Coherent-State Quantum Random Number Generation

• Coherent-state QRNG is a technique that utilizes the inherent quantum uncertainty of optical coherent states (vacuum and laser) to generate fundamentally unpredictable random numbers.
• It employs advanced measurement architectures—such as homodyne, heterodyne, phase noise sampling, and photon-number parity—to ensure robust entropy extraction and high-speed bit generation.
• Integration with telecom-grade and photonic integrated circuits enables output rates from Mbps to tens of Gbit/s, supporting critical applications in cryptography, simulation, and satellite communications.

Coherent-state quantum random number generation (QRNG) leverages the intrinsic quantum uncertainty of optical coherent states—predominantly vacuum or laser-generated states—to generate fundamentally unpredictable random numbers. Relying on quantum-optical measurement processes such as homodyne detection, heterodyne detection, phase noise sampling, photon-number parity measurements, and multi-mode interference, coherent-state QRNGs reach bit rates from several Mbps up to tens of Gbit/s, with rigorous entropy estimation and robust statistical security. Advances in integration enable deployment on telecom-grade transceivers and photonic integrated circuits, with applications ranging from cryptography to simulation and satellite payloads.

1. Quantum Principles Underlying Coherent-State QRNG

Coherent-state QRNG exploits the quantum-mechanical description of the electromagnetic field: the fundamental annihilation and creation operators $a$ and $a^{\dagger}$, and their conjugate quadrature operators,

$$X = \frac{a + a^{\dagger}}{\sqrt{2}}, \quad P = \frac{a - a^{\dagger}}{i\sqrt{2}},$$

with $[X, P] = i\hbar$. The vacuum state $|0\rangle$ exhibits Gaussian statistics with zero mean and variances $$\langle \Delta X^2 \rangle = \langle \Delta P^2 \rangle = \hbar / 2$$—no measurement can surpass this unpredictability due to Heisenberg's uncertainty relation (Milovančev et al., 2020). When a coherent state $|\alpha\rangle$ is measured, the quantum observables $X$ and $P$ possess intrinsic, irreducible uncertainty, forming a natural entropy source.

Coherent-state QRNGs harness quantum randomness from:

• Vacuum fluctuation quadratures (via balanced homodyne/heterodyne detection) (Milovančev et al., 2020, Kincaid et al., 21 Oct 2025, Samsonov et al., 2020, Rao et al., 10 Dec 2025)
• Laser phase noise (phase diffusion and reconstruction) (Li et al., 16 Jan 2024, Sun et al., 2017, Álvarez et al., 2019)
• Photon-number parity (even/odd photon number in coherent pulses) (Gerry et al., 2021)
• Multi-mode interference and bunching (enhanced valid output via Hong–Ou–Mandel-type effects) (Silva et al., 2016)

2. Measurement Architectures and Implementation Strategies

2.1 Homodyne and Heterodyne Detection

In balanced homodyne detection, a strong local oscillator (LO) is mixed with the vacuum or signal mode in a $90^\circ$ optical hybrid, and the resulting output currents from balanced photodetectors encode the vacuum quadrature ($X$ or $P$) with shot-noise-limited variance. Heterodyne detection, typically via a 4x4 multimode interference coupler, simultaneously samples both quadratures ($X$, $P$) (Kincaid et al., 21 Oct 2025).

Key implementation paradigms include:

• Receiver-centric architectures, reusing coherent receivers in telecom transceivers to sample vacuum fluctuations at GHz bandwidths, yielding extracted rates $\sim$1.5–6 Gbit/s (Milovančev et al., 2020).
• Transmitter-centric architectures, utilizing monitor photodiodes in polarization-multiplexed I/Q modulators for in-situ quadrature measurement during blanked data slots, enabling lower-cost QRNG integration at Mbit/s rates (Milovančev et al., 2020).

2.2 Phase Noise and Phase Reconstruction

Spontaneous emission induces quantum phase diffusion in semiconductor lasers, described by

$$\sigma^2 = \langle [\Delta \phi_0(t)]^2 \rangle = 2 T_{\text{delay}} / \tau_c,$$

where phase difference distributions become uniform as $\sigma^2 \gg 10$ (Li et al., 16 Jan 2024). Simultaneous I/Q sampling in a $90^\circ$ hybrid allows direct reconstruction of the phase, avoiding the entropy halving seen in single-quadrature sampling. The reconstructed phase is then discretized into bins, yielding $n$ bits/sample with min-entropy $n$ (Li et al., 16 Jan 2024, Álvarez et al., 2019).

2.3 Photon-Number Parity and Interferometric Enhancement

Parity QRNGs utilize projective measurements of Fock-space even/odd number states, where for moderate coherent amplitudes ($|\alpha|^2 \gtrsim 6$), $P_{\text{even}} = P_{\text{odd}} \approx 0.5$ (Gerry et al., 2021). Implementation employs cascaded 50:50 beam splitters and photon-number-resolving detectors.

Interference of two indistinguishable weak coherent states at a beam splitter (Hong–Ou–Mandel regime) enhances the valid bit fraction to $P_{\text{gen}} \approx 0.66$ for optimized parameters, a 32% improvement over standard path-splitting QRNGs (Silva et al., 2016).

