New concepts and construction of quantum random number generators

Abstract

A new concept of a quantum random number generator related to the application of a quantum entanglement to produce several mutually coupled in a random manner bit strings is proposed. This allows for an entropy assessment in real time in the public domain by testing of one disclosed string without compromising the coupled by quantum entanglement dual strings remaining fully secret and as random as the one tested. Additionally, a new prototype of a miniaturized quantum random number generator is presented, which utilizes quantum transitions along the Fermi golden rule as the entropy source and is developed for an application to reduce size of cryptography systems on continuous variables and for a usage in portable IT devices.

Appendices

Appendix: Quantum sources of entropy other than von Neumann projection

Implementations of von Neumann projection scheme of quantum measurement, though transparent physically, are not easy to be implemented for probing and register outcomes of quantum measurements [15]. Moreover, are usually heavily biased by classical noise components. Therefore, to construct QRNGs some alternative sources of the entropy are considered. A wide class of other than a direct measurement quantum effects attributed to quantum randomness is related to probabilistic interpretation of wave functions and quantum transitions. A wave function provides only statistical information on the behavior of a system, which can be observed classically. For example, the probing of the quantum particle position will give a random sequence according to the probability defined by the modulus square of the wave function of this particle, which, in the position representation, describes random quantum localization of a particle. The same holds for the decomposition of the wave function in the basis of some orthonormal states, which span a Hilbert space and can be used to a change of a representation. Coefficients ($c_i$ complex, in general) of this decomposition ($\Psi =\sum _ic_i\Phi _i$) define only probability ($\vert c_i\vert ^2$, $\sum _i \vert c_i\vert ^2=1$) for projection onto the corresponding i-th state from the basis of eigen-functions (or, more generally, eigen-subspaces) of the Hermitian operator corresponding to the measured variable. However, it arises here a question, which probability type occurs here – frequentional or conditional ones. The former assumes infinite, in principle, repeating the measurement of the quantum state, which is, however, impossible, because each quantum measurement is demolishing and the original quantum state disappears at the measurement and cannot be repeatably measured. The solution would be cloning of an initial quantum state, but according to the non-cloning theorem by Wootters and Żurek, any unknown quantum state cannot be cloned [39]. The exception of this rule is only for known states—these states should be considered as states of the known fixed basis. Taking an arbitrary set of qubits and measuring each in a known basis, statistically half of qubits will be projected onto the same known state (one of two possible in the measurement). In the case when the known states were stationary states, by the application of some external field (e.g., of a magnetic field for spin-like qubits), one can select qubits (particles) in the same known state, via separation in energy (e.g., due to Zeeman splitting for spin qubits).

The alternative notion for probability is the conditional probability, which seems even more natural in quantum mechanical terms. Employing properties of Bayesian conditional probability to quantum mechanics changes its foundations and the related quantum mechanics interpretation is called as QBism [55]. The consequences for formal quantum information processing are, however, not explored upon this approach.

In the case of a system which is affected by time-dependent perturbation, the system response can be accounted for in terms of the probability of induced by the perturbation transitions between original stationary states of the systems. This probability is again of the unconditional quantum type, as it corresponds to the decomposition of the non-stationary state of the system with inclusion of time-dependent perturbation in the basis of original stationary states. The time-dependent coefficients of this decomposition define the probability amplitudes for the induced quantum transitions. The modulus square of these coefficients are truly random numbers and can be easily estimated in linear approximation with respect to the perturbation strength. The resulting Fermi golden rule for these transitions probability is directly applicable to a wide class of physical observable phenomena in micro-systems and their characteristics possible to be measured in a classical manner. The latter property is related to a step toward a classical behavior assuming quantum transitions in the continuous spectrum of stationary states (similar to classical distribution of energy). To be more specific, the transition probability between n-th and m-th stationary states of the system upon the perturbation ${\hat{V}}$ switched on at $t=0$ is given for the passage of time T by the formula,

$$\begin{aligned} w_{n,m}=\vert \langle n\vert {\hat{V}}\vert m\rangle \vert ^2 \frac{sin^2(\omega _{n.m}T/2)}{(\hbar \omega _{n.m}/2)^2}, \end{aligned}$$

(26)

where $\omega _{n,m}=\frac{E_n-E_m}{\hbar }$. In the limit $T\rightarrow \infty$ the above equation attains a form of Fermi golden rule (assuming that the perturbation field ${\hat{V}}$ periodically varies in time with the frequency $\omega$),

$$\begin{aligned} w_{n,m}/T=\frac{2\pi }{\hbar }\vert \langle n\vert {\hat{V}}\vert m\rangle \vert ^2\delta (E_n-E_m\pm \hbar \omega ), \end{aligned}$$

(27)

Dirac delta has arisen from the limiting property of sinc function ($lim_{x\rightarrow \infty }\frac{sin^2(ax/2)}{(a/2)^2x}=2\pi \delta (a)$), $\pm \omega$ corresponds to the emission or absorption of perturbation field quantum, respectively. Probability must be limited by 1; thus, the Dirac delta has to be quenched upon some integral—alternatively, upon an integral with respect to a continuous spectrum of final states m, or initial states n, or with respect to a Fourier variable related to the spectral picture of a non-monochromatic perturbation. The probability $w_{n,m}$ after integration removing the Dirac delta meets with macroscopic properties of quantum systems manifesting themselves at, e.g., a conductivity or other transport phenomena, at optical transitions (absorption and emission of light), at nuclear transitions and others—in fact, at all macroscopically noticeable properties of quantum systems. The probability expressed by Fermi golden rule still possesses, however, an absolute random quantum character hidden in tiny fluctuations of quantum system response according to this rule.

