Space division multiplexing can be also implemented by using multicore fibers (MCFs), that is, by using codirectional modes, therefore, we must also consider these optical fibers for QKD. We must stress that there are many configurations of MCFs [31], that is, different groupings of the cores as for example at the vertices of triangle, a square, or even grouping in pairs [32]. These groupings are enough separated to consider that non modal coupling is produced, however, groups of cores can undergo modal coupling and then are processed by MIMO (Multiple-input Multiple-output) technique. The goal of the groups of cores is to enlarge the transmission capacity by increasing the density of cores. In this work we consider grouping in pairs of single-mode cores, and propose a method for autocompensating the modal coupling between the corresponding modes of two cores [32]. Moreover, an integrated device to measure the Bell states is also proposed. We must stress that sometimes the mentioned cores also are far enough apart and therefore only relative phases are compensated, however, the proposed method can also compensate some possible residual coupling, in fact, mechanical and thermal perturbations can induce mode coupling. The matrix describing an arbitrary perturbation *k*(=1,...,*q*) is also given by the matrix *S*_{k} shown in Eq. (17) where now the modes *e*_{1} and *e*_{2} correspond to the fundamental modes of the single-mode cores. Finally, we must stress that polarization is ignored because it is assumed that due to the proximity between cores the polarization changes are common to both cores, that is, a single photon state can be represented as follows

$$ {}\vert L\rangle= c_{1}\vert 1_{1}\rangle+ c_{2}\vert 1_{2}\rangle, \quad \vert 1_{j}\rangle=c_{H}\vert 1_{{jH}}\rangle +c_{V}\vert 1_{{jV}}\rangle, $$

(30)

with *j*=1,2. This indicates that the polarization state is the same for both cores. From a classical point of view it means that the total optical field can be factorized as *e*=(*c*_{1}*e*_{1}+*c*_{2}*e*_{2})*u*, with *u*=(*c*_{H}*u*_{H}+*c*_{V}*u*_{V}) an unpredectible polarization unitary vector but non relevant to spatial coupling. In short, the autocompensating method consists of implementing once more the matrix *M*=*i**σ*_{y} used in collinear modes. However, we must stress that unlike of a single-photon QKD we have two photons coming from Alice and Bob, therefore a polarization autocompensating is also required. Such autocompensation will be obtained as indicated in the case of polarization modes, that is, by a HWP rotated *π*/4.

blackOn the other hand, the way to encode information obeys the following spatial scheme: if the photon travels path 1 (core 1) that corresponds to bit 0 and if it travels path 2 (core 2) that would correspond to bit 1. The result is that a photon can propagate in a superposition of path 1 and path 2. Note that this corresponds to the usual dual-rail logic. Formally, |1_{1}〉→*path* 1 and |1_{2}〉→*path* 2. The experimental implementation parallels that of the previous collinear case, as seen in Fig. 6. There are however a number of changes, especially regarding both the autocompensating process and the Bell-state measurement device (BMD), which in this case is implemented by a optical integrated circuit. Now, each state travels its own path, there being two paths for Alice and two paths for Bob. Each path corresponds by the above encoding to a quantum state. If we label Alice’s paths by *a* and Bob’s paths by *b*, we have, for Alice, paths *a*_{1},*a*_{2} and *b*_{1},*b*_{2} for Bob.

In an analogous fashion with respect to the collinear case, Charlie sends a pair of qubits from his ISG, using either a single-photon source or WCPs (decoy states are also required, as usual, for protection against PNS attacks, if WCPs are used). We need to introduce an OFD device in order to introduce a delay between the fundamental states |1_{1}〉 and |1_{2}〉, for the same reason as before. An example of such a device is shown in Fig. 7. It consists on a pair of photonic lanterns (PL) that extract the modes from the MCF into two parallel SMFs. In one of them, a fiber loop producing a delay *τ* is located, as in the MZI. Another photonic lantern takes the photons and puts them into the two cores of the MCF again. Therefore, after the OFDs we have the biphoton state

$$ \left(\frac{\vert 1_{1} \rangle + \vert 1_{2\tau}\rangle}{\sqrt{2}}\right)_{a}\otimes \left(\frac{\vert 1_{1} \rangle + \vert 1_{2\tau}\rangle}{\sqrt{2}}\right)_{b}. $$

(31)

Light travels along two cores of an MCF to meet Alice’s and Bob’s laboratories. There, local loops contain autocompensating devices and phase modulators. Optical circulators are also required for the local loops of Alice and Bob and to redirect photons to the Bell-state measurement apparatus. Note that circulators do not mix the paths 1 and 2. Although we draw only one circulator, a pair of circulators each one operating in one path is to be understood.

