A Robust Quantum Random Number Generator Based on Bosonic Stimulation
AA Robust Quantum Random Number Generator Based on Bosonic Stimulation
Akshata Shenoy H
Dept. of Electrical and Communication Engg., Indian Institute of Science, Bangalore, India
S. Omkar
Poornaprajna Institute of Scientific Research, Sadashivnagar, Bengaluru- 560080, India.
R. Srikanth ∗ Poornaprajna Institute of Scientific Research, Sadashivnagar, Bengaluru- 560080, India. andRaman Research Institute, Sadashivnagar, Bengaluru- 560060, India.
T. Srinivas
Dept. of Electrical and Communication Engg., Indian Institute of Science, Bangalore, India.
We propose a method to realize a robust quantum random number generator based on bosonicstimulation. A particular implementation that employs weak coherent pulses and conventionalavalanche photo-diode detectors (APDs) is discussed.
I. INTRODUCTION
Random numbers are crucial for various tasks, among them generating cryptographic secret keys, authentication,Monte-Carlo simulations, digital signatures, statistical sampling, etc. Random number generators can be classified intotwo types: pseudo-random number generators (PRNG) [1] and true random number generators (TRNG). A PRNGis an algorithm, computational or physical, for generating a sequence of numbers that approximates the propertiesof random numbers. A physical or hardware version is typically based on stochastic noise or chaotic dynamics in asuitable physical system [2]. Computational PRNGs are based on computational algorithms that generate sequences ofnumbers of very long periodicity, making them look like true random numbers for sufficiently short sequences. Carefulobservation over long periods will in principle reveal some kind of pattern or correlation, suggestive of non-randomness.As far as is known today, the inherent indeterminism or fluctuations in quantum phenomena is the only sourceof true randomness, an essential ingredient in quantum cryptography [3]. Various proposed underlying physicalprocesses for quantum random number generators (QRNGs) include: quantum measurement of single photons [4, 5],an entangled system [6], coherent states [7, 8] or vacuum states [9]; phase noise [10], spin noise [11], or radioactivedecay or photonic emission [12].In this work, we propose a novel method of QRNG that is a quite different indeterministic paradigm from the abovetwo. It uses bosonic stimulation to randomly amplify weak coherent pulses to intense pulses that can be easily detectedby a conventional APDs. Bosons (integer-spin quantum particles) obey Bose-Einstein statistics, which entails that thetransition probability of a boson into a given final state in enhanced by the presence of identical particles in that state.If there are N particles in a given quantum state, the probability that an incoming boson makes a transition into thatstate is proportional to N + 1. This effect is called bosonic stimulation, and is responsible for coherent matter waveamplification in atomic lasers, as well as the sustenance of a particular mode in a laser cavity, whereby the presence ofphotons in a particular lasing mode stimulates the emission of more photons into that mode. It provides a new wayto combine quantum indeterminism with the tendency of bosons to congregate indistinguishably. We may call thisQRNG a random bosonic stimulator . Perhaps the practical merit of our proposed QRNG, apart from its harnessinga novel version of quantum indeterminism, is that it simplifies the detection module to the point where it may beaccessible to an advanced undergraduate laboratory. II. BOSONIC STIMULATION AS A REALIZATION OF THE P ´OLYA URN PROBLEM
Consider an urn with b blue balls and r red balls. A ball is picked at random and replaced with c balls of thesame color or d balls of different color. The addition of same color balls results in positive feedback whereas that of ∗ Electronic address: [email protected] a r X i v : . [ qu a n t - ph ] F e b FIG. 1: The relative population of the two ball populations of the P´olya urn, for 4 different runs with the same initial condition b (0) = r (0) = 3. different color balls results in negative feedback. During a run of trials, the fractional population of each urn mayinitially fluctuate randomly, but eventually settles down to a randomly selected limiting value t , giving an instanceof symmetry breaking (Figure 1).Let the initial population of two states labelled as “blue” and “red” be b (0) and r (0), respectively. As the incomingballs start populating the two states, the subsequent growth in population of the modes exhibits P´olya urn behaviour.The probabilistic law of evolution of the fractional population at i th instance is given by bosonic stimulation to be( b ( i ) , r ( i )) −→ (cid:40) ( b ( i ) + 1 , r ( i )) with probability b ( i )+1 b ( i )+ r ( i )+2 ( b ( i ) , r ( i ) + 1) with probability r ( i )+1 b ( i )+ r ( i )+2 . (1)The limiting value t ≡ b ( i ) / ( b ( i ) + r ( i )) for large i itself varies randomly in the range [0 ,
1] from run to run, havinga beta distribution f ( t ; β, ρ ) = 1 B ( β, ρ ) t β − (1 − t ) ρ − , (2)where B is the beta function that normalizes f , β = b (cid:48) /c , ρ = r (cid:48) /c , b (cid:48) ≡ b (0) + s b , r (cid:48) ≡ r (0) + s r . For bosonicstimulation, the shifts s b = s r = 1. If the seeding is symmetric, then so is the asymptotic distribution of the limitingvalues, thereby restoring symmetry, as it should be [13].Depending on the number of trial runs, the final state can have an arbitrarily large number of bosons. Dependingon whether t > . t < . III. PRACTICAL REALIZATION
