\zeta (\theta )}{e}^{i\varphi}{\rangle}_{\psi ,{d}_{3}}$ given by Equation (6) over all angles θ, ϕ, and ψ. In particular, we have that the output state σ in this scenario (see Figure 1) can be written as

_{3}is suitable for QKD and Alice sends it to Bob through the quantum channel. In addition, she uses the strong intensity signal available in mode d

_{3}to measure both its intensity and polarization. This last measurement can be realized, for example, by means of a passive BB84 detection scheme where the basis choice is performed by a 50 : 50 BS, and on each end there is a PBS and two classical photodetectors. From the different intensities observed in each of these four photodetectors, Alice can determine both the value of the angle ψ and the total intensity of the signal. Note that, by assumption, we have that the intensity of the input states ρ

_{i}is very high.

_{1}coherent states of random amplitude. This is done by combining two coherent states or equal amplitude but random phase at a BS. This part follows the approach proposed in [37,38] to passively prepare decoy states. Then, in a second step, the setup generates signals whose photons are randomly polarized within the X-Z plane of the Bloch sphere. For this, it first prepares two phase-randomized coherent states of the same amplitude in +45° and −45° linear polarization, and then interferes them at a PBS. This approach is inspired by the solution introduced in [42]. Indeed, this is the role of the interferometric part of the scheme, where the SFG process is actually used to imprint a random phase to each state. Finally, in a third step, the setup attenuates the generated signals to the single-photon level before they are sent to Bob, and it also determines both the intensity and polarization of the signals prepared.

_{3}. The first intensity interval, ξ

_{d}= [0, Λ], can be associated, for instance, to the generation of a decoy state in output mode c

_{3}(that we shall denote as σ

_{d}), while the second intensity interval, ξ

_{s}= [Λ, 4µ(1−t)], corresponds to the case of preparing a signal state (σ

_{s}). Note, however, that the analysis presented in this section can be straightforwardly adapted to cover as well the case of several intensity intervals ξ

_{i}(i.e., the generation of several decoy states). Figure 2 (case A) shows a graphical representation of the intensity (1 − t)ζ(θ) in mode d

_{3}versus the angle θ, together with the threshold value Λ and the intensity intervals ξ

_{d}and ξ

_{s}.

_{Λ}that satisfies (1 − t)ζ(θ

_{Λ}) = Λ is given by

_{acc}, is given by

_{acc}favors Ω ≈ 0, but this action also results in an increase of the quantum bit error rate (QBER) of the protocol. A low QBER favors Ω ≈ π/4, but then p

_{acc}≈ 0. Note that in the limit where Ω tends to π/4 we recover the standard decoy-state BB84 protocol.

## 3. Lower Bound on the Secret Key Rate

_{i}denotes the probability to generate a state associated to the intensity interval ξ

_{i}(i.e., p

_{s}= θ

_{Λ}/π and p

_{d}= 1 − p

_{s}), and

^{i}denotes the gain, i.e., the probability that Bob obtains a click in his measurement apparatus when Alice sends him a signal σ

_{i}; f(E

^{i}) represents the efficiency of the error correction protocol as a function of the error rate E

^{i}, typically f(E

^{i}) ≥ 1 with Shannon limit f(E

^{i}) = 1 [58]; Y

_{n}is the yield of an n-photon signal, i.e., the conditional probability of a detection event on Bob’s side given that Alice transmits an n-photon state; e

_{n}denotes the error rate of an n-photon signal; and H(x) = −x log

_{2}(x) − (1 − x) log

_{2}(1 − x) represents the binary Shannon entropy function.

^{i}and E

^{i}. These quantities are given in Appendix B. Our results, however, can also be straightforwardly applied to any other channel model or detection setup, as they depend only on the observed gain and QBER.

_{0}and Y

_{1}, together with the single-photon error rate e

_{1}, by solving the following set of linear equations:

_{0}= 1/2).