### Bell-state measurement device

Finally, the light that returns to Charlie is directed to a Bell-state measurement device. In this case, such device will be an integrated optical circuit. It is shown in Fig. 8. It is a four-port device; its input fed with the emerging light that comes back to Charlie from Alice’s and Bob’s sites. Synchronous directional couplers (DC) and phase shifters (PS) implement the required transformations. Finally, at the end of each port, a detector is located. The result is a integrated device that is totally analogous to an Innsbruck scheme but on a spatial encoding. Recall that a general synchronous directional coupler implements the following transformation

$$ D(\alpha)= {\left(\begin{array}{cc} \cos\alpha & i\sin\alpha \\ i\sin\alpha& \cos\alpha\\ \end{array}\right)}. $$

(32)

The value of *α*=*κ**L* can be tailored to our choice according to the value of the coupling coefficient *κ* and the coupling length *L*. In this case, the integrated device requires the use of *D*(*π*/4) and *D*(*π*/2) couplers, along with a pair of *ϕ*=−*π*/2 phase shifters.

A first *D*(*π*/2) coupler is required; it couples inputs \(\hat {a}_{o2}\) and \(\hat {b}_{o1}\). Next, two *D*(*π*/4) couplers are needed. They couple, respectively, inputs *a*_{o1} and *b*_{o2} to intermediate outputs *a*2′ and *b*1′, respectively. In terms of the photonic absorption operators, the implemented transformations are

$$a_{a'2}=\hat{a}_{ao1}; \ \ \ a_{a'1}=\hat{a}_{ao2}, $$

$$ a^{\dagger}_{ao1}=\frac{\hat{a}^{\dagger}_{a1}+i\hat{a}^{\dagger}_{b1}}{\sqrt{2}}; \ \ \ \hat{a}^{\dagger}_{a'2}=\frac{i\hat{a}^{\dagger}_{a1}+\hat{a}^{\dagger}_{b1}}{\sqrt{2}}, $$

(33)

$$\hat{a}^{\dagger}_{b'1}=\frac{\hat{a}^{\dagger}_{a2}+i\hat{a}^{\dagger}_{b2}}{\sqrt{2}}; \ \ \ \hat{a}^{\dagger}_{bo2}=\frac{i\hat{a}^{\dagger}_{a2}+\hat{a}^{\dagger}_{b2}}{\sqrt{2}}. $$

As in the previous cases, we will need suitable bases for autocompensation purposes. Therefore, we shall use the following two MUBs

$$\begin{array}{@{}rcl@{}} \mathcal{B}_{D}:\left\{\vert 1_{D}\rangle=\frac{\vert 1_{1}\rangle+\vert 1_{2}\rangle}{\sqrt{2}}, \vert 1_{A}\rangle=\frac{\vert 1_{1}\rangle-\vert 1_{2}\rangle}{\sqrt{2}}\right\}, \end{array} $$

(34)

$$\begin{array}{@{}rcl@{}} \mathcal{B}_{C}:\left\{\vert 1_{L}\rangle=\frac{\vert 1_{1}\rangle+i\vert 1_{2}\rangle}{\sqrt{2}}, \vert 1_{R}\rangle=\frac{\vert 1_{1}\rangle-i\vert 1_{2}\rangle}{\sqrt{2}}\right\}. \end{array} $$

(35)

Moreover, the transformation between absorption operators \(\hat {a}\) in both bases is given by

$$ {}\hat{a}_{\left({D\atop A}\right)}=\frac{1}{\sqrt{2}}\left(\hat{a}_{1}\pm\hat{a}_{2}\right);\quad \hat{a}_{\left({L\atop R}\right)}=\frac{1}{\sqrt{2}}\left(\hat{a}_{1}\mp i\hat{a}_{2}\right). $$

(36)

Let us assume, for instance, that Alice and Bob prepare the input state \(\vert L_{{nf}}\rangle =\vert 1_{{aoD}}1_{{boD}}\rangle =a^{\dag }_{{aoD}}a^{\dag }_{{voD}}\vert 00\rangle \), then by taking into account Eq. (36) we obtain the state

$$ {}\frac{1}{2}\{\vert 1_{ao1}1_{bo1}\rangle + \vert 1_{ao1}1_{bo2}\rangle+ \vert 1_{ao2}1_{bo1}\rangle+\vert 1_{ao2}1_{bo2}\rangle\}. $$

(37)

With this in hand, together with the transformations in Eq. (33), we can make a Bell-state analysis of the device in Fig. 8. Thus, after a long but straightforward calculation, we will find that the output state is a superposition of states which do not produce coincidences and states producing coincidences, that is, |*L*〉=|*L*_{nc}〉+|*L*_{c}〉, with

$$ \vert L_{{nc}}\rangle = i\frac{\sqrt{2}}{4}(\vert 2_{a1}\rangle+ \vert 2_{b1}\rangle+ \vert 2_{a2}\rangle+\vert 2_{b1}\rangle), $$

(38)

$$ \vert L_{c}\rangle =\frac{i}{2}(\vert 1_{a1}1_{a2}\rangle+\vert 1_{b1}1_{b2}\rangle)\equiv i\frac{\sqrt{2}}{2}\vert \Phi^{+}\rangle. $$

(39)

Now, let us assume that Alice and Bob send back to Charlie orthogonal states but in the same (diagonal basis), i.e the input state is

$$ \vert L_{f}\rangle=a^{\dag}_{{aoD}}a^{\dag}_{{boA}}\vert 00\rangle. $$

(40)

In this case, by identical procedure, we will obtain number states |2〉 and the following Bell state |*Ψ*^{−}〉, with probability 1/2:

$$ \vert \Psi^{-}\rangle=\frac{1}{\sqrt{2}}(\vert 1_{b1}1_{a2}\rangle-\vert 1_{a1}1_{b2}\rangle). $$

(41)

Thus, the optical integrated circuit we propose works as a Bell-state analyser, as it should, and therefore the MDI protocol can be implemented. Next, we present the autocompensating method.

### Autocompensating method

As in the collinear case, unpredictable coupling between modes of adjacent cores of an MCF results in a series of perturbations *S*_{k}, each which can be characterized again as a SU(2) matrix with the same form as Eq. (17). It represents, again, the perturbation associated to a short length, in the *z*-direction, of the core the photons travel along. The total matrix is again given by *S*=*S*_{q}*S*_{q−1}...*S*_{2}*S*_{1}. Importantly, these matrices have the same symmetries as the ones of collinear modes. Consequently, we need to find a way to implement the transformation *M*=*i**σ*_{y} on codirectional modes, so that we obtain *S*_{k}*MS*_{k}=*M* for each perturbation *k*. There are a number of ways to implement such transformation. Here, we choose a simple one, consisting of a directional coupler together with phase shifter introducing a phase *π*, placed in a closed circuit, as shown in Fig. 9. Specifically, for the DC, we set *α*=*π*/2, so from Eq. (32) we obtain

$$ D(\kappa)= i {\left(\begin{array}{cc} 0 & 1 \\ 1& 0\\ \end{array}\right)}. $$

(42)

Next, place a *π* phase shifter on the path 2 and thus we obtain the transformation

$$ U(\pi)D(\pi/2)=i {\left(\begin{array}{cc} 0 & 1 \\ -1 & 0 \\ \end{array}\right)}, $$

(43)

which is just what we wanted, up to a harmless global phase *π*/2. Note that the DC implements a swapping operation. This means that it exchanges the physical modes (not their identity), in the sense that the retarded mode will now follow the path which has no delay, and the advanced mode will be delayed. Thus, modes reach Charlie’s Bell-state measurement apparatus at the same time and autocompensation is achieved. This is analogous to what happened in the collinear case.

Finally, a HWP is also included to autocompensate polarization because the perturbations in the optical fiber linking Alice & Charlie and the optical fiber linking Bob & Charlie will be different. Obviously, we can take advantage of this autocompensating polarization technique to use two different QKD channels, that is, collinear modes in H and V polarizations (channels), and therefore doubling the bit rate. We only need to separate the polarization channels (by using, for instance, PBSs) and to place two BMDs in the A-MDI-QKD system.

black

### Secret key rate analysis

We shall now perform a secret key rate analysis by presenting a simulation of the impact on the key rate *R* when perturbations (cross-talk and phase shifting) are considered in the optical link and a comparison with the autocompensated case is shown. We consider cross-talk between modes of multicore optical fibers. This is a very interesting case [3–6] because, among other reasons, the use of this optical link avoids the requirement of alignment between the Alice and Bob bases, which is needed with both collinear modes and polarization modes, in order to avoid misalignment errors that reduce the secure key rates. Moreover, simulation will be made with optical perturbations modelized by a generalized optical error function *E*_{opt}(*L*), where *L* is the propagation distance. We start by recalling that the expression of the lower bound of the key rate *R* of a MDI-QKD protocol, in the case of an infinitely long key and involving the use of decoy states, is given by [12, 33]

$$ R\geq Q^{R}_{11}\left[1-H\left(e^{D}_{11}\right)\right]-Q^{R}_{\mu_{a}\mu_{b}}fH\left(E^{R}_{\mu_{a}\mu_{b}}\right). $$

(44)

Here, \(Q^{R}_{11}\) is the single-photon gain in the rectilinear (*R*) or *Z* basis, \(\mathcal {B}_{R}=\{\lvert 1_{1} \rangle, \lvert 1_{2}\rangle \}\); \(e^{D}_{11}\) is the single-photon bit error in the diagonal or *X* basis \(\mathcal {B}_{D}\); \(Q^{R}_{\mu _{a}\mu _{b}}\) and \(E^{R}_{\mu _{a}\mu _{b}}\) are, respectively, the total gain and total error rate in the rectilinear basis when signal WCP states of mean photon number *μ*_{a} and *μ*_{b} are sent by Alice and Bob, respectively; *f* is the error correction inefficiency and *H* is the binary Shannon entropy function, given by *H*(*a*)=−*a* log2(*a*)−(1−*a*) log2(1−*a*). It is assumed that the WCPs are phase randomized.

Note that for our protocol, because it implements autocompensation, we need to encode the states in the diagonal and circular bases \(\mathcal {B}_{D}\) and \(\mathcal {B}_{C}\), respectively. However, the above key rate formula involves terms in the rectilinear and diagonal bases. We can use this equation, nonetheless, if we change the basis just before detection. This is, we encode in the diagonal and circular bases but we detect in the rectilinear and diagonal bases, thus Eq. (44) is still valid in our setting. The final expression of the key rate is general, and can be used in our case without modification. In fact, our experimental setting is similar to that of [33]. To change from the \(\mathcal {B}_{D}\) and \(\mathcal {B}_{C}\) bases to \(\mathcal {B}_{D}\) and \(\mathcal {B}_{R}\) we can use a simple 3dB synchronous directional coupler, putting *α*=*π*/4 in Eq. (32), so that we obtain, up to global phases, the following map: *D*(*π*/4)|1_{L}〉→|1_{1}〉; *D*(*π*/4)|1_{R}〉→|1_{2}〉; *D*(*π*/4)|1_{D}〉→|1_{D}〉; *D*(*π*/4)|1_{A}〉→|1_{A}〉. We must stress that the detection devices already presented remain unchanged.

The multi-photon terms in the key rate formula are given by [33]

$$ Q^{R}_{\mu_{a}\mu_{b}}=Q_{C}+Q_{E}, $$

(45)

$$ E^{R}_{\mu_{a}\mu_{b}}=\frac{{e_{{opt}}}Q_{C}+(1-{e_{{opt}}})Q_{E}}{Q^{R}_{\mu_{a}\mu_{b}}}, $$

(46)

where *Q*_{C} and *Q*_{E} are given by the following expressions

$$ \begin{aligned} Q_{C}&=2(1-P_{d})^{2}e^{-\mu'/2}\left[1-(1-P_{d})e^{-\eta_{a}\mu_{a}/2}\right]\\ &\quad\times [1-(1-P_{d})e^{-\eta_{b}\mu_{b}/2}], \end{aligned} $$

(47)

$$ Q_{E}=2P_{d}(1-P_{d})^{2}e^{-\mu'/2}\left[I_{0}(2\xi)-(1-P_{d})e^{-\mu'/2}\right], $$

