The world of cybersecurity is developing exceedingly fast, and data encrypted by the traditional encryption methods are facing the danger of being breaked. Random numbers have always been a necessary part of every encryption system: higher quality in the random numbers implies higher security in the encrypted data. Random number generators exploits the intrinsic randomness of quantum physics. Semi-device independent (semi-DI) quantum random number generators (QRNG) offer an excellent trade-off between security, ease-of-implementation, and performance, making them a promising candidate for applications. Semi-DI QRNG certify the quantum randomness of the generated numbers by few assumptions and manageable experimental characterization of the used devices.

In this work, we illustrate a many-outcomes scheme with the binary phase-shift keying (BPSK) for a semi-DI protocol based on the energy assumption. The protocol is based on two untrusted devices, the preparation and measurement, and a single assumption corresponding to an upper bound on the prepared state's energy. A general scheme of this protocol is shown in the figure above. We showed that the min-entropy (namely the number of certified random bits) that can be obtained by the d-outcomes system outperforms the standard scheme (binary-outcomes). Furthermore, we compared the results of two well-known measurement schemes, homodyne and heterodyne detection. Lastly, considering the experimental imperfections, we discussed the practical feasibility of the d-outcome design. Our work shows that by using the same system, the improvement of post-processing scheme allows to certify more random numbers.

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