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Laser Physics

Homodyne detection in quantum optics: deterministic extractors and quantum random number generators on ‘vacuum fluctuations’

S. N. Molotkov

Published Apr 7, 2022• 2 citations
Publisher

Abstract

Abstract Quantum random number generators with a continuous variable are considered based on a primary randomness of the outcomes of homodyne measurements of a coherent state. A deterministic method of extraction of truly random 0 and 1 from the primary sequence of measurements of the quadrature of the field in homodyne detection is considered. The method, in the case of independence of successive measurement outcomes, in the asymptotic limit of long sequences, allows us to extract with a polynomial complexity all the true randomness contained in the primary sequence. The method does not require knowledge of the probability distribution function of the primary random sequence, and also does not require additional randomness in the extraction of random 0 and 1. The approach with deterministic randomness extractors, unlike other methods, contains fewer assumptions and conditions that need to be satisfied in the experimental implementation of such generators, and is significantly more effective and simple in experimental implementation. The fundamental limitations dictated by nature for achieving statistical independence of successive measurement outcomes are also considered. The statistical independence of the measurement outcomes is the equivalent of true randomness, in the sense that is possible in the case of the independence of the measurement outcomes, provably, with deterministic extractor, to extract a ‘truly random sequence of 0 and 1’. It is shown that in the asymptotic limit it is possible to extract all the true randomness contained in the outcomes of physical measurements.

