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11Real time demonstration of high bitrate quantum random number generation

with coherent laser lightT. Symul,1 S. M. Assad,1 and P. K. Lam1, a)

Centre for Quantum Computation and Communication Technology, Department of Quantum Science,

Australian National University, Canberra, ACT 0200, Australia

(Dated: 25 July 2011)

We present a random number generation scheme that uses broadband measurements of the vacuum fieldcontained in the radio-frequency sidebands of a single-mode laser. Even though the measurements maycontain technical noise, we show that suitable algorithms can transform the digitized photocurrents into astring of random numbers that can be made arbitrarily correlated to a subset of the quantum fluctuations(High Quantum Correlation regime) or arbitrarily immune to environmental fluctuations (High EnvironmentalImmunity). We demonstrate up to 2 Gbps of real time random number generation that were verified usingstandard randomness tests.

Reliable and unbiased random numbers (RNs) areneeded for a range of applications spanning from numer-ical modeling to cryptographic communications. Withthe numerous improvements in quantum key distribu-tion (QKD) protocols1,2, fast and reliable RN generationis now one of the main technical impediment to high-speed QKD. Whilst there are algorithms that can gener-ate pseudo-RNs, they can never be perfectly random norindeterministic. True RNs from physical processes mayoffer a surefire solution.Several physical RN generation schemes have been pro-

posed and demonstrated35, including schemes based onsingle photon detections611. The limit in speed of thesesystems are in the dead time of photon counters. Analternative quantum approach to photon counting is touse the vacuum fluctuations of an electromagnetic fieldfor RN generation12,13. In this letter, we demonstratea simple scheme to measure and convert vacuum fieldfluctuations into RNs.The schematic of the quantum RN generator is shown

in Fig. 1. A single-mode laser beam at 1550 nm is used asthe light source. A few mW of light is split into two equalintensity beams and detected by a pair of photodetectorsin a balanced homodyne scheme. When the average laserfield amplitude is significantly larger than the vacuumfield fluctuation the subtracted photo-current from thepair of detectors is proportional to Xv(), where Xvis the quadrature amplitude of the vacuum field. Thebalanced homodyne setup therefore measures the ampli-fied quadrature amplitude of the vacuum field fluctua-tions. Only sideband frequencies well above the technicalnoise frequencies of the laser are used for RN generation(shaded region of the radio frequency (RF) spectrum ofFig. 1(a)). This is achieved by demodulating the pho-tocurrent with an RF frequency (1.6 GHz) followed by alow pass filter. The undulations in the spectra are dueto non-uniform RF electronic gain in the photodetectorsamplification stages (Fig. 1(b)). Nevertheless, the quan-tum noise has a constant clearance above the electronic

a)Electronic mail: Ping.Lam@anu.edu.au

noise level of 8.5 dB. Using a Field-programmable GateArray (FPGA) a filter function can be programmed toneutralize the non-uniform electronic gain as shown inFig. 1(c). Finally, using suitable numerical processes,the quantum noise is converted into a sequence of ran-dom digital bits as depicted by the 8-bits colour code inFig. 1(d).In practice, vacuum fluctuations cannot be detected

in complete isolation. The electronic noise of the photo-detector will be superimposed onto the measured pho-tocurrents. While it is reasonable to assume that thequantum noise of a vacuum field Xv is perfectly randomover all frequencies and cannot be tampered with, elec-tronic noise may not possess these ideal properties. Forthis letter, we would like to consider two possible scenar-ios.In the first scenario, we assume that the electronic

noise is untampered. We wish to find a protocol to gen-erate RNs solely from closely tracking the quantum fluc-tuations of the vacuum field. We will show that in thisscenario, a 1-bit digitized encoding of the vacuum fluctu-ations, together with thresholding, can indeed allow arbi-trarily high quantum correlation. In the second scenariowe assume that the electronic noise may be tamperedand is untrustworthy. We desire an algorithm that willgenerate RNs that are tamper-proof, even in the presenceof possibly forged electronic noise. Provided that quan-tum noise remains the dominant source of noise, we willshow that digitizing the vacuum fluctuations into mul-tiple bits, and then discarding the most-significant bits,can indeed allow arbitrarily high environmental immu-nity. Correlations between environmental noise and thegenerated RNs can be made arbitrarily small.We describe the measured signal as Xm = Xv + Xe,

where Xe corresponds to the electronic noise superim-posed onto the vacuum fluctuations Xv (see Fig. 2(a)).Xe and Xv can be modeled as two uncorrelated Gaussiandistributions of zero mean and variance Ve and Vv, re-spectively. Xm is also Gaussian with a conditional prob-ability given by

P (Xm|Xv) = 12piVe

e(XmXv)

2

2Ve . (1)

2FIG. 1. Random number generation schematic. Top left figure shows the spectra of the quadrature amplitudes for the vacuumfluctuations, Xv and the laser mode Xa. The shaded frequency range, , shows a region where the laser source is quantumnoise limited. (a) Sum (S) and difference (D), of the laser vacuum fluctuations, as well as electronic noise (E) from a pair ofhomodyne photodetectors. The shaded frequency range is used for generating random numbers. (b) Digitized and demodulatedshot noise and electronic noise spectra. (c) A filter function is used to correct for the non-uniform electronic gain. The quantumnoise at this stage has an unbiased Gaussian distribution with a normalized mean at 0 dB and dark noise clearance of 8.5 dB.(d) High randomness of the final digital random numbers depicted by the featureless color plot. BS: 50/50 beamsplitter; D:photodiode; ADC: Analog to digital converter; NUM. PROC.: Numerical processing algorithm; Q-RN: Final generated randomnumbers.

In order to extract n-bit RNs from Xm with uniformprobabilities, we need to transform the measured Gaus-sian distributed photocurrent into a uniform distributionof Ym =

[1 + erf(Xm/

2Vm)

]/2. The n-bit RNs require

the distribution Ym to be divided into 2n equal and non-

overlapping domains (as shown in Fig. 2 for the case ofn = 3). We can then index each domain, preferably byusing Grays binary encoding14, to minimize the pertur-bations due to unaccounted technical noise so that the2n domains now correspond directly to the n-bits RNs(see Fig. 2(b)).

We introduce a thresholding condition to reject datapoints that fall within a certain range t between twoadjacent domains (see Fig. 2). We digitize continu-ous variable quantities into a binary n-bits with Ym =

(Y(1)m Y

(2)m ...Y

(n)m ), where Y

(i)m {0, 1}. Y (1)m is the most

significant bit (MSB) and Y(n)m the least significant bit

(LSB) in Grays binary decomposition. We define a prob-

ability of error, Pe,q = P (Y(i)m 6= Y (i)v ). This quantifies

the discrepancy between a measured bit and the vac-uum fluctuations. Perturbation from the electronic noise,which causes the digitized bit to differ from the vacuumfluctuation, is referred to as an error. Pe,q = 0 meansthat the generated RNs are the perfect digitization ofpure vacuum fluctuations.

We also introduce the notion of information leakagefrom the electronic noise IE. This information leakageis the amount of information that can be imposed onthe final random bits by somebody having access to the

electronic noise. Using Shannons binary information for-mula, we obtained

IE = 1 + Pe,e log2 (Pe,e) + (1 Pe,e) log2 (1 Pe,e) (2)where analogous to Pe,q , Pe,e is the probability of errorthat the measured ith-bit differs from the electronic noise.IE = 0 means that the generated RNs are indeterminableeven with full knowledge of the electronic noise.Figure 3 shows IE and Pe,q plotted as a function of the

RN generation rateR = n[1Pthr(t)] for n = 1, 2, 3, 4, 5bits extracted per measurement, where Pthr(t) is theprobability that a bit is rejected because it lies withinone of the thresholding areas. From Fig. 3(a) we notethat the amount of information leaked through the elec-tronic noise decreases for the least significant bit (LSB)with high bit encoding. Moreover, thresholding paradox-ically increases IE. Minimizing IE therefore requires highbit encoding, no thresholding, and omission of all but theLSB. In the case of 5-bits encoding, the best result ob-tained using this protocol was IE < 10

6 for the LSB.In Fig. 3(b) we see that the MSBs are the most accu-

rate representation of the vacuum fluctuations. In con-trast to information leakage, thresholding in general re-duces the probability of error Pe,q since thresholding ef-fectively discards all data where electronic noise is thedominant source of error, leaving behind only data withlarge vacuum fluctuations. Thresholding is therefore fa-vorable for increasing the correlations between the RNsand the vacuum fluctuations. For multi-bit rate encod-ing, however, Pe,q is nonzero even with very large thresh-

3-4 -3 -2 0 1 2 3 4

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Y m

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m)

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-1

t

(c)

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P(Xm)

P(Xm|Xv)

(i)0

FIG. 2. (a) Gaussian distribution of the measured and vac-uum fluctuations P (Xm,v), and the conditional probabilitydistribution of P (Xm|Xv) for an arbitrary value of Xv = 1.The figure is plotted for a Xe that is 8.5 dB below Xv . (b) 1Gsamples of the Gaussian distributed Xm after 12-bits digitalfiltering. Experimental data is plotted with blue bars and itscorresponding theoretical expectation with purple lines. (c)The Gaussian error function is used to transform Xm into auniform distribution of P (Ym). Thresholding (represented bythe green bars) can be introduced to reject data points thatfall within a t range from the boundary of two adjacentdomains.

olding. In fact, at high bit rates Pe,q is no longer criticallydependent on thresholding. An ideal protocol for maxi-mizing the quantum correlation of RN is therefore a sin-gle bit encoding with threshold value significantly largerthan the electronic noise. From this we can define tworegimes of operation for our quantum RN generator: (i)A regime of high quantum correlation where vacuum fluc-tuations are accurately converted to digital RNs. (ii) Aregime of high environmental immunity where tamperingof photodetector electronic noise does not compromisethe indeterminacy of the RN generation.We implemented our proposed algorithm in real-time

using an integrated 12 bits 250 Msamples per secondanalog to digital converter (ADC) and a FPGA. Ourresults show a uniformly distributed random binary se-quences where 8 bits are extracted for each measurementwithout thresholding, corresponding to a real-time ran-dom bit rate generation of 2 Gbps. This demonstrationconsistently passes the NIST15 and Diehard randomnesstests16.In the context of the two previously introduced

regimes, we also implemented a high quantum correla-tion single-bit encoding with thresholding rejecting 90%of the samples, and a high environmental immunity zerothreshold 8-bit encoding that only keeps the 4 LSBs. In

10-6

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R

R

(a)

(b)

IE

Pe,q

Ym(1) Ym

(1)

Ym(2)

Ym(3)

Ym(4)

Ym(5)

Ym(4)

Ym(3)

Ym(2)

Ym(1)

Ym(1)

Ym(5)

Pe,q

I E

FIG. 3. (a) Information leakage, IE, for multi-bits divisionof vacuum fluctuations as a function of RN generation rate,R corresponding to the number of bits extracted times theprobability of not being rejected by the thresholding proce-dure. The LSB from the Gray code shows the least amountof information leakage. (b) Probability of error, Pe,q , plottedas a function of R. Fewer bits encoding gives smaller Pe,q.

the high quantum correlation mode of operation the bi-nary random sequence is produced at a rate of 25 Mbps,with Pe,q < 10

6, whilst in the high environmentalimmunity regime the random bit-rate is 1 Gbps withIE < 10

6.In conclusion, we have demonstrated the generation

of continuous random bit sequences at a rate of 2 Gbpsin real time by sampling the broadband vacuum fluc-tuations. We proposed two methods of generating RNswhere: (i) quantum correlation is optimized for a nearideal representation of a thresholded random subset ofthe vacuum fluctuations, and (ii) environmental immu-nity is optimized to combat against possible tamperingof the electronic noise.We thank QuintessenceLabs, M. Neharkar and K. L.

Chong for technical assistance. This research was con-ducted by the Australian Research Council Centre of Ex-cellence for Quantum Computation and CommunicationTechnology (project number CE110001029).

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