Random numbers are important resources in scientific and practical applications. Classical random number generators deny the existence of unpredictability, which cannot provide secure randomness. In contrast, quantum random number generators (QRNGs) can generate genuine randomness from the inherent indeterminacy of quantum mechanics,1,2 which have been applied in various quantum information processing tasks.3
In the last decades, the generation of quantum random numbers has been extensively studied. Various high-speed and real-time QRNGs have been developed6
For practical semi-DI QRNGs, security, generation rate, and practicality are highly desirable in applications. Particularly, any deviation of the realistic source from its theoretical modeling may affect the security and generation rate of true randomness. Source-DI QRNGs generating true randomness from an untrusted source provided convenient and characterized measurement devices, offer distinct advantages in semi-DI QRNGs, and have been extensively studied.
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One kind of approach is based on measurement of the vacuum noise via homodyne detection.23,29
2 Source-DI QRNG Protocol
In our protocol, we suppose an untrusted source produces a tripartite state with the reduced state , where and are distributed to two noncommunicating observers named Alice and Bob, respectively, and is held by the underlying eavesdropper Eve as a quantum memory or considered as the environment. In the ideal case, is a pure time–energy entangled photon pair state generated via SPDC. Here we suppose that the SPDC source is pumped by a pulsed laser with a center frequency of and a coherence time of and that the generated photon pairs have a correlation time of determined by phase-matching bandwidth. The ideal state can be written in the time and frequency domains, respectively, as follows:
Alice and Bob both have two trusted positive operator-valued measures (POVMs), denoted by and with and . The measurement is the direct photon arrival time detection, expressed as
However, in practice, we perform measurements and in a range from to , where is the frame size (dimensionality); thus the null measurements and can be defined when the photon arrives before or after the range, which limits the characterization of entanglement in high-dimensional quantum systems.39 The null measurements can be expressed by
Then the refined POVMs can be written as and .
Alice and Bob choose two measurements, and , separately, which are switched through a classical random signal with probabilities and , respectively. Before extracting random numbers, Alice and Bob record the joint outcomes of the measurements to estimate the detection precision of the system. Then the outcomes of measurement in Alice are recorded as the raw random bits, whereas the joint outcomes of the measurements for Alice and Bob are utilized to certify the entanglement of source and estimate the amount of randomness.
In the process of certification for the source, the NDC35 is available as a nonlocal test of the time–energy entanglement, where the dispersion effect can be nonlocally canceled when two time–energy entangled photons propagate in two media with equal magnitudes and opposite dispersion signs, respectively. We define the code distance associated with the outcomes of measurement as a testing value given by38
Since the source device is untrusted, the input state might be controlled by an eavesdropper, Eve, who can obtain the side information through system . The amount of genuine randomness that can be extracted from Alice in measurement is quantified by the conditional quantum min-entropy40 defined as , where is the maximum probability that Eve guesses correctly the outcome of conditional on her side information. In previous works, the lower bound of conditional quantum min-entropy can be given by exploiting the EUR.41,42
In practical implementations, the finite measurement range problem will significantly compromise the evaluation of secure min-entropy. To further improve security, we explore the extractable randomness lower bound with the modified EUR39 based on smooth entropy by taking into account the finite measurement range. The -smooth conditional min- and max-entropies are defined as
Additionally, in Eq. (12) is the maximum overlap for the POVMs and , excluding the null measurement POVM elements, satisfying39
Finally, we extract the secure random bits from the raw random bits by the Toeplitz-hashing extractor and claim that our QRNG scheme successfully generates a string of genuine random bits if all statistical tests are passed.
3 Experimental Demonstration
The experimental setup comprises an entanglement source and measurement devices, as shown in Fig. 1. The pump light is a pulsed laser with a repetition rate of 10 MHz and a measured coherence time of 2.1 ns, which is extracted from a continuous-wave laser at 774.9 nm through a lithium niobate electro-optic modulator. It is adjusted to horizontal polarization by a polarization controller, then coupled into a 5-cm Ti-diffused periodically poled lithium niobate (Ti:PPLN) waveguide with a poling period of . The time–energy entangled photon pairs are produced via the type-II SPDC process. After blocking out the pump by a long-pass filter and a 3-nm bandpass filter centered at 1550 nm, the output orthogonally polarized entangled photon pairs are spatially separated by a polarization beam splitter (PBS) and distributed to Alice and Bob, respectively. The wavelength-degenerate photon pairs are centered at 1549.8 nm with 0.7 nm full width at half-maximum (FWHM). The overall detection efficiencies are 20.5% for the photon to Alice and 20% for the photon to Bob, respectively. When the pump power coupled into the waveguide is 1 mW, the single-photon counting rates measured by superconducting nanowire single-photon detectors (SNSPDs) at Alice and Bob are 5 and 4.85 MHz, respectively, with the dark counting rate observed around 500 Hz and thus are ignored. The two-photon coincidence counting rate obtained by the time-to-digital converter (TDC) (PicoHarp-300) is 1 MHz. Thus the proportion of genuine entangled photons in Alice’s detection can be estimated to be 97%.
Figure 1.Experimental setup of the source-DI QRNG. (a) Entanglement source: the time–energy entangled photon pairs are generated from the Ti:PPLN waveguide pumped by a pulsed laser with a duration of 5 ns, which are separated by a PBS. (b) Measurement device: photons are passively selected for measurement
Alice and Bob both randomly perform measurement or by a passive beam splitter, i.e., in protocol. Explicitly, the measurement is implemented by directly measuring the arrival time at the SNSPD, while for the measurement , arrival time detection is performed after the photons to Alice (Bob) propagate through a dispersion module composed of an optical circulator and a chirped (antichirped) Bragg grating with a GVD coefficient of (). The arrival time is detected by the SNSPDs, then recorded by the TDCs with the total time jitters estimated approximately as (1 standard deviation). The outcome rate of measurement in Alice is .
To explore the performance of the source and certify the security of the scheme, we plot the coincidence curves of four combinations for two observers’ measurements, as illustrated in Fig. 2. If Alice and Bob both make measurement , the FWHM of the coincidence peak is , as shown in Fig. 2(a), and thus the detection precision is calculated to be based on the assumption that the resolution of all detectors is identical. If the measurements performed by Alice and Bob are different, coincidence peaks are broadened to 750 ps in Fig. 2(b) and 760 ps in Fig. 2(c) due to the dispersion effect. The slight difference between two peaks is caused by the slight difference in magnitude of GVD coefficients in Alice and Bob. If two observers both choose measurement , as shown in Fig. 2(d), the peak recovers with a narrow FWHM of , as shown in Fig. 2(d), corresponding to [ for Gaussian function] due to the NDC effect. In this case, the testing value is calculated to be 0.64 according to Eq. (9), which is much smaller than the classical bound (see Appendix C).
Figure 2.Photon coincidence counts (CCs) recorded for four measurement combinations of two observers (denoted as
The preset value is set to be 0.64, since it is the upper bound in the vast majority of the measurement runs in our experiment. If from the experimentally observed results, the protocol is passed, implying that we can evaluate and extract true randomness from the raw random bits to generate genuine random numbers.
4 Randomness Evaluation and Extraction
From the above results, we could calculate the randomness from the raw random bits according to Eqs. (12)–(20). The null probabilities and can be obtained with and in our experiment, where is the error function.44 The statistical fluctuation defined in Eq. (20) is obtained by setting the smooth entropy parameter , where the total count is deduced by the count rate and the cumulative time as .
We plot the smooth min-entropy with respect to and , as shown in Fig. 3. It can be seen that increases with , while for a given , with the increasing , first keeps growing due to increasing measurement range and then declines for larger statistical fluctuation, where the maximum value can be obtained by optimizing . The maximal entropy values are obtained to be 0.778, 0.877, 0.903, and 0.913 for four processing units with frame size , 246, 250, and 256, respectively.
Figure 3.Smooth entropy
As a trade-off between the entropy bound and practicality, the processing unit is set as , corresponding to the highest min-conditional entropy of 0.917 bit per count with , , , and . Considering the proportion of genuine entangled photons of the SPDC is measured to be 97%, we can extract 0.900-bit genuine randomness per -bit sample. Hence, we generate a Toeplitz matrix with a scale of to extract genuine random numbers. As the outcomes rate is , the final generation rate of random numbers is 4 Mbps.
To test the quality of random numbers, we perform an autocorrelation coefficient test between the raw and final random data, where the raw data and final random data satisfy the Gaussian distribution and uniform distribution, respectively. As shown in Fig. 4, the final autocorrelation coefficients are below 0.001 within the 200-bit delay, which are significantly lower than the raw data. Furthermore, we perform a standard NIST test suite using 1000 samples of 1 Mb; the significant level is set as . The NIST test is passed if values are higher than 0.01 and the proportion value within the confidence interval of for all tests. As shown in Fig. 5, the random bits in our scheme pass all 15 tests.
Figure 4.Autocorrelation coefficients of raw random data and final random data.
Figure 5.Results of NIST statistical test suite.
5 Conclusions and Discussions
In conclusion, we have proposed and experimentally demonstrated a scheme for a source-DI QRNG, where the random bits are generated by measuring the arrival time of single photons from an untrusted time–energy entangled photon pair source. The NDC effect is employed to testify the entanglement source and thus guarantee the security of true random number acquisition. With a high-quality PPLN waveguide SPDC source, we realized a fast generation of true random numbers with a generation rate of 4 Mbps, which were extracted by utilizing the modified EUR. In Table 1, we list several semi-DI QRNGs as a comparison. It shows that our work achieves a trade-off among security, speed, and practicality.
|Refs.||Uncharacterized Source||Uncharacterized Measurement||Finite-size Analysis||Finite Measurement Ranges Considered||Generation Rate|
|This work||√||×||√||√||4 Mbps|
Table 1. Features of our protocol as compared to the features of existing semi-DI QRNG protocols.
The generation rate of our protocol can be further increased to Gbps provided we use state-of-the-art single-photon detectors. For instance, the single-photon detector45 with a temporal resolution of 29 ps could theoretically achieve optimal ; combining with its maximum count rate of 2 GHz, the random number generation rate can reach 5.16 Gbps. Moreover, the source-DI QRNG we realized is based on the PPLN waveguide SPDC source, which may be further developed to be an integrated chip-scale device by exploring on-chip photon generation, manipulation, and detection techniques. We hope our approach can stimulate more such investigations.
Furthermore, our scheme provides a secure certification for quantum information and quantum communication tasks with an untrusted source based on dispersion cancellation. Recently, the work on the QKD protocol where the source is trusted but imperfect was proposed.36 Our approach offers a way to certify the untrusted source via dispersion cancellation for this protocol, which enables us to access the source-DI QKD tasks.
6 Appendix A: The Definition of Testing Value
In this section, we provide the proof that the testing value defined in Eq. (9) as the code distance for systems and in basis can be used to certify the time–energy entanglement for the ideal state in Eq. (1).
Let us consider the case that systems and are two separable photons or classical pulses. The spectrum and temporal functions of the photon can be written as, respectively,
The joint detection probability that Alice’s detector clicks at time and Bob’s clicks at time simultaneously is , and the overall probability of detecting two photons at a time lag can be calculated as
It has been proved that the origin correlation time and standard deviation in the spectrum intensity of the sum of frequency for two separable photons satisfy the following inequality:46,47
A violation of this inequality implies the presence of entanglement, which is able to be used as a witness for the certification of time–energy entanglement. We denote the right-hand side of Eq. (28) as the classical bound .
Let us now consider the case that the source device distributes the entangled photon pairs with the state given by Eq. (1) to Alice and Bob, and they both choose measurement , i.e., the arrival time after two photons traveled through the dispersive elements. The joint detection rate between two detectors is proportional to the Glauber second-order correlation function,
Then the correlation time of outcomes in measurement can be calculated as
In the limit of large coherence time , the testing value reduces to
7 Appendix B: The Maximum Overlap of
We recall the measurements and , which can be expressed as
The associated observables of and can be, respectively, written as
Based on the derivation in Ref. 38, the observable can be further simplified as
8 Appendix C: The Classical Bound of Experimental Testing Value
In our source-DI QRNG framework, the security of the scheme relies on the observation of in experiment. To certify the entanglement, we need to calculate the classical bound of testing value in our experiment.
Taking into account the time jitter of our detection systems in practice, the correlation time in Eq. (27) can be rewritten in a modified form,
Recall that we measured the coincidence distribution and obtained with in Fig. 2(a), i.e., . Then combining the GVD coefficient in our system, we can calculate the modified correlation time ps and the corresponding classical bound .
Ji-Ning Zhang is now a PhD student at the School of Physics of Nanjing University. Her current research interests include quantum optics and quantum information.
Ran Yang is now a PhD student at the School of Physics of Nanjing University. His current research interests include quantum optics and quantum tomography.
Xinhui Li obtained her PhD in cryptography from Beijing University of Posts and Telecommunications in 2020. She was awarded a scholarship from the State Scholarship Fund which was selected through a rigid academic evaluation process organized by the China Scholarship Council to pursue her studies at the National University of Singapore from August 2017 to August 2018. She is now a postdoctoral fellow at the School of Physics of Nanjing University. She is currently working on the security of quantum information processing and the foundations of quantum correlations.
Chang-Wei Sun obtained his PhD from the School of Physics at Nanjing University in 2021. He works on nonlinear optics and quantum optics.
Yi-Chen Liu received his PhD from the School of Physics at Nanjing University in 2021. In 2021, he joined as a senior researcher at Qingdao University of Technology. His current research interests include nonlinear optics and quantum optics.
Ying Wei is now a PhD student at the School of Physics of Nanjing University. His current research interests include quantum simulation and quantum tomography.
Jia-Chen Duan is now a PhD student at the School of Physics of Nanjing University. His current research interests include nonlinear optics and integrated optical quantum technologies.
Zhenda Xie obtained his PhD from Nanjing University in 2011. From 2011 to 2016, he joined as a postdoctoral fellow at Columbia University in the City of New York and a research fellow at University of California, Los Angeles, respectively. He is now a professor at the School of Electronic Science and Engineering of Nanjing University. He is currently working on solid-state laser technology, nonlinear optics, and quantum optics.
Yan-Xiao Gong obtained his PhD in optics from the University of Science and Technology of China in 2009. In 2009, he joined as a postdoctoral fellow at Nanjing University. From 2011 to 2017, he worked in the Department of Physics of the Southeast University. He is now a professor at the School of Physics of Nanjing University. He is currently working on nonlinear optics, quantum optics and integrated optical quantum technologies, and quantum information.
Shi-Ning Zhu obtained his PhD from Nanjing University in 1996 and is the group leader of Dielectric Superlattice Laboratory at Nanjing University. His research interests include condensed matter optics, quasiphase matching physics and nonlinear optics, optoelectronic functional materials, quantum optics, and metamaterials.
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