DOI
10.1088/1555-6611/ac5ccc

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Abstract Quantum random number generators with a continuous variable are considered based on a primary randomness of the outcomes of homodyne measurements of a coherent state. A deterministic method of extraction of truly random 0 and 1 from the primary sequence of measurements of the quadrature of the field in homodyne detection is considered. The method, in the case of independence of successive measurement outcomes, in the asymptotic limit of long sequences, allows us to extract with a polynomial complexity all the true randomness contained in the primary sequence. The method does not require knowledge of the probability distribution function of the primary random sequence, and also does not require additional randomness in the extraction of random 0 and 1. The approach with deterministic randomness extractors, unlike other methods, contains fewer assumptions and conditions that need to be satisfied in the experimental implementation of such generators, and is significantly more effective and simple in experimental implementation. The fundamental limitations dictated by nature for achieving statistical independence of successive measurement outcomes are also considered. The statistical independence of the measurement outcomes is the equivalent of true randomness, in the sense that is possible in the case of the independence of the measurement outcomes, provably, with deterministic extractor, to extract a ‘truly random sequence of 0 and 1’. It is shown that in the asymptotic limit it is possible to extract all the true randomness contained in the outcomes of physical measurements.
Quantum random number generators (QRNGs) harness the intrinsic randomness in measurement processes: the measurement outputs are truly random, given the input state is a superposition of the eigenstates of the measurement operators. In the case of trusted devices, true randomness could be generated from a mixed state ρ so long as the system entangled with ρ is well protected. We propose a random number generation scheme based on measuring the quadrature fluctuations of a single mode thermal state using an optical homodyne detector. By mixing the output of a broadband amplified spontaneous emission (ASE) source with a single mode local oscillator (LO) at a beam splitter and performing differential photo-detection, we can selectively detect the quadrature fluctuation of a single mode output of the ASE source, thanks to the filtering function of the LO. Experimentally, a quadrature variance about three orders of magnitude larger than the vacuum noise has been observed, suggesting this scheme can tolerate much higher detector noise in comparison with QRNGs based on measuring the vacuum noise. The high quality of this entropy source is evidenced by the small correlation coefficients of the acquired data. A Toeplitz-hashing extractor is applied to generate unbiased random bits from the Gaussian distributed raw data, achieving an efficiency of 5.12 bits per sample. The output of the Toeplitz extractor successfully passes all the NIST statistical tests for random numbers.
Quantum random number generators (QRNGs) are currently in high demand across a large number of cryptographic applications as secure sources of true randomness. In this work, we characterize the conditions from which randomness can be extracted in a QRNG based on homodyne measurements of vacuum fluctuations by assessing the impact of experimental limitations, such as the digitizer resolution or the presence of excess local oscillator (LO) noise due to an unbalanced detection. Moreover, we propose an estimation method to quantify the excess entropy contribution introduced by an unbalanced detection and analyze the implementation of the post-processing algorithm. Finally, we submitted the generated numbers to a set of statistical tests to assess the quality of its output randomness and verified that it passes the standard libraries.
A quantum random number generator (QRNG) based on the quantum nature of vacuum fluctuations allows one to obtain random bit sequences that can be used in applications that require a high degree of randomness.In that type of quantum random generation system, optical beam splitters with two inputs and two outputs are normally used.A comparison of Y-splitter and spatial beam splitters shows that for two types of optical splitters, the quantum mathematical description of output signals is identical.This allows the use of fiber Y-splitters in practical QRNG schemes.The possibility of generating true random bits was demonstrated experimentally by using quantum random number generator based on homodyne detection.
One of the main problems in the construction of quantum random number generators Ð obtainment of a provably random output sequence from physical measurements, i.e., from an initial sequence generated by a physical random number generator Ð is investigated.The conceptual feasibility and the conditions under which randomness can be `reached,' as well as the meaning of the `provable randomness' term, are discussed.We consider the methods of extracting provably random bit sequences from stationary Markov chains of finite order, i.e., under the assumption of the finite depth of the dependence of the results of physical measurements on the prehistory, which is an adequate approximation of the real situation.The extraction of the output provably random bit sequence from the initial sequence of the results of physical measurements using V F Babkin's effective arithmetic coding method is demonstrated.It is shown that random bit sequences can be provably obtained even from primary sequences of the results of physical measurements, which are dependent on (correlated to) any finite depth (prehistory).We aim to reveal the connection of various approximations employed in the development and description of methods for obtaining random bit sequences with the basic physical limitations of Nature.The mathematical proofs are detailed to the level of practical algorithms applied in real random number generators.The necessary mathematical propositions are presented at an intuitive level for physicists, do not require prior knowledge in this area, and are comprehensible to undergraduate university students.
Abstract The nature of randomness and constructive and provable methods to obtain (extract) it from observations of physical systems are discussed. True randomness, which exists only in a microcosm in the quantum-mechanical description of physical systems, is a fundamental property of quantum systems, which manifests itself in the outcomes of measurements upon quantum systems. The classical description of physical systems does not include any randomness and, in fact, it is introduced ‘manually’ by means of uncertainty — unknown initial conditions. Methods to really ‘feel’ quantum randomness are discussed using the example of a quantum device, a random number generator. Issues related to the ‘proof’ of randomness — testing of numerical sequences — are reviewed, and logical constructions that underlie such testing are analyzed. A mathematical apparatus is used to this end, which does not require special academic training, so standard knowledge from university courses on quantum mechanics and probability theory is sufficient. The authors aim to track a unified logical path from the origin of randomness in the quantum domain to its extraction, physical implementation, and testing.
It is a well-known fact in classical information theory that no deterministic procedure can extract close-to-ideal randomness from an arbitrary entropy source. On the other hand, if additional knowledge about the source is available -- e.g., that it is a sequence of independent Bernoulli trials -- then deterministic extractors do exist. For quantum entropy sources, where in addition to classical random variables we consider quantum side information, the use of extra knowledge about their structure was pioneered in a recent publication [C. Foreman and L. Masanes, Quantum 9, 1654 (2025)]. In that work, the authors provide deterministic extractors for device-independent randomness generation with memoryless devices achieving a sufficiently high CHSH score. In this work, we port their construction to the prepare-and-measure scenario. Specifically, we prove that the considered functions are also extractors for memoryless devices in settings with partial trust, either in the state preparation or in the measurement, as well as in a semi-device-independent setting under an overlap assumption on the prepared quantum states. Within this last setting, we simulate the resulting randomness generation protocol on a novel and experimentally relevant family of behaviors, observing positive key rates already for 7×1037\times 10^37×103 rounds.