_{B}= 3.2×10

^{−7}, the overall transmittance of Bob’s detection apparatus is η

_{B}= 0.045, and the loss coefficient of the channel is α = 0.2 dB/km. We further assume that q = 1/2, and f(E

^{i}) = 1.22. With this configuration, it turns out that the optimal value of the parameter µt decreases with increasing distance, while the optimal value of the parameter Ω increases with the distance. A similar behavior was also observed in the passive BB84 transmitter (without decoy states) proposed in [42]. In particular, µt diminishes from ≈ 0.175 to ≈ 0.125, while Ω augments from ≈ 0.393 to ≈ 0.7. At long distances the gain of the protocol is very low and, therefore, it is important to keep both the multi-photon probability of the source (related with the parameter µt) and the intrinsic error rate of the signals sent by Alice (related with the parameter Ω) also low. Figure 3 includes as well an inset plot with the optimized parameters µt (dashed line) and Ω (solid line). The optimal value for the parameter Λ turns out to be constant with the distance; it is given by Λ = 2µ(1 − t), i.e., the threshold angle θ

_{Λ}is equal to π/2. This figure also shows a lower bound on the secret key rate for the cases of an active decoy-state BB84 system with infinite decoy settings (black line) [6], and a passive transmitter with infinite intensity intervals ξ

_{i}(red line). The cutoff points where the secret key rate drops down to zero are ≈ 181 km (passive setup with two intensity settings), ≈ 183 km (passive setup with infinite intensity settings), and ≈ 192 km (active transmitter with infinite decoy settings). From the results shown in Figure 3 we see that the performance of the passive transmitter presented in Section 2, with only two intensity settings, is similar to that of an active asymptotic setup, thus showing the practical interest of the passive scheme. The relatively small difference between the achievable secret key rates in both scenarios is due to two main factors: (a) the intrinsic error rate of the signals accepted by Alice, which is zero only in the case of an active source; and (b) the probability p

_{acc}to accept a pulse emitted by the source, which is p

_{acc}< 1 in the passive setup and p

_{acc}= 1 in the active scheme. For instance, we have that for most distances Ω ≈ 0.393, which implies p

_{acc}≈ 0.5. This fact reduces the key rate on logarithmic scale of the passive transmitter by a factor of log

_{10}p

_{acc}≈ 0.3. The additional factor of ≈ 0.45 that can be observed in Figure 3 arises mainly from the intrinsic error rate of the signals.

## 4. Phase Encoding

_{i}, with i ∈ {1,…, 4}, are pure coherent states with arbitrary phase relationship: $|\sqrt{2\mu}{e}^{i{\theta}_{1}}{\rangle}_{{a}_{0},+45\xb0}$ and $|\sqrt{2\mu}{e}^{i{\theta}_{2}}{\rangle}_{{b}_{0},+45\xb0}$ (of frequency w

_{1}), and $|\sqrt{\mu}{e}^{i{\theta}_{3}}{\rangle}_{{a}_{2},+45\xb0}$ and $|\sqrt{\mu}{e}^{i{\theta}_{4}}{\rangle}_{{b}_{2},+45\xb0}$ (of frequency w

_{2}). Let Δt denote the time difference between two consecutive pulses generated by the sources. Then, from Section 2 we have that the signals in modes c

_{2}and d

_{2}at time instances t and t + Δt/2 can be written as

_{4}− θ

_{3}. Similarly, we find that the quantum states in modes c

_{3}and d

_{3}are given by, respectively,

_{3}are used to measure both their phases, relative to some local reference phase, and their intensities by means of an intensity and phase measurement, while Alice sends the weak signals in mode c

_{3}to Bob. Again, just like in the passive source with polarization encoding shown in Figure 1, Alice can now select some valid regions for the measured phases and also distinguish between different intensity settings. Then, we have that the analysis and results presented in Section 3 also apply straightforwardly to this scenario.