A concrete idea for realizing a random bosonic stimulator is to use a lasing medium that supports two radiationmodes, for example by vertical and horizontal polarization of the same frequency [14]. A scheme of the proposedexperiment is given in Figure 2. Two equal intensity, highly attenuated modes of coherent states are input into a lasingmedium. To ensure that the two inputs are synchronized and of equal intensity, a calibrated Mach-Zehnder set-up isused with an attentuated coherent laser pulse fed into one of its input ports. This results in an ouput consisting oftwo (unentangled) coherent pulses with half the intensity. A quarter wave plate in one of the arms ensures that thepolarization in one arm rotated to be 90 ◦ with respect to the other. FIG. 2: At the input port, a weak coherent pulse | α, H (cid:105) with horizontal polarization and average photon number | α | entersthe experiment. The upper arm branch is rotated to vertical polarization by the quarter-wave plate QWP. The product state | α , V (cid:105)| α , H (cid:105) enters the lasing medium, where the V and H modes participate in bosonic stimulation. In the symmetric case,the detector complex generates a 0 or 1 bit depending on whether I H > I V or the converse, where I H , I V are intensities of thehorizontal and vertical polarization output components. Each mode in a pulse corresponds to a ball color in the P´olya urn problem. Because of bosonic stimulation, theoutput intensity will randomly favor vertical or horizontal polarization. Let I H ( I V ) denote the intensity of theoutcoming light in the horizontal (vertical) polarization mode.The incoming photon emitted as a result of the coherent de-excitation of the atoms in the medium is assumed to beequally coupled to both modes. Suppose the two modes start in the state | n, m (cid:105) , where the first register correspondsto the ‘blue’ mode and the second to the ‘red’ mode. Further let the series of atoms in the population-inverted statebe in the initial excited state | e, e, · · ·(cid:105) . By giving up a photon into the blue or red mode (which could be angularmomentum states), the atom is left in the state | b (cid:105) or | r (cid:105) , assumed to be mutually orthogonal. The joint system ofthe modes and atoms evolves in a manner analogous to a quantum walk, given by: | n, m (cid:105)| e, e, · · ·(cid:105) → √ n + m + 2 (cid:0) √ n + 1 | n + 1 , m (cid:105)| b (cid:105) + √ m + 1 | n, m + 1 (cid:105)| r (cid:105) (cid:1) | e · · ·(cid:105)→ (cid:112) ( n + m + 2)( n + m + 3) (cid:16)(cid:112) ( n + 1)( n + 2) | n + 2 , m (cid:105)| b, b (cid:105) + (cid:112) ( n + 1)( m + 1) | n + 1 , m + 1 (cid:105)× ( | b, r (cid:105) + | r, b (cid:105) ) + (cid:112) ( m + 1)( m + 2) | n, m + 2 (cid:105)| r, r (cid:105) (cid:17) | · · ·(cid:105) , (3)and so on. Each term in the superposition is rendered incoherent because it is entangled with a distinct state ofatoms, and thus the probability for scattering into a given urn state is quantitatively the same as the classical P´olyaurn situation.Because of the entanglement with the polarization degrees of freedom of the atoms (Eq. (3)), the state of theatoms bears an imprint of the final outcome of laser light. However, as the laser atoms are not individually accessible,the random bit generated is practically unique. From the view point of Monte-Carlo simulations, etc., only therandomness from the laser light read-out will be used. From a cryptographic perspective, the laser system will remainphysically well within the encoder unit, preventing its access to a malevolent eavesdropper. Assuming that anypossible information leakage through side-channels (like heat radiations from the laser) are reasonably blocked out tothe outside world, the assumption of practical uniqueness of the generated randomness applies here, too.A random bit x is generated by the detector module, depending on which mode dominates, for example, accordingto the recipe: I H > I V = ⇒ x = 0 ,I H < I V = ⇒ x = 1 . (4)The mean and variance of the distribution f ( t ; β, ρ ) are given, respectively, by µ = b (0) b (0) + r (0) ;∆ = b (0) r (0)( b (0) + r (0)) ( b (0) + r (0) + 1) . (5)The value t obtained will tend to peak towards the mean, with ever lower variance if one or both of the initialpopulations are large. If the instrument function of the detector is denoted by a normal distribution e with FWHM s ,the observed distribution is the convolution p ( x ) = (cid:82) f ( t ; β, ρ ) e ( x − t ) dt . It is important that we use sufficiently weakpulses obtained by attenuating coherent light sources, so that s (cid:28) ∆. This ensures that the quantum randomnessdominates over stochastic noise in the detector’s reading. The distribution f ( t ; β = 1 , ρ = 1) is uniform over [0 , H mode stronger than to the V mode(because of an atomic or beam-splitter feature) by a factor (1 + (cid:15) ), then the new distribution can be shown to be Eq.(2), but with β → β (cid:48) = β (1 + (cid:15) ). Let t / be the median of the distribution f , defined such that (cid:82) t / t =0 f ( t ; β (cid:48) , ρ ) = ,where f is the beta distribution (2). Then, we obtain our uniformly random bit by replacing prescription (4) by: t < t / = ⇒ x = 0 t ≥ t / = ⇒ x = 1 (6)To be precise, the above numbers assume that the input modes are pure number states.More realistcally, taking into account that they are coherent states, we must replace Eq. (2) by f (cid:48) ( t ; λ ) = (cid:88) β,ρ B ( β, ρ ) t β − (1 − t ) ρ − P ( λ, β ) P ( λ, ρ ) , (7)where P ( λ, x ) is the Poisson distribution of x with mean λ . Furthermore, in practice we may have to let λ to rangeover an interval because of practical difficulties of producing the same exact degree of attentuation on each run. Itcan be shown that this added complication does not affect our main results.For sufficiently low noise in each run, two bits may also be generated per run according to the recipe: t < t / = ⇒ x = 00 t / ≤ t < t / = ⇒ x = 01 t / ≤ t < t / = ⇒ x = 10 t ≥ t / = ⇒ x = 11 , (8)where t ξ is defined such that (cid:82) t ξ f ( t ; β (cid:48) , ρ ) = ξ . More generally, to generate n bits, the noise level should be lowerthan 2 − n . IV. DISCUSSION AND CONCLUSIONS