亂數被廣泛運用於現今社會當中，如電腦模擬、賭博遊戲以及密碼學 等。現在廣泛使用的偽亂數產生器 (pseudo random number generators)，將稱 為種子 (seed) 的初始值，利用某種演算法的計算，生成出具有近似於隨機 性的數列。
雖然偽亂數產生器具有快速和可再生性的優勢，但因為其存在週期性的 問題。隨著量子電腦的發展，電腦的運算能力將大幅提升，偽亂數在例如 密碼學的應用當中，就會產生被破解的安全性的疑慮，因此能夠有效率的 產生真正隨機的亂數，變成一個很重要的議題。
真亂數產生器 (true random number generators)，是藉由物理世界裡的隨 機過程，如電路雜訊、混沌系統、量子現象等來產生真正隨機的亂數。其 中量子現象，是利用量子力學內在具有的機率性質，來當作亂數隨機性的 來源。
本研究利用平衡零差偵測器 (balanced homodyne detectors) 量測真空態， 實現可產生速度約 80Mbps 之量子亂數產生器 (quantum random number gen- erator)，並探討不同實驗參數對原始資料隨機性的影響，以優化原始資料的 隨機性。

Random numbers have lots of application these days, such as simulations, gam- bling and cryptography. Widely used pseudo random number generators expands a random seed by some deterministic algorithm to random-like sequences. Since it's generated by a deterministic algorithm, there exist patterns behind these sequences. The development of quantum computers poses a threat to the security of pseudo random number generators because of its powerful computing power. Therefore, it is important to develop an efficient way to generate quality random numbers using true random processes.
The true random number generators are realized by unpredictable physical processes such as electronic noise in a circuit, chaotic systems, and quantum phenomena. Among these processes, quantum phenomena exploits the probabilistic nature of quantum mechanics as the source of randomness, which means that it is naturally random.
In this work, we demonstrate a quantum random number generator that can reach a 80Mbps generation rate by sampling vacuum state with a balanced homodyne detector. We also study the influence of different experiment parameters on the autocorrelation function of the raw data in order to improve the randomness.

摘要 i
Abstract ii
1 序論 1
1.1 研究背景..................................... 1
1.2 量子亂數產生器的應用............................. 2
1.3 研究目的..................................... 3
1.4 文獻回顧..................................... 3
2 研究原理 5
2.1 真空態簡介.................................... 5
2.1.1 真空態.................................. 5
2.1.2 量子態量測 ............................... 9
2.1.3 BHD表現對亂數產生之重要性 .................... 11
2.2 隨機性 ...................................... 13
2.2.1 真空態隨機性.............................. 13
2.2.2 最小熵.................................. 14
2.2.3 自相關函數 ............................... 16
2.2.4 隨機性檢定 ............................... 17
3 研究方法與實驗架構 19
3.1 熵源的製備.................................... 19
3.2 資料擷取..................................... 21
3.3 資料後處理.................................... 26
3.3.1 隨機性量化 ............................... 27
3.3.2 隨機性萃取 ............................... 28
3.4 實驗步驟整理 .................................. 30
4 實驗結果與討論 31
4.1 鎖相放大器解調訊號 .............................. 31
4.2 高通濾波器訊號................................. 33
4.2.1 訊噪比.................................. 33
4.2.2 取樣頻率和截止頻率 .......................... 37
5 壓縮態製備 45
5.1 壓縮態 ...................................... 45
5.2 實驗架構..................................... 47
5.3 壓縮態量測結果與分析............................. 49
6 總結與未來工作 55
6.1 總結........................................ 55
6.2 未來工作..................................... 55
A Wiener-Khinchin theorem 計算 57
參考資料 61