Registering of these fluctuations is a central problem for practical realizations of QRNGs, easiest to be probed for a very weak perturbations, like at tunneling currents across p-n semiconductor junctions in diodes or transistors or for extremely weak light emission of photo-diodes. In the latter case, this emission is frequently referred to single photon emission, which is, however, only popular terminology, because for really fainting light signals, energy emitted per some period of time is often only a fraction of a single photon energy $\hbar \omega$. The popular view on the quantum character of light is originated in quantum optics based on the quantization of classical e-m field in some volume being the collection of non-interacting oscillators with frequencies of e-m field Fourier spectrum. The planar wave components of the Fourier series for the vector potential of the e-m field serve as annihilation and creation operators of introduced in this way ’photons’, and the whole spirit of quantum optics is related to the non-commutation of the operator of photon numbers with the quantized in the same manner electric and magnetic fields. This non-commutation is caused by the quantum oscillator properties including its zero oscillations, and it leads to a quantum uncertainty principle of the photon number and the field phase [56]. Probabilistic character of optical transitions used for QRNGs is not related thus with single photons, but rather with the quantum randomness expressed by the Fermi golden rule for quantum transitions in some other system interacting with light.

The same holds for key distribution (QKD) systems in quantum cryptography utilizing photons as flying qubits [3], where through the dark quantum channel are not pushing single photons but rather weak e-m waves with the phase randomly assigned (in a macroscopic way) according to a true random sequence of bits produced by a QRNG. This enhances the significance of QRNGs because QKD over large distances is most prospective for such continuous variable randomly selected for flying qubits, being in fact imprinted in a classical e-m wave according to QRNG bit sequence (both for dark channels in optical fibers or in open air of space, the latter for quantum cryptographic communication between e.g., satellites).

Classical methods for testing of randomness

There are many statistical tests being utilized to testing of randomness of bit sequences. To most popular belong,

• NIST test [10, 57, 58]—developed by The National Institute of Standards and Technology of the United States at 2010. The NIST kit includes: test for the frequency of single bits 0 and 1 occurrences in the studied sequence, test for the frequency of single 0 and 1 occurrences in the block of the studied sequence, mileage test, test of the longest run in the block of the studied sequence, binary matrix order test—test of linear relationships between subsequences of the same length of the tested sequence, spectral test—discrete Fourier transform test, test for occurrence of non-overlapping patterns, test for overlapping patterns, universal statistical Maurer test [59], linear complexity test, serial test - pattern distribution test, entropy estimation test, cumulative sum test, random trip test and variant random trip test [60]. The NIST test partially overlaps with the Diehard (Dieharder) tests listed below. The NIST test, similarly to other statistical tests, is effective in the assessment of randomness of digital sequences produced by classical pseudo-random generators or QRBGs, but is unable to confirm absolute quantum randomness.

• Diehard and Dieharder tests [61,62,63]—is a set of randomness tests created by the American mathematician and computer scientist George Marsaglia in 1964 and revised in 2006 by Robert G. Brown, who named the modified Dieharder kit. The related kit embraces: birthday spacing test - a statistical test of the number of duplicate intervals between successive properly defined birthdays, overlapping permutation test, binary order test for matrices, search for missing 20 letter (letters 0 and 1) words in overlapping 20 letter words, into which the analyzed sequence is divided, test for the rare occurrence of overlapping short sequences, DNA - a test based on two-bit sequences encoding the letters C, G, A, T forming 10 letter words, test to count ones in the byte stream, parking lot test, minimum distance test, squeeze test, overlapping sums test, runs test, craps test. Similarly to NIST, the Diehard/Dieharder test is unable to confirm the absolute quantum randomness.

• Test U01—is a C language library for empirical testing of random number generators created by a team at the University of Montreal in 2007. The kit includes: tests of the distribution of t-dimensional vectors in a unit hypercube, tests based on discrete and continuous empirical entropies [64], test of the distance between the nearest points in a sample on a unit torus in t-dimensional space, Knuth tests in the book [24], Diehard/Dieharder tests, test of discrete random walks, compressibility test, tests using spectral methods (mainly discrete Fourier transform [65, 66]). This test is also effective in assessment of the randomness quality, but is unable to distinguish a true quantum randomness.

Fig. 5

Generator Quantis by the Swiss company IDQuantique [ idquantique.com]

Examples of QRNGs in the market

There are many QRNGs commercially proposed in the market, below some of them are listed and compared.

