## Experimental implementation

### QKD protocol

The protocol implemented in the current experiment is the well known BB84 (with decoy states). By using the spatial dimension as a degree of freedom, instead of the standard way of using polarization, we encode the qubits on multiple cores of the MCF in such a way that for every two cores (cores A and B and cores C and D and so on), two mutually unbiased bases can be generated. In particular, for cores A and B, the basis $${{\mathscr{X}}}_{1}$$ is defined as $$(|A\rangle ;|B\rangle )$$ and basis $${{\mathscr{Z}}}_{1}$$ as $$(|A+B\rangle ;|A-B\rangle )$$. Similarly for cores C and D the states $$\{|C\rangle ,|D\rangle \}\in {{\mathscr{X}}}_{2}$$ and $$\{|C+D\rangle ,|C-D\rangle \}\in {{\mathscr{Z}}}_{2}$$. The final secret key rate is established using3:

$$R\,\ge \,{I}_{AB}-min({I}_{AE},{I}_{BE})$$
(1)

I AB represents the classical mutual information between Alice and Bob (I XY  = H(X) − H(X|Y)), with the marginal entropy is defined as $$H(X)={\sum }_{x\in X}\,p(x)logp(x)$$. The right term of equation (1) min(I AE and $${I}_{BE}$$), is related to the quantum mutual information between Alice and Eve or Bob and Eve. Note that using the same chip structure a slightly different implementation is possible, i.e. asymmetric BB84 with decoy-states18,19. This protocol relies on two mutually unbiased bases, but does not use an equal probability for all quantum states. In other words, one of the two bases ($${\mathscr{X}}$$), is chosen more often than the other ($${\mathscr{Z}}$$) $${p}_{{\mathscr{X}}}\ne {p}_{{\mathscr{Z}}}$$. In this way $${\mathscr{X}}$$ is used for the key generation process and $${\mathscr{Z}}$$ for security check. It follows that this protocol is more efficient compared to the standard BB84 (efficiency of 50%), and it allows a higher final secret key rate. In the current experiment we selected an equal probability both for the bases choice and for the state preparation, so the overall efficiency.

### Decoy-state weak coherent pulse generation

Most practical QKD systems today are implemented with weak coherent pulses (WCP), generated by an attenuated laser. This scheme however, is not completely secure against particular kinds of attack, like photon-number splitting (PNS). In PNS attack, Eve blocks and discards all the single photon pulses while she only measures the multi-photon ones after the information reconciliation process. In this way, Bob and Eve measure the same quantum state, and at the end of the process Eve shares the same key. The decoy-state technique was introduced in order to overcome this problem. A controlled real-time fluctuation of the mean photon number per pulse (μ) is used, in order to ensure the complete security of the final secret key. This technique is implemented in our experiment, where Alice’s silicon chip, constituted by multiple Mach-Zehnder interferometers (MZIs), allows a complete freedom in terms of photon per pulse. By tuning the VOA 1 (variable optical attenuator) and the MZI 00 (the first index 0 represents the level of the MZI starting from left, while the second index is related to the number of the cores of the fiber) with a specific voltage, different values of μ can be obtained (see Fig. 1). In Table 1 we reported all the different cases for a 2-keys example. The MZI operates like a tunable ratio (transmittance/ reflectance) beam-splitter where Alice randomly decides which values to use. In such a way, it is possible to create two independent quantum channels, which will generate two quantum keys. The example can be easily extended to a generic case where N cores generate N/2 different keys.

### Generation of the quantum states

The quantum states used in the current implementation are based on spatial encoding, exploiting different cores of a MCF. As shown in Fig. 1, a 1550 nm continuous wave (CW) laser, has its light carved out to pulses by an intensity modulator at 5 kHz repetition rate and pulse width of 10 ns, which is coupled through a vertical coupler into the transmitter silicon chip (Alice). The quantum states are randomly prepared, by tuning the various MZIs, with a pseudorandom binary sequence (PRBS) sequence created by an FPGA board. Two PRBS seeds were used in order to create two parallel independent keys. In particular, by applying a different voltage on the MZI in Alice chip is possible to control the outputs of the integrated interferometers. After a first characterization of Alice’s chip, we fixed a 0 V level corresponding to having light only in one output (upper or lower). Consequently, a V π V value determines a reverse exit and a value of V π /2 V represents the fifty-fifty case with light in both outputs.