[1] Quantum random number generators, https://doi.org/10.48550/arXiv.1604.03304.
[2] T. Symul, S. M. Assad, and P. K. Lam, “Real time demonstration of high bitrate quantum random number generation with coherent laser light,” Applied Physics Letters, vol. 98, 2011.
[3] Z. Zheng, Y. Zhang, W. Huang, and H. Guo, “6 gbps real-time optical quantum ran- dom number generator based on vacuum fluctuation,” Review of Scientific Instruments, vol. 90, 2019.
[4] D. G. Marangon, G. Vallone, and P. Villoresi, “Source-device-independent ultrafast quan- tum random number generation,” Physical Review Letters, vol. 118, 2017.
[5] F. Xu, B. Qi, X. Ma, H. Xu, H. Zheng, and H.-K. Lo, “Ultrafast quantum random number generation based on quantum phase fluctuations,” Optics Express, vol. 20, 2012.
[6] Y.-Q. Nie, L. Huang, Y. Liu, F. Payne, J. Zhang, and J.-W. Pan, “The generation of 68 gbps quantum random number by measuring laser phase fluctuations,” Review of Scientific Instruments, vol. 86, 2015.
[7] C. R. S. Williams, J. C. Salevan, X. Li, R. Roy, and T. E. Murphy, “Fast physical random number generator using amplified spontaneous emission,” Optics Express, vol. 18, 2010.
[8] A. Argyris, E. Pikasis, S. Deligiannidis, and D. Syvridis, “Sub-tb/s physical random bit generators based on direct detection of amplified spontaneous emission signals,” Journal of Lightwave Technology, vol. 30, 2012.
[9] M. Stipčević and B. M. Rogina, “Quantum random number generator based on photonic emission in semiconductors,” Review of Scientific Instruments, vol. 78, 2007.
[10] M. Furst, H. Weier, S. Nauerth, D. G. Marangon, C. Kurtsiefer, and H. Weinfurter, “High speed optical quantum random number generation,” Optics Express, vol. 18, 2010.
[11] M. Ren, E. Wu, Y. Liang, Y. Jian, G. Wu, and H. Zeng, “Quantum random-number generator based on a photon-number-resolving detector,” Physical Review A, vol. 83, 2011.
[12] Samsung galaxy quantum 4 smartphone, https : / / www . idquantique . com / random - number-generation/qrng-use-cases/samsung-qrng-use-case/.
[13] Docusign qscd appliance, https://www.idquantique.com/random-number-generation/ qrng-use-cases/docusign-esignature-use-case/.
[14] Ezquant security key, https://www.idquantique.com/random-number-generation/ qrng-use-cases/ezquant-qrng-use-case/.
[15] Quantum fingerprint security key, https://octatco.com/eng_product6.
[16] How fido works, https://fidoalliance.org/how-fido-works/.
[17] C. Gabriel, C. Wittmann, R. D. D. Sych, U. L. A. W. Mauerer, C. Marquardt, and G. Leuchs, “A generator for unique quantum random num- bers based on vacuum states,” Nature Photonics, vol. 4, pp. 711–715, 2010.
[18] F. Raffaelli, G. Ferranti, D. H. Mahler, et al., “A homodyne detector integrated onto a photonic chip for measuring quantum states and generating random numbers,” Quantum Science and Technology, vol. 3, 2018.
[19] L. Huang and H. Zhou, “Integrated gbps quantum random number generator with real- time extraction based on homodyne detection,” Journal of the Optical Society of America B, vol. 36, 2019.
[20] B. Bai, J. Huang, G.-R. Qiao, et al., “18.8 gbps real-time quantum random number gen- erator with a photonic integrated chip,” Applied Physics Letters, vol. 118, 2021.
[21] T. Michel, “Real-time source-independent quantum random-number generator with squeezed state,” Physical Review Applied, vol. 12, 2019.
[22] M. Fox, “Quantum optics an introduction,” Oxford Press University, pp. 127–133, 138– 141, 2006.
[23] X. Ma, “Postprocessing for quantum random-number generators: Entropy evaluation and randomness extraction,” Physical Review A, vol. 87, 2013.
[24] Y. Shen, “Practical quantum random number generator based on measuring the shot noise of vacuum states,” Physical Review A, vol. 81, 2010.
[25] F. Raffaelli, “Quantum random number generators in integrated photonics,” Ph.D. dis- sertation, University of Bristol, 2019.
[26] J. Y. Haw, “Continuous variable optimisation of quantum randomness and probabilistic linear amplification,” Ph.D. dissertation, National University of Singapore, 2018.
[27] G. E. Box, “Time series analysis,” John Wiley & Sons, pp. 21–25, 28, 30–33, 2016.
[28] A. Rukhin, “A statistical test suite for random and pseudorandom number generators for cryptographic applications,” National Institute of Standards and Technology, 2016.
[29] Introduction to eo modulators, https : / / www . thorlabs . com / newgrouppage9 . cfm ? objectgroup_id=2729.
[30] F. Castanié, “Digital spectral analysis: Parametric, non-parametric and advanced meth- ods,” Wiley, pp. 127–129, 2011.
[31] Squeezed light, https://doi.org/10.48550/arXiv.1401.4118.
[32] Spontaneous parametric down-conversion, https://en.wikipedia.org/wiki/Spontaneous_ parametric_down-conversion.