(48)

with *P*_{d} the dark count rate of an individual detector (*Y*_{0}/2,*Y*_{0} the background yield); *μ*^{′}=*η*_{a}*μ*_{a}+*η*_{b}*μ*_{b}, with *η*_{a} and *η*_{b} the transmission efficiency of Alice and Bob’s channels, which we set equal *η*_{a}=*η*_{b} (symmetric scenario). This is given, in turn, by, \(\phantom {\dot {i}\!}\eta _{a,b}=10^{-\alpha _{{att}} L/10}\eta _{d}\eta _{C}\), where *α*_{att} is the attenuation of the fiber link measured in dB/km, *L* the length of the fiber link (Alice/Bob to Charlie) and *η*_{d} and *η*_{C} are the detector’s efficiency and internal transmittance of Charlie’s devices, respectively. Moreover, *ξ* is given by \(\xi =\sqrt {\mu _{a}\mu _{b}\eta _{a}\eta _{b}}/2\) and *I*_{0}(*ξ*) is the modified Bessel function of the first kind. Finally, *e*_{opt} in Eq. (46) is the so-called optical misalignment-error probability [15, 33]. Importantly, it is the error to be modified when optical perturbations are considered along the optical fiber.

On the other hand, in a practical situation, with a finite number of decoy states, \(Q^{R}_{11}\) and \(e^{D}_{11}\) in Eq. (44) need to be estimated from the total gains and error rates. In that case, a good estimation is provided by a decoy setting employing a signal state, a weak decoy and a vacuum. For that, we use the bounds for \(\phantom {\dot {i}\!}Q^{R}_{11}=\mu _{a}\mu _{b}e^{-\mu _{a}\mu _{b}}Y^{R}_{11}\) and \(e^{D}_{11}\) obtained by [34]. First of all, we need to reproduce the total gains and errors in the diagonal basis, from [33], that is,

$$ Q^{D}_{\mu_{a}\mu_{b}}=2\gamma^{2}[1+2\gamma^{2}-4\gamma I_{0}(\xi)+I_{0}(2\xi)], $$

(49)

$$ Q^{D}_{\mu_{a}\mu_{b}}E^{D}_{\mu_{a}\mu_{b}}=e_{0} Q^{D}_{\mu_{a}\mu_{b}}-2(e_{0}-e_{{opt}})\gamma^{2}[I_{0}(2\xi)-1], $$

(50)

where \(\phantom {\dot {i}\!}\gamma =(1-P_{d})^{-\mu ^{\prime }/4}\) and *e*_{0}=1/2 is the dark count error, that is, the random dark count in a detector which is not expected to fire. Equations (45) to (50) can be particularized for any intensity setting by simply substituting the intensities’ values. Now, say that Alice sends a signal state *μ*_{a}, a weak decoy *ν*_{a} and a vacuum. Bob sends *μ*_{b},*ν*_{b} and a vacuum. Following [34], define *m*= min(*a,b,c*), where

$$ \begin{aligned} a&=\frac{\mu_{a} \mu_{b}^{2}-\nu_{a}\nu_{b}^{2}}{\mu_{a}\nu_{b}^{2}+\nu_{a} \mu_{b}^{2}},\\ b&=\frac{\mu_{a}^{2} \mu_{b}-\nu_{a}^{2}\nu_{b}}{\mu_{a}^{2}\nu_{b}+\nu_{a}^{2} \mu_{b}},\\ c&=\frac{\mu_{a}^{2} \mu_{b}^{2}-\nu_{a}^{2}\nu_{b}^{2}}{\mu_{a}^{2}\nu_{b}^{2}+\nu_{a}^{2} \mu_{b}^{2}}. \end{aligned} $$

(51)

Furthermore, with *β*={*R,D*}, the following parameters are defined

$$ \begin{aligned} g^{\beta}_{1}&=e^{\mu_{b}}Q^{\beta}_{0\mu_{b}}+e^{\mu_{a}}Q^{\beta}_{\mu_{a}0}-e^{\nu_{b}}Q^{\beta}_{0\nu_{b}}-e^{\nu_{a}}Q^{\beta}_{\nu_{a}0},\\ g^{\beta}_{2}&=m\left(e^{\mu_{a}+\nu_{b}}Q^{\beta}_{\mu_{a}\nu_{b}}-e^{\nu_{b}}Q^{\beta}_{0\nu_{b}}-e^{\mu_{a}}Q^{\beta}_{\mu_{a}0}+Q^{\beta}_{00}\right),\\ g^{\beta}_{3}&=m\left(e^{\nu_{a}+\mu_{b}}Q^{\beta}_{\nu_{a}\mu_{b}}-e^{\mu_{b}}Q^{\beta}_{0\mu_{b}}-e^{\nu_{a}}Q^{\beta}_{\nu_{a}0}+Q^{\beta}_{00}\right),\\ g^{\beta}_{4}&=e^{\nu_{b}}Q^{\beta}_{0\nu_{b}}E^{\beta}_{0\nu_{b}}+e^{\nu_{a}}Q^{\beta}_{\nu_{a}0}E^{\beta}_{\nu_{a}0}-Q^{\beta}_{00}E^{\beta}_{00}. \end{aligned} $$

(52)

With these in hand, the lower bound of \(Y^{\beta }_{11}\) and upper bound of \(e^{\beta }_{11}\) are given by [34]

$$ Y^{\beta}_{11}\geq\frac{g^{\beta}_{1}+g^{\beta}_{2}+g^{\beta}_{3}-e^{\mu_{a}+\mu_{b}}Q^{\beta}_{\mu_{a}\mu_{b}}+e^{\nu_{a}+\nu_{b}}Q^{\beta}_{\nu_{a}\nu_{b}}}{\nu_{a}\nu_{b}-\mu_{a}\mu_{b}+m \mu_{a}\nu_{b}+ m\nu_{a}\mu_{b}}, $$

(53)

$$ e^{\beta}_{11}\leq\frac{e^{\nu_{a}+\nu_{b}}Q^{\beta}_{\nu_{a}\nu_{b}}E^{\beta}_{\nu_{a}\nu_{b}}-g^{\beta}_{4}}{\nu_{a}\nu_{b}Y^{\beta}_{11}}. $$

(54)

Next, by particularizing for *β*=*R* in \(Y^{\beta }_{11}\) and for *β*=*D* in \(e^{\beta }_{11}\) one can obtain the remaining parameters of Eq. (44).

As commented above, in order to take into account the perturbations on the fiber producing undesired coupling (cross-talk) we present the following model for the optical error, already introduced in [35] for a high-dimensional QKD analog of the BB84 protocol. The expression of the optical error is modelled as follows

$$ e_{{opt}}\rightarrow E_{{opt}}=e_{{opt}}+\left(\frac{1}{2}-e_{{opt}}\right)\left(1-e^{-\alpha_{{opt}}L}\right), $$

(55)

where *α*_{opt} is the perturbation coefficient along the fiber. Note that this error increases monotonically with *L* and it is reduced to *e*_{opt} when *α*_{opt}=0, and it goes to 1/2 for *α*_{opt}≫1, that is, for large perturbations all states have the same probability 1/2 to be detected (two states in each base) and therefore the error will be also equal to 1/2 as it also occurs for *e*_{0}. In short, *E*_{opt} generalizes the misalignment error *e*_{opt} above.

To perform a numerical simulation of the key rate, given by Eq. (44), we use the equations above, substituting *e*_{opt} by *E*_{opt}, for a series of realistic values of *α*_{opt}, including *α*_{opt}=0, which is the case when the perturbations have been successfully autocompensated. Thus, we will use values of *α*_{opt} in the interval (0.5·10^{−3},2·10^{−3}) which are compatible with those ones found in the technical literature about modal cross-talking due to perturbations in optical fibers. For instance, for *α*_{opt}=2·10^{−3}km^{−1} we obtain an optical error about 1.0·10^{−3} which corresponds approximately to -28 dB. This value is a realistic one because both spatial and polarization mode cross-talking in optical fibers can take values around -30 dB or even larger [36, 37]. In short, cross-talking provides an estimation of errors due to the loss of information (bits) carried by an optical mode.

As one can see in Fig. 10, the range at which we can transmit secure information depends critically on the value of *α*_{opt}. The values for the parameters used in the simulation were taken across the relevant literature [15, 30, 33, 34] and are detailed in Fig. 10. Moreover, we have imposed the optimal conditions *μ*_{a}*η*_{a}=*μ*_{b}*η*_{b} and *ν*_{a}=*ν*_{b} [15, 33]. The results under full autocompensation gives a secure key rate up to a distance about 130 km between Alice (Bob) and Charlie, therefore 260 km between Alice and Bob. If we consider a perturbation coefficient of *α*_{opt}=0.5·10^{−3} a reduction of 38% in such a distance is obtained, that is, a secure key rate distance about 160 km between Alice and Bob. In the case *α*_{opt}=2.0·10^{−3} a dramatic reduction of the secure key rate distance is obtained. Therefore, these results show that the optical perturbations in the links for MDI protocol are an important source of errors what contributes to a remarkable reduction of the secure key rate distance, which is more important than in the case of protocols based on a single photon, like, for example, the BB84 one [35].