Moreover, a real-time individual decoy states value is prepared for each pulse. Different voltages applied to the MZI00 correspond to a specific decoy value, as reported in Table 1. Subsequently to the preparation of the quantum states, we used a grating coupler array to couple from the silicon integrated circuit to a 7-cores fiber20,21.

By exploiting this technique, we obtained a negligible cross talk between cores, around −30 dB, and stable transmission can be achieved. The insertion and coupling losses attributed to Alice’s chip are around 15 dB. In this way, we created two independent quantum channels based on the principle of space division multiplexing.

### Detection

Once the quantum states are created and sent through the MCF, Bob measures the states in order to extract the quantum keys. In the experimental setup, two independent quantum keys, $${k}_{1}$$ and k 2, as reported in Fig. 1, are generated and the keys can be extracted by creating interference between the cores at the output22,23. In particular, tuning MZI11 to MZI1N , on Bob’s side, it is possible to project the quantum states in different bases. Separate MZIs are used to measure in the mutually unbiased bases. In this way the randomness is maintained on the measurement side. The other MZIs (MZI01 to MZI0N ), present on Bob’s chip are used for phase stabilization between cores. In Fig. 2 we show the tomography of the two independent MUBs measured with weak laser pulses, repetition rate of 10 kHz, and average mean photons number of 0.4. By using the classical definition of fidelity (F(x, y) = ∑ i (p i q i )1/2 with x and y random variables and q i and p i vectors of probability distribution) we measured 93% and 96%. Another important parameter on the detection part is represented by the losses on the Bob’s side. The insertion loss attributed to Bob’s chip are measured to be around 8 dB (from the output of the MCF, just before the facet, to the output of the chip). This loss can be further decreased in future chips by introducing an Al mirror below the grating area24. The four different outputs are coupled using a grating coupler array to four InGaAs single photon detectors, two ID230 and two ID220 respectively. In Fig. 3 we report the measured QBER for the two independent keys. Stable and low QBER well below the coherent attack limit are obtained for more than 12 minutes. The two plots show the different independent keys extracted in the experiment.

### Secret key rate

After the measurement process it is possible to define a bound on the final secret key rate. This rate, given in Equation (1), depends on the strategy of the eavesdropper. We here included the case of collective attacks (CAs), where Eve can store the quantum states in her quantum memories and postpone the measurement till same future time. Alice and Bob discard the unmatched bases measurements, and subsequently perform error correction and privacy amplification, to extract the final key rate. In the case of decoy-state quantum key distribution it is possible to derive the following equation for the secret key rate:

$${R}_{sk}\ge \tfrac{1}{2}\{-{Q}_{u}f({E}_{u}){h}_{2}({E}_{u})+{Q}_{1}\mathrm{[1}-{h}_{2}({e}_{1})]\}$$
(2)

Here 1/2 is the probability related to the bases choice, h 2 is the binary Shannon information function, u denotes the intensity of the signal states, Q u is the gain of the signal states, E u is the overall quantum bit error rate (QBER), e 1 is the error rate of the single-photon states and f(x) is the bidirectional error correction efficiency, usually upper bounded with the value of 1.22. The parameter Q u and E u can be measured directly from the experiment, while $${Q}_{1}$$ and e 1 can be estimated. Following the approach reported in Ma et al.4, it is possible to derive a secret key rate bound. To be noted that in a practical implementation of this system, a different bound including the statistical fluctuation can be used5. In Fig. 4, a real time measurement of the decoy state gain is reported. An average value of $${Q}_{{\mu }_{1}}\,=\,3.32\cdot {10}^{-2}\pm 1.2\cdot {10}^{-3}$$ and $${Q}_{{\mu }_{2}}\mathrm{=1.67}\cdot {10}^{-2}\pm 0.1\cdot {10}^{-3}$$ are measured on Bob’s side for the two independent keys, corresponding to a secret key rate generation of 113 bit/s for k 1 and 60 for k 2. Note that for a complete QKD system realization, where Eve cannot steal any information from the link, the gain value should be measured on Alice’s side. However, in the current chip realization an extra output to do this measurement was not available. Nonetheless, in order to prove the real-time decoy state technique, we characterized the chip before the transmission over the multicore fiber channel in order to estimate the expected values.