3. Randomness Extraction and Entropy Quantification

Quantum min-entropy per digitized sample is bounded by

$$H_{\min} \geq -\log_2[p_{\max}] \approx -\log_2(\Delta x / (\sqrt{2\pi}\sigma_{\text{tot}})),$$

where $\Delta x$ is ADC bin width and $\sigma_{\text{tot}}$ is the total measured variance (Milovančev et al., 2020, Rao et al., 10 Dec 2025, Kincaid et al., 21 Oct 2025, Zhou et al., 2018). Adversarial knowledge of all classical noise is conservatively assumed. Strong (two-universal) hash extractors, in particular Toeplitz-matrix hashing, compress raw samples to near-uniformity at a ratio dictated by estimated $H_{\min}$ (Milovančev et al., 2020, Li et al., 16 Jan 2024).

Source-device-independent constructions rigorously bound $H_{\min}$ against arbitrary quantum source-side attacks, relying only on the calibrated, trusted measurement device (Kincaid et al., 21 Oct 2025). In multi-mode and phase-reconstruction schemes, entropy per sample approaches the digitizer capacity without $1/2$ entropy loss (Li et al., 16 Jan 2024, Samsonov et al., 2020).

4. Performance Metrics and Integration in Telecom/Integrated Platforms

Recent experimental and prototype implementations achieve secure QRNG bit rates as outlined:

Architecture	Extracted Bit Rate	Implementation Features
Intradyne receiver-centric	$1.5$–$6$ Gbit/s	Standard PM 90° hybrid receivers, no extra optics
Transmitter-centric (monitor PDs)	$5.6$ Mbit/s	On-modulator monitors, no added passives
Photonic integrated circuit (PIC)	$35$ Gbit/s	Monolithic InP, source-device-independence (Kincaid et al., 21 Oct 2025)
Multi-mode, phase-modulated	$400$ Mbit/s	LiNbO$_3$ modulator, balanced detection (Samsonov et al., 2020)
Reconstructed phase (I/Q)	$1.96$ Gbit/s	90° hybrid, 200 MSa/s, no entropy halving (Li et al., 16 Jan 2024)
Parity/counting, beamsplitter	$100$ Mbit/s	SNSPD array, parity of photon count (Gerry et al., 2021)
Homodyne CV-QRNG (sat. payload)	$19.5$ Kbit/pass ($\sim$2 bits/samp)	Balanced homodyne on flight hardware (Rao et al., 10 Dec 2025)

All reported implementations utilize strong randomness certification: NIST SP 800-22, DieHard, and TestU01 suites are routinely passed after extraction. Real-time entropy estimation is embedded in hardware or firmware.

5. Security Models and Loss/Device Side-Channel Resistance

Coherent-state QRNG security analyses encompass:

• Trusted-device models, where only quantum entropy sources (vacuum, phase noise) matter and all classical noise is adversarially assigned (Milovančev et al., 2020, Li et al., 16 Jan 2024, Samsonov et al., 2020, Rao et al., 10 Dec 2025).
• Measurement-device-independent (MDI) models, such as qubit-POVM tomography, which self-test the measurement by random basis choices and admit arbitrarily low detection efficiency (Cao et al., 2015).
• Source-device-independent models, where the entropy source (vacuum port) may be adversarial, and only the detection chain is trusted; lower bounds on $H_{\min}$ are guaranteed from the measurement POVM and ADC calibration (Kincaid et al., 21 Oct 2025).

Digital instantiations replicate quantum statistics (Poisson counting, modular projection) using system-jitter entropy, with mathematically rigorous min-entropy proofs and robust resistance to timing-side channels (Kuang, 11 Dec 2025).

6. Practical Limitations, Robustness, and Integration Opportunities

Physical performance is constrained by:

• Bandwidth of balanced detectors ($\gtrsim$10 GHz for multi-Gbit/s rates), LO power, and phase noise characteristics.
• Electronic noise and DC offsets, requiring continuous calibration, PID feedback, and tuning (e.g. phase modulator index in multi-mode QRNGs).
• Interferometer stability; phase-reconstruction schemes mitigate requirements for active stabilization by extracting phase modulo $2\pi$ directly (Li et al., 16 Jan 2024).

Coherent-state QRNG is uniquely compatible with telecom standards, enabling direct embedding in coherent transceivers and photonic integrated circuits (Milovančev et al., 2020, Kincaid et al., 21 Oct 2025). Dual-use platforms—for instance, QRNG modules in quantum key distribution payloads or satellite communications—are realized by repurposing existing optical and electronic resources (Rao et al., 10 Dec 2025).

7. Prospects and Advanced Developments

Coherent-state QRNG continues to scale in rate and integration:

• Monolithic integration achieves source-device-independence, >30 Gbit/s rates, and plug-and-play black-box operation in InP platforms (Kincaid et al., 21 Oct 2025).
• Phase-reconstruction and multi-quadrature techniques nearly double the extractable entropy relative to single-quadrature sampling (Li et al., 16 Jan 2024).
• Space-qualification for CV-QRNG on satellite payloads demonstrates resilience across extreme operational conditions (Rao et al., 10 Dec 2025).
• Digital approaches using computational permutation and OS-level entropy extend the design space for purely software-based implementations with provable uniformity (Kuang, 11 Dec 2025).

Open technical questions center on scaling secure rates to $>100$ Gbit/s, further minimizing photonic/electronic footprints for mass-deployment, and unifying classical and quantum entropy modeling in hybridized platforms.