• IDQ Quantis Random Number Generator – generator by the Swiss company idQuantique, which is a pioneer in the commercialization of quantum cryptography (Clavis series) and quantum random generators (Quantis). The company was founded as a spin-off of the University of Geneva. The QRNGs were initially components of the cryptographic Clavis quantum key distribution sets, only after some time they were introduced to the offer as separate devices. Speed: PCIe 4/16 Mb/s, USB 4 Mb/s, size more than 10 cm (cf. Fig. 5), source of randomness: the light source illuminates the semi-transmissive mirror and hits the two detectors.

• ComScire PureQuantum—generator of the American company ComScire specializing in creating generators based on physical phenomena. The company was established for this purpose in 1994. Speed: USB 4/32/128 Mb/s Entropy source: current noise associated with leakage of carriers due to tunneling in MOS transistor, size > 10 cm.

• Toshiba UFICS-QRNG—generator of the Japanese Toshiba Corporation, ultra-fast, speed 8 Gb/s, entropy source: phase noise from spontaneous emission from a pulsed laser, size ca. 20 cm.

• PicoQuant PQRNG 150—generator by the German company PicoQuant, founded in 1996, specializing in the production of optical and electronic devices and components. Its offer includes a generator that uses a high-speed quantum process. The manufacturer does not publish the detailed parameters of the device. Speed: 150 Mbps USB, entropy source: generation process based on photon arrival times, size—macroscopic.

• Whitewood Entropy Engine—generator from the American company Whitewood, established to commercialize its own quantum random number generator for cryptographic applications. Speed: 350 Mbps PCI Express, entropy source: lensing a photon beam onto the detector, size—macroscopic.

• QuintessenceLabs qStream—generator of the Australian company QuintessenceLabs established as a spin-off of the Australian National University. The company offers comprehensive solutions in the field of modern cryptography, including quantum cryptography. The proposed solutions, both the quantum key distribution (QKD) and QRNG systems, are characterized by very high speed parameters (related to the physical model used, which, however, on the other hand, may be questioned as not fully quantum). Speed: 1 Gbits/s RJ-45 Entropy source: dividing a multi-photon laser beam into two and “subtracting” them from each other, size—macroscopic.

• QuantumNumbersCorp QNG2—generator by the Canadian company Quantum Numbers Corporation (QNC) established for the commercialization of security systems based on QRNG. The company’s activities are aimed at creating microprocessor solutions for the broad financial, banking, military, mobile devices and telecommunications markets. They propose the first QRNG in the form of a chip on the commercial market. Speed: 1 Gbits/s, entropy source: quantum tunneling, size—miniaturized to a micro-chip (not fully quantum—electron transport may have a significant classical component).

• EYL Micro Quantum Random Number Generator—product of the South Korean company EYL (Everywhere in Your Life, founded in 2015), which promotes its technology based on the decay of radioactive isotopes, miniaturized to the size of a chip (sub-centimeter size). The company was awarded the title of the most promising startup in 2016 in the international competition in Boston, USA. The offer is focused on the integration of QRNG with all possible IT-based solutions, in particular with intelligent factories, smart homes or intelligent mobility (e.g., intelligent cars). Speed: 40 Gb/s, entropy source: decay of radioactive isotopes (probability of classical noise contribution).

• qutools quRN—generator of the German company qutools, which is a spin-off of the University in Munich from 2005. The company specializes in the production and distribution of quantum computing devices, including sources of entangled photons, QRNGs, quantum cryptography systems and optics in the quantum regime. Speed: 50 Mb/s USB, entropy source: uncertainty of detection times with single-photon detectors of photons emitted from the LED, size—macroscopic.

• Micro Photon Devices Quantum Random Number Generator – product of the Italian company Micro Photon Devices, which is a spin-off of the Polytechnic University of Milan, founded in 2004. The company manufactures specialized optical equipment in the quantum regime (e.g., single-photon SPAD detectors), including QRNG based on the phenomena of quantum optics. Speed: 128 Mb/s USB, entropy source: internal characteristics of the illuminated quantum detector in the form of an avalanche diode, size—macroscopic.

• QuantumCTek Quantum Random Number Generator QRNG100E – generator of the Chinese company QuantumCTek Co., Ltd., a Chinese pioneer and leader in commercialized quantum information technology. The product is claimed to be one of the fastest commercial QRNG. Speed: 600 Mb/s USB, entropy source: based on the phase noise of laser spontaneous emission (size—macroscopic, probable classical noise component).

• uside FMC 400 – generator of the Spanish company Quside, which is a spin-off of The Institute of Photonic Sciences (ICFO) in Barcelona, founded in 2017. In 2020 the company launched its first QRNG Quside FMC 400, based on Quside’s proprietary phase-diffusion quantum random number generation technology. Speed: 400 Mb/s USB, entropy source: phase-diffusion process for light emitter due to spontaneous emission events (size—macroscopic, probable classical noise component) [40,41,42,43, 67].