⚛️ quantum physics

Semi-device-independent randomness certification on discretized continuous-variable platforms

This paper proposes a semi-device-independent scheme for certifying genuine quantum randomness on continuous-variable optical platforms by restricting state preparations to a two-level Fock subspace, demonstrating that simple homodyne-based setups can achieve dimension-witness violations and positive min-entropy even under realistic experimental imperfections.

Original authors: Moisés Alves, Vitor L. Sena, Santiago Zamora, Tailan S. Sarubi, A. de Oliveira Junior, Alexandre B. Tacla, Rafael Chaves

Published 2026-04-10

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CC BY 4.0

Original authors: Moisés Alves, Vitor L. Sena, Santiago Zamora, Tailan S. Sarubi, A. de Oliveira Junior, Alexandre B. Tacla, Rafael Chaves

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to generate a truly random number, like rolling a perfect die, but you don't trust the dice manufacturer. Maybe the dice are weighted, or maybe the manufacturer is secretly controlling the outcome with a hidden script. In the world of quantum physics, this is the challenge of Randomness Certification: proving that the numbers you get are truly unpredictable and not just a clever trick.

This paper is about building a better, more practical "dice factory" using light, and proving that the randomness it produces is genuine, even if you don't know exactly how the factory works inside.

Here is the breakdown of their work using simple analogies:

1. The Problem: The "Black Box" Dilemma

Usually, to prove a quantum device is truly random, you have to either:

Trust the manufacturer completely (Device-Dependent): "I believe the company says this is a quantum die."
Assume the laws of physics are perfect and the setup is flawless (Device-Independent): This requires incredibly complex, expensive labs with lasers and detectors that must be perfectly aligned, like trying to catch a specific raindrop in a hurricane.

The authors wanted a middle ground: Semi-Device-Independent (SDI).

The Analogy: Imagine you are buying a magic coin. You don't trust the seller, but you do trust that the coin is small enough to fit in your pocket (a size limit). You don't need to see the seller's workshop; you just need to know the coin isn't a giant, hidden mechanism. If the coin behaves in a way that a "pocket-sized" object shouldn't be able to do, you know it's magic (quantum).

2. The Platform: Light as a "Continuous River"

Most quantum experiments use single particles (like individual photons) which are like distinct marbles. This paper uses Continuous Variable (CV) systems, which are like a flowing river of light.

The Challenge: Rivers are continuous; you can't just count "1, 2, 3" easily. You have to measure the water level, which gives you a smooth number like 3.14159...
The Solution: The authors figured out how to "bucket" this river. They take the smooth flow of light and chop it into two buckets: "High" and "Low." This turns the continuous river into a simple "Heads or Tails" coin flip that computers can use.

3. The Trick: The "Two-Level" Rule

To prove the randomness, they impose a strict rule: The light can only exist in two states.

The Metaphor: Imagine a piano. Usually, a piano has 88 keys. But for this experiment, the authors say, "We are only allowed to press the C and the D keys."
By restricting the light to just these two "keys" (the lowest two energy levels of light, called Fock states), they create a strict size limit. If the light behaves in a way that requires more than two keys to explain, they know it's genuinely quantum and not a classical trick.

4. The Tools: Homodyne vs. Displacement

They tested two different ways to measure this light:

Homodyne Detection (The "Ruler"): This measures the wave of light very precisely. It's like using a super-accurate ruler to measure the height of the water. It's very efficient (rarely misses a drop) but harder to turn into a simple "Heads/Tails" result.
Displacement Detection (The "Push"): This involves gently pushing the light wave and seeing if a detector "clicks." It's like pushing a swing to see if it goes over a bar. It's easier to turn into a binary click/no-click, but it's more prone to missing things (losses).

The Discovery: They found that even using just the "Ruler" (Homodyne), they could prove the randomness! This is huge because rulers are easier to build and less likely to fail than the complex "click" detectors.