## 5. Conclusions

## Acknowledgments

## Appendix

## A. Sum-Frequency Generation

_{3}= w

_{1}+ w

_{2}[45]. The parameter χ is a coupling constant that is proportional to the second-order susceptibility χ

^{(2)}of the nonlinear material, and H.c. denotes a Hermitian conjugate. When the pump mode at frequency w

_{2}is kept strong and undepleted, then this mode can be typically treated classically as a complex number. With this assumption, we have that the effective Hamiltonian above can now be written as $H\phantom{\rule{1em}{0ex}}=\phantom{\rule{1em}{0ex}}i\hslash \chi (\sqrt{\mu}{e}^{i{\theta}_{3}}{c}_{1}{c}_{2}^{\u2020}-\mathrm{H}.\mathrm{c}.)$. Using the Heisenberg equation of motion, it is straightforward to obtain the following coupled-mode equations:

_{c}we find that the resulting output state at frequency w

_{3}from the SFG process is given by

## B. Gain and QBER

^{i}and error rates E

^{i}, with i ∈ {s, d}, for the passive QKD transmitter with two intensity settings introduced in Section 2. For that, we employ the typical channel model in the absence of eavesdropping [6,59]; it just consists of a BS of transmittance ${\eta}_{\text{channel}}={10}^{-\frac{\alpha d}{10}}$, where α denotes the loss coefficient of the channel measured in dB/km and d is the transmission distance. Moreover, for simplicity, we consider that Bob employs an active BB84 detection setup with two threshold detectors.

_{B}of Bob’s detectors and we neglect other background contributions like, for instance, stray light arising from timing pulses which are not completely filtered out in reception. The operators ${F}_{\text{VAC}}^{\beta}$, ${F}_{0}^{\beta}$, ${F}_{1}^{\beta}$, and ${F}_{\mathrm{dc}}^{\beta}$ have the form

_{l}(|n, m〉

_{c}) represent the state which has n photons in the horizontal (circular left) polarization mode and m photons in the vertical (circular right) polarization mode. The parameter η

_{sys}denotes the overall transmittance of the system. This quantity can be written as η

_{sys}= η

_{B}η

_{channel}, where η

_{B}is the overall transmittance of Bob’s detection apparatus, i.e., it includes the transmittance of any optical component within Bob’s measurement device together with the efficiency of his detectors.

^{i}are independent of the actual polarization of the signals σ

_{i,ψ}given by Equation (9) and the basis β used to measure them. We obtain

_{s}= θ

_{Λ}/π, p

_{d}= 1 − p

_{s}, ${\theta}_{{\xi}_{s}}=[0,{\theta}_{\mathrm{\Lambda}}]$, and ${\theta}_{{\xi}_{d}}=[{\theta}_{\mathrm{\Lambda}},\pi ]$.

_{Λ}= π/2, which is the value that maximizes the secret key rate formula given by Equation (12), we have that the gains Q

^{i}can be written as

_{q,z}represents the modified Bessel function of the first kind, and L

_{q,z}denotes the modified Struve function. These functions are defined as [61,62]

^{i}depend on the value of the angle ψ. By symmetry, we can restrict ourselves to evaluate the QBER in only one of the valid regions for ψ. Note that is the same in all of them. For instance, let us consider the case where ψ ∈ [7π/4 + Ω, π/4 − Ω] (which corresponds to the horizontal polarization interval), and let ${E}_{\psi}^{i}$ denote the error rate of a signal σ

_{i,ψ}in that region. This quantity can be written as

_{Λ}= π/2, these expressions can be simplified as

_{±}(ψ) and ϵ

_{±}(ψ) have the form

^{i}are then given by

## C. Estimation procedure

_{1}. Hence, for our purposes, it is enough to obtain a lower bound on the quantities ${p}_{1}^{i}{Y}_{1}+{p}_{0}^{i}{Y}_{0}$ for all i ∈ {s, d}, together with ${e}_{1}^{U}$. For that, we can directly use the results obtained in [38], which we include in this Appendix for completeness. The probabilities ${p}_{n}^{i}$ given by Equation (10) need to satisfy certain conditions that we confirm numerically. In particular, we have that

_{0}given by

_{0}= 1/2. The single-photon error rate e

_{1}can be upper bounded as

_{1}and the background rate Y

_{0}. These quantities are given by

_{Λ}= π/2, we obtain

## D. Asymptotic Passive Decoy-State BB84 Transmitter

_{s}≈ 1, and we assume that Alice and Bob can estimate the relevant parameters Y

_{0}, Y

_{1}, and e

_{1}perfectly. Moreover, we use the channel and detection models introduced in Appendix B. In this situation, it turns out that the yields Y

_{0}and Y

_{1}are given by Y

_{0}= ϵ

_{B}(2 − ϵ

_{B}) and Y

_{1}= 1 − (1 − Y

_{0})(1 − η

_{sys}).

_{1}can be calculated using Equations (28)–(32) with σ

_{i,ψ}= |1

_{ψ}〉〈1

_{ψ}|. After a short calculation, we obtain

## Author Contributions

## Conflicts of Interest

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**Figure 1.**Basic setup of a passive decoy-state BB84 QKD source with polarization encoding using phase-randomized strong coherent pulses. The mean photon number of the signal states ρ

_{i}, with i ∈ {1,…, 4}, can be chosen very high; for instance, ≈ 10

^{8}photons. BS denotes a beamsplitter, PBS represents a polarizing beamsplitter in the ±45° linear polarization basis, such PBS transmits −45° linear polarization and reflects +45° linear polarization [54]. These two orthogonal linear polarizations have creation operators ${a}_{\pm 45\xb0}^{\u2020}=1/\sqrt{2}({\alpha}_{H}^{\u2020}\pm {a}_{V}^{\u2020})$. F is an optical filter, R denotes a polarization rotator changing +45° linear polarization to −45° linear polarization, |vac〉 represents the vacuum state, and t denotes the transmittance of a BS; it satisfies t ≪ 1.

**Figure 2.**(Case A) Graphical representation of the intensity (1 − t)ζ(θ) in mode d

_{3}(see Figure 1) versus the angle θ. Λ represents the threshold value of the classical intensity measurement, θ

_{Λ}is its associated threshold angle, and ξ

_{d}and ξ

_{s}denote the resulting intensity intervals. (Case B) Graphical representation of the valid regions for the angle ψ. These regions are marked in gray. They depend on an acceptance parameter Ω ∈ [0, π/4].

**Figure 3.**Lower bound on the secret key rate R given by Equation (12) in logarithmic scale for the passive transmitter with two intensity settings illustrated in Figure 1 (green line). For simulation purposes, we consider the following experimental parameters: the dark count rate of Bob’s detectors is ϵ

_{B}= 3.2 × 10

^{−7}, the overall transmittance of Bob’s detection apparatus is η

_{B}= 0.045, the loss coefficient of the channel is α = 0.2 dB/km, q = 1/2, and the efficiency of the error correction protocol is f(E

^{i}) = 1.22. We further assume the channel model described in [6,59], where we neglect any misalignment effect. Otherwise, the actual secure distance will be smaller. The inset figure shows the value for the optimized parameters µt (dashed line) and Ω (solid line) in the passive setup. The optimal value for the threshold parameter Λ turns out to be constant with the distance and equal to 2µ(1 − t), i.e., the threshold angle θ

_{Λ}satisfies θ

_{Λ}= π/2. The black line represents a lower bound on R for an active asymptotic decoy-state BB84 system with infinite decoy settings [6], while the red line shows the case of a passive transmitter with infinite intensity intervals ξ

_{i}(see Appendix D).

**Figure 4.**Basic setup of a passive decoy-state BB84 QKD source with phase encoding. The delay introduced by one arm of the interferometer is equal to half the time difference Δt between two consecutive pulses.

© 2015 by the authors; licensee MDPI, Basel, Switzerland This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Curty, M.; Jofre, M.; Pruneri, V.; Mitchell, M.W.
Passive Decoy-State Quantum Key Distribution with Coherent Light. *Entropy* **2015**, *17*, 4064-4082.
https://doi.org/10.3390/e17064064

**AMA Style**

Curty M, Jofre M, Pruneri V, Mitchell MW.
Passive Decoy-State Quantum Key Distribution with Coherent Light. *Entropy*. 2015; 17(6):4064-4082.
https://doi.org/10.3390/e17064064

**Chicago/Turabian Style**

Curty, Marcos, Marc Jofre, Valerio Pruneri, and Morgan W. Mitchell.
2015. "Passive Decoy-State Quantum Key Distribution with Coherent Light" *Entropy* 17, no. 6: 4064-4082.
https://doi.org/10.3390/e17064064