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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: [email protected]
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-
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3-4 -3 -2 0 1 2 3 4
1
2
3
4
5
6
7
8
X m
Y m
P(X
m)
000
100
111
010
001
011
101
110 000
100
111
010
001
011
101
110
P(Y m )1/8
(a) (b)P
(Xv,
m)
P(X
m|X
v)
4 2 0 2 4
0
0.1
0.2
0.3
0.4
0.5
0
0.2
0.4
0.6
0.8
1
Xv,m
-1
t
(c)
P(Xv)
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
10-5
10-4
0.001
0.01
0.1
1
0
0.1
0.2
0.3
0.4
0.5
0.2 0.4 0.6 0.8 1 0.5 1 1.5 2 0.5 1 1.5 2 2.5 3 1 2 3 4 1 2 3 4
5
10-6
10-5
10-4
0.001
0.01
0.1
1
0
0.1
0.2
0.3
0.4
0.5
0.2 0.4 0.6 0.8 1 0.5 1 1.5 2 0.5 1 1.5 2 2.5 3 1 2 3 4 1 2 3 4
5
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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