5. The "Free Spirit" Experiment (Reference Frames)

In many quantum experiments, Alice (the sender) and Bob (the receiver) must be perfectly aligned, like two people holding compasses that must point North. If one person rotates their compass, the experiment fails.

The Paper's Breakthrough: They showed that even if Alice and Bob are totally disoriented—like two people trying to play a game while spinning in circles on a merry-go-round—they can still prove the randomness exists.
The Analogy: Imagine Alice sends a secret message using a specific hand gesture. Usually, Bob needs to be facing the same way to understand it. The authors showed that even if Bob is spinning wildly, as long as he has a few different ways to interpret the gesture, he can still figure out the message is real and random.

6. Why This Matters

Security: Randomness is the foundation of encryption. If your "random" numbers are predictable, your bank account is vulnerable.
Simplicity: This method doesn't require a million-dollar lab. It uses standard optical equipment (lasers and mirrors) that are already used in fiber-optic internet cables.
Robustness: It works even if the equipment isn't perfect (it has some "loss" or noise).

Summary

The authors built a simple, robust, and trustworthy "randomness machine" using light. They proved that by restricting the light to a tiny "two-key" range and using standard measuring tools, they can generate numbers that are mathematically guaranteed to be unpredictable. They did this without needing perfect alignment or expensive, complex setups, making true quantum randomness accessible for everyday technology like secure communications and future quantum computers.

1. Problem Statement

Randomness is critical for secure communication and information processing. While quantum mechanics offers intrinsic unpredictability, certifying that generated randomness is genuine (and not merely pseudorandom due to device imperfections) is challenging.

Device-Independent (DI) vs. Semi-Device-Independent (SDI): DI protocols (based on Bell inequalities) are robust but experimentally demanding (requiring loophole-free detection). SDI protocols relax assumptions by bounding a specific resource (e.g., system dimension) while treating devices as black boxes.
The Gap: Most SDI randomness certification has focused on Discrete-Variable (DV) systems (qubits). Continuous-Variable (CV) optical systems (using homodyne and displacement measurements) are attractive for scalability and high detection efficiency but are underexplored for SDI certification.
Specific Challenges in CV:
1. Homodyne measurements yield continuous outcomes, which must be discretized (binned) to fit dimension-witness frameworks.
2. Enforcing a dimension bound (limiting the system to a qubit subspace) in a CV platform requires operational constraints.
3. Establishing a robust link between witness violations and extractable randomness (min-entropy) in the presence of realistic losses and reference frame misalignments.

2. Methodology

The authors propose an SDI Prepare-and-Measure (PAM) scheme tailored for CV optics.

Operational Dimension Bound: Instead of assuming a hard dimension limit, the authors restrict state preparations to the two-level Fock subspace ( $\{|0\rangle, |1\rangle\}$ ). This effectively encodes a qubit within the CV platform.
Measurement Schemes:
1. Homodyne Detection: Measures field quadratures. Outcomes are continuous but are binned into dichotomic results ( $+1/-1$ ) based on a defined interval. This offers near-unity detection efficiency.
2. Displacement-based Photodetection: Uses a displacement operator followed by an on/off photon detector. This is inherently binary but suffers from lower detection efficiency.
3. Hybrid Schemes: Combinations of both measurement types.
Witnesses: The study analyzes linear dimension witnesses for three scenarios:
- $(3, 2, 2, 2)$ : 3 preparations, 2 measurements, 2 outcomes.
- $(4, 2, 2, 2)$ : 4 preparations, 2 measurements, 2 outcomes.
- $(3, 3, 2, 2)$ : 3 preparations, 3 measurements, 2 outcomes.
Randomness Metrics: The authors calculate the min-entropy ( $H_\infty = -\log_2 p_{guess}$ ), where $p_{guess}$ is the maximum probability an adversary can correctly guess the outcome. They derive both numerical bounds (via optimization) