• Advanced Photonics
  • Vol. 5, Issue 3, 036003 (2023)
Ji-Ning Zhang1、2、3, Ran Yang1、2、3, Xinhui Li1、2、3、*, Chang-Wei Sun1、2、3, Yi-Chen Liu1、3、4, Ying Wei1、2、3, Jia-Chen Duan1、2、3, Zhenda Xie1、3、5, Yan-Xiao Gong1、2、3、6、*, and Shi-Ning Zhu1、2、3
Author Affiliations
  • 1Nanjing University, National Laboratory of Solid State Microstructures, Nanjing, China
  • 2Nanjing University, School of Physics, Nanjing, China
  • 3Nanjing University, Collaborative Innovation Center of Advanced Microstructures, Nanjing, China
  • 4Qingdao University of Technology, School of Science, Qingdao, China
  • 5Nanjing University, School of Electronic Science and Engineering, Nanjing, China
  • 6Hefei National Laboratory, Hefei, China
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    DOI: 10.1117/1.AP.5.3.036003 Cite this Article Set citation alerts
    Ji-Ning Zhang, Ran Yang, Xinhui Li, Chang-Wei Sun, Yi-Chen Liu, Ying Wei, Jia-Chen Duan, Zhenda Xie, Yan-Xiao Gong, Shi-Ning Zhu. Realization of a source-device-independent quantum random number generator secured by nonlocal dispersion cancellation[J]. Advanced Photonics, 2023, 5(3): 036003 Copy Citation Text show less


    Quantum random number generators (QRNGs) can provide genuine randomness by exploiting the intrinsic probabilistic nature of quantum mechanics, which play important roles in many applications. However, the true randomness acquisition could be subjected to attacks from untrusted devices involved or their deviations from the theoretical modeling in real-life implementation. We propose and experimentally demonstrate a source-device-independent QRNG, which enables one to access true random bits with an untrusted source device. The random bits are generated by measuring the arrival time of either photon of the time–energy entangled photon pairs produced from spontaneous parametric downconversion, where the entanglement is testified through the observation of nonlocal dispersion cancellation. In experiment, we extract a generation rate of 4 Mbps by a modified entropic uncertainty relation, which can be improved to gigabits per second by using advanced single-photon detectors. Our approach provides a promising candidate for QRNGs with no characterization or error-prone source devices in practice.

    1 Introduction

    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.35

    In the last decades, the generation of quantum random numbers has been extensively studied. Various high-speed and real-time QRNGs have been developed69 and started to become commercial.10,11 However, these QRNGs can only extract true randomness based on the strong assumption that the source and measurement devices are trusted. The device-independent QRNG (DI QRNG)4,12,13 is able to access true randomness without any assumptions on the source and measurement devices, but it requires a loophole-free Bell test, resulting in great challenges in implementation and low efficiency. An alternative technique is semi-DI QRNG, where high speed and low-cost information-provable randomness can be generated based on a few justifiable assumptions on the system operation and its critical components, such as trusted sources,1417 the characterized measurement settings,1824 assumptions on the indistinguishability, or dimension of the input states.2528

    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.

    One kind of approach is based on measurement of the vacuum noise via homodyne detection.23,2931 Benefiting from the fast detection speed, such a technique has achieved a random number generation rate as high as gigabits per second (Gbps); however, the homodyne detection requires a well modeled and calibrated local oscillator. In contrast, the single-photon detection technique, despite the drawback on detection speed, has the merit of easy operation and simple structure. With such a technique, source-DI QRNGs have also been reported18,20 based on an assumption of the squashing model32 in the detection devices. In this paper, we propose and experimentally demonstrate a secure and fast source-DI QRNG based on single-photon detection and entangled photons. The random bits are generated via the measurement of photon arrival time that is beneficial for producing high-dimensional QRNGs.33,34 In our scheme, we use either photon of time–energy entangled photon pairs produced from spontaneous parametric downconversion (SPDC) as the entropy source. The security of our scheme relies on the observation of nonlocal dispersion cancellation (NDC),35 which has been applied to guarantee the security of quantum key distribution tasks.3638 Moreover, we employ a modified entropic uncertainty relation (EUR)39 to quantify the randomness to improve security. The experiment results show that the genuine quantum randomness can be extracted at a rate of 4 Mbps (megabits per second), which could reach the level of Gbps if using the advanced single-photon detectors with faster detection speed and lower temporal resolution.

    2 Source-DI QRNG Protocol

    In our protocol, we suppose an untrusted source produces a tripartite state ρABE with the reduced state ρAB=TrE[ρABE], where A and B are distributed to two noncommunicating observers named Alice and Bob, respectively, and E is held by the underlying eavesdropper Eve as a quantum memory or considered as the environment. In the ideal case, ρAB 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 ωp and a coherence time of σcoh and that the generated photon pairs have a correlation time of σcor determined by phase-matching bandwidth. The ideal state can be written in the time and frequency domains, respectively, as follows: ΨABt=ψ(tA,tB)eiωp(tA+tB)/2|tAA|tBBdtAdtB,ΨABω=ϕ(ωA,ωB)|ωAA|ωBBdωAdωB,where the joint time function ψ(tA,tB) and joint frequency function ϕ(ωA,ωB) are given by ψ(tA,tB)=12πσcohσcore(tAtB)2/4σcor2(tA+tB)2/16σcoh2,ϕ(ωA,ωB)=1π/2σcohσcore(ωAωB)2σcor2/4(ωA+ωB)2σcoh2,where |tAA(|tBB) and |ωAA(|ωBB) represent photons A(B) at time tA(tB) and frequency ωA(ωB).

    Alice and Bob both have two trusted positive operator-valued measures (POVMs), denoted by Tδj={Tkj} and Dδj={Dkj} with j{A,B} and kN. The measurement Tδj is the direct photon arrival time detection, expressed as Tkj=kδ(k+1)δ|XtjXt|jdt,where |Xtj=dω2πeiωt|ωj and δ is the detection precision of the system. The other measurement, Dδj, is the arrival time detection after the photons in Alice and Bob, respectively, undergo normal and anomalous dispersion with equal magnitudes, which can be written as Dkj=kδ(k+1)δ|YtjYt|jdt,where |Ytj=dω2πei(ωt+βjω2/2)|ωj and βA(B) is the group-velocity dispersion (GVD) coefficient in Alice (Bob) satisfying βA=βB.

    However, in practice, we perform measurements Tδj and Dδj in a range from Ndδ/2 to Ndδ/2, where Nd is the frame size (dimensionality); thus the null measurements TjØ and DjØ 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 TjØ=Ndδ/2|XtjXt|jdt+Ndδ/2|XtjXt|jdt,DjØ=Ndδ/2|YtjYt|jdt+Ndδ/2|YtjYt|jdt.

    Then the refined POVMs can be written as Tδj={Tkj}k=Nd/2Nd/2TjØ and Dδj={Dkj}k=Nd/2Nd/2DjØ.

    Alice and Bob choose two measurements, Tδ and Dδ, separately, which are switched through a classical random signal S with probabilities q and 1q, respectively. Before extracting random numbers, Alice and Bob record the joint outcomes of the measurements Tδ to estimate the detection precision δ of the system. Then the outcomes of measurement Tδ in Alice are recorded as the raw random bits, whereas the joint outcomes of the measurements Dδ 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 Dδ as a testing value d given by38d=2πσcoh,Dδ,where σcoh,D is the correlation time of the photon pairs when Alice and Bob both perform measurement Dδ, and σcoh,D=σcor2+β2/4σcoh2 for the ideal state. The source can be certified to be time–energy entangled if d is less than the classical bound determined by the actual experimental parameters (see Appendix A for details). A preset value d0 is selected here that is not larger than the classical bound, and the protocol is aborted when d>d0.

    Since the source device is untrusted, the input state might be controlled by an eavesdropper, Eve, who can obtain the side information through system E. The amount of genuine randomness that can be extracted from Alice in measurement Tδ is quantified by the conditional quantum min-entropy40 defined as Hmin(TδA|E)=log2Pguess(TδA|E), where Pguess(TδA|E) is the maximum probability that Eve guesses correctly the outcome of Tδ conditional on her side information. In previous works, the lower bound of conditional quantum min-entropy Hmin(TδA|E) 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 ϵ-smooth conditional min- and max-entropies are defined as Hminϵ(A|B)ρ=maxρBϵ(ρ)Hmin(A|B)ρ,Hmaxϵ(A|B)ρ=maxρBϵ(ρ)Hmax(A|B)ρ,where Bϵ(ρ)={ρ|12ρρtrϵ} is the set of operators within an ϵ distance of ρ. Then the modified EUR is written as39Hminϵ(TδA|E)ρHlowϵ(TδA|E)ρ=2log2(f+(pTδAØ(ρ),ϵ)+f+(pDδAØ(ρ),ϵ)+1f(pDδAØ(ρ),ϵ)c<(TδA,DδA)(2Hmaxϵ(DδA<|B)ρ)),where f±(piØ(ρ),ϵ)=2ϵϵ2+2piØ(ρ)ϵ24piØ(ρ)ϵ±2(1ϵ)piØ(ρ)ϵ[1piØ(ρ)](2ϵ)+piØ(ρ),and pTδAØ(ρ)=Tr[ρATAØ], pDδAØ(ρ)=Tr[ρADAØ] are the null probabilities for measurement TAØ and DAØ, respectively, which can be written as pTδAØ(ρ)=112πσcohNdδ/2Ndδ/2etA22σcoh2dtA,pDδAØ(ρ)=112πσcohNdδ/2Ndδ/2etA22σcoh2dtA,where σcoh is the standard deviation of arrival-time distribution photon A after propagating through the dispersive medium.

    Additionally, c<(TδA,DδA) in Eq. (12) is the maximum overlap for the POVMs TδA and DδA, excluding the null measurement POVM elements, satisfying39c<(TδA,DδA)=maxTδA,DδAØTδADδA2,where · denotes the maximum singular value. c<(TδA,DδA) can be the upper bound by the c(TδA,DδA)=maxTδA,DδATδADδA2 because the sets of POVMs over which the former is maximized are subsets of the sets over which the latter is maximized. Thus we obtain c<(TδA,DδA)c(TδA,DδA)=δ24π2β,where β=|βA| (see Appendix B for details). The smooth conditional max-entropy Hmaxϵ(DδA<|B)ρ in Eq. (12) represents Bob’s lack of knowledge about the measurement results of DδA after Alice discards the null measurements, which can be bounded by43Hmaxϵ(DδA<|B)ρlog2γ(d0+Δ),where function γ(·) is formulated as γ(x)=(x+1+x2)(x1+x21)x,and the statistical fluctuations Δ can be written as Δ=Nd1q(q1)NTAln(ϵ/422(1(1pTδAØ(ρ))NTA)),where NTA is the total number of detections for TδA in a processing unit.

    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 9.2  μm. 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%.

    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 Tδ or Dδ by a 90:10 beam splitter (BS) after being coupled to fiber in Alice and Bob sides. PC, polarization controller; FI, filter; C-BG, chirped Bragg grating; OC, optical circulator; SNSPD, superconducting nanowire single-photon detector; and TDC, time-to-digital converter.

    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 Tδ or Dδ by a 90:10 beam splitter (BS) after being coupled to fiber in Alice and Bob sides. PC, polarization controller; FI, filter; C-BG, chirped Bragg grating; OC, optical circulator; SNSPD, superconducting nanowire single-photon detector; and TDC, time-to-digital converter.

    Alice and Bob both randomly perform measurement Tδ or Dδ by a passive 9010 beam splitter, i.e., q=0.9 in protocol. Explicitly, the measurement Tδ is implemented by directly measuring the arrival time at the SNSPD, while for the measurement Dδ, 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 1440  ps2 (1440  ps2). The arrival time is detected by the SNSPDs, then recorded by the TDCs with the total time jitters estimated approximately as σj34  ps (1 standard deviation). The outcome rate of measurement Tδ in Alice is nTA=4.5  MHz.

    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 Tδ, the FWHM of the coincidence peak is ΔT=120  ps, as shown in Fig. 2(a), and thus the detection precision is calculated to be δ=ΔT/2=84  ps 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 Dδ, as shown in Fig. 2(d), the peak recovers with a narrow FWHM of ΔD=160  ps, as shown in Fig. 2(d), corresponding to σD=68  ps [σD=ΔD/(22ln2) for Gaussian function] due to the NDC effect. In this case, the testing value d is calculated to be 0.64 according to Eq. (9), which is much smaller than the classical bound d¯c=1.35 (see Appendix C).

    Photon coincidence counts (CCs) recorded for four measurement combinations of two observers (denoted as A and B) in 10 s.

    Figure 2.Photon coincidence counts (CCs) recorded for four measurement combinations of two observers (denoted as A and B) in 10 s.

    The preset value d0 is set to be 0.64, since it is the upper bound in the vast majority of the measurement runs in our experiment. If dd0 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 pTδAØ(ρ)=1ferr(0.0140Nd) and pDδAØ(ρ)=1ferr(0.0138Nd) can be obtained with σcoh=2.1 and σcoh=2.15  ns in our experiment, where ferr is the error function.44 The statistical fluctuation Δ defined in Eq. (20) is obtained by setting the smooth entropy parameter ϵ=1010, where the total count NTA is deduced by the count rate nTA and the cumulative time τ as NTA=nTAτ(1pTδAØ(ρ)).

    We plot the smooth min-entropy Hlowϵ(TδA|E)ρ with respect to NTA and Nd, as shown in Fig. 3. It can be seen that Hlowϵ(TδA|E)ρ increases with NTA, while for a given NTA, with the increasing Nd, Hlowϵ(TδA|E)ρ first keeps growing due to increasing measurement range and then declines for larger statistical fluctuation, where the maximum value can be obtained by optimizing Nd. The maximal entropy values are obtained to be 0.778, 0.877, 0.903, and 0.913 for four processing units with frame size Nd=232, 246, 250, and 256, respectively.

    Smooth entropy Hlowϵ(TδA|E)ρ with respect to the frame size Nd for different processing units NTA. The dotted lines represent the entropy evaluated from the experimental data. The red triangles represent optimal results.

    Figure 3.Smooth entropy Hlowϵ(TδA|E)ρ with respect to the frame size Nd for different processing units NTA. The dotted lines represent the entropy evaluated from the experimental data. The red triangles represent optimal results.

    As a trade-off between the entropy bound and practicality, the processing unit is set as NTA=4.5×108, corresponding to the highest min-conditional entropy of 0.917 bit per count with Nd=256, pTδAØ(ρ)=4×107, pDδAØ(ρ)=6×107, and τ=100  s. Considering the proportion of genuine entangled photons of the SPDC is measured to be 97%, we can extract 0.900-bit genuine randomness per log2(256)-bit sample. Hence, we generate a Toeplitz matrix with a scale of 80,000×9000 to extract genuine random numbers. As the outcomes rate is nTA=4.5  Mcounts/s, 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 α=0.01. The NIST test is passed if P values are higher than 0.01 and the proportion value within the confidence interval of (1α)±3(1α)α/n=0.99±0.00944 for all tests. As shown in Fig. 5, the random bits in our scheme pass all 15 tests.

    Autocorrelation coefficients of raw random data and final random data.

    Figure 4.Autocorrelation coefficients of raw random data and final random data.

    Results of NIST statistical test suite.

    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 SourceUncharacterized MeasurementFinite-size AnalysisFinite Measurement Ranges ConsideredaGeneration Rate
    15××5.7 kbps
    17××47.8 Mbps
    20×1 Mbps
    21××8.05 Gbps
    24×b1 Mbps
    25c23 bps
    27d1.25 Mbps
    31××17 Gbps
    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 Hlowϵ(TδA|E)ρ=2.66; 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 d defined in Eq. (9) as the code distance for systems A and B in Dδ basis can be used to certify the time–energy entanglement for the ideal state in Eq. (1).

    Let us consider the case that systems A and B are two separable photons or classical pulses. The spectrum and temporal functions of the photon A can be written as, respectively, ϕAc(ω)eω24σν2,ψAc(t)et24σt2,where σν is the spectrum bandwidth (1 standard deviation) of the photon, and σt is the temporal bandwidth. Meanwhile, ϕBc(ω) and ψBc(t) for photon B are defined similarly with photon A. After two photons propagate through the dispersive medium, the intensity detected at Alice and Bob can be written as IA(tA)=|dωA2πϕAc(ωA)ei(ωAtA+βωA2/2)|2,IB(tB)=|dωB2πϕBc(ωB)ei(ωBtB+βωB2/2)|2.

    The joint detection probability that Alice’s detector clicks at time tA and Bob’s clicks at time tB simultaneously is P(tA,tB)=IA(tA)IB(tB), and the overall probability P(Γ) of detecting two photons at a time lag Γ=tAtB can be calculated as P(Γ)=IA(tA)IB(tB)dtAeΓ22σcor,c2,where the correlation time thus given by σcor,c2=σcor2+2β2σν2,and σcor=2σt is the origin correlation time.

    It has been proved that the origin correlation time σcor and standard deviation in the spectrum intensity of the sum of frequency Δ(ωA+ωB) for two separable photons satisfy the following inequality:46,47σcorΔ(ωA+ωB)1,where Δ(ωA+ωB) can be calculated to be 2σν. Hence, substituting this inequality into Eq. (25), we can obtain σcor,c2σcor2+β2σcor2,which defines the minimum broadening of temporal correlations between two separable photons after they propagate through two dispersive media with equal and opposite dispersion. By normalizing the correlation time σcor,c into the detection precision δ, the testing value d for a pair of separable photons can be written as d2σcor2πδ2+2β2πδ2σcor2.

    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 dc.

    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 Dδ, 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, G(2)(tA;tB)=|YtA(tA)YtB(tB)|ΨABω|2=|ψD(tA,tB)|2,where the joint time function becomes ψD(tA,tB)=12πϕAB(ωA,ωB)eiβ2(ωA2ωB2)i(ωAtA+ωBtB)dωAdωB.

    Then the correlation time of outcomes in measurement Dδ can be calculated as σcor,D2=(tAtB)2|ψD(tA,tB)|2dtAdtB=(tAtB)2|ψAB(tA,tB)|2dtAdtB+β2(ωA+ωB)2|ϕAB(ωA,ωB)|2dωAdωB=σcor2+β2σω2,and σω=1/(2σcoh) is the pump spectrum bandwidth. Thus the theoretical d for the ideal state given by Eq. (1) is achieved by d=2σcor2πδ2+β22πσcoh2δ2.

    In the limit of large coherence time σcoh, the testing value d reduces to d=2σcor2πδ2,which is obviously smaller than the classical bound dc.

    7 Appendix B: The Maximum Overlap of TδA and DδA

    We recall the measurements TδA={TkA} and DδA={DkA}, which can be expressed as TkA=kδ(k+1)δ|XtAXt|Adt,DkA=kδ(k+1)δ|YtAYt|Adt,where |XtA=a(t)|0 satisfies the orthonormality condition Xt1|Xt2=δ(t1t2). Note that the measurements DδA and TδA can be transformed by the dispersion operator U36 as DδA=UTδAU,where U=12πβ+dt1+dt2ei(t1t2)2/2β|Xt1AXt2|A.

    The associated observables of TδA and DδA can be, respectively, written as OTA=+dtt|XtAXt|A,ODA=12πβ+dt+dt1+dt2tei(t12t22)/2β+i(t1t2)t/β|Xt1AXt2|A.

    Based on the derivation in Ref. 38, the observable ODA can be further simplified as ODA=+dtt|XtAXt|A+βi+dt|XtAtXt|A,=OTA+2πβOωA,where OωA=+dω2πω|ωAω|A is the observable of frequency. According to the commutation relation [OTA,OωA]=i,48 we can derive the commutation relation of OTA and ODA as follows: [OTA,ODA]=i2πβ.

    Using the overlap result for maximally incompatible observables,38,49 we can obtain c(TδA,DδA)=δ24π2β.

    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 d 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, σ¯cor,c22σj2+σcor2+β2σcor2.

    Recall that we measured the coincidence distribution and obtained σ0=ΔT/(22ln2) with β=0 in Fig. 2(a), i.e., σ02=2σj2+σcor2. Then combining the GVD coefficient β in our system, we can calculate the modified correlation time σcor,c100 ps and the corresponding classical bound d¯c=1.35.

    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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    Ji-Ning Zhang, Ran Yang, Xinhui Li, Chang-Wei Sun, Yi-Chen Liu, Ying Wei, Jia-Chen Duan, Zhenda Xie, Yan-Xiao Gong, Shi-Ning Zhu. Realization of a source-device-independent quantum random number generator secured by nonlocal dispersion cancellation[J]. Advanced Photonics, 2023, 5(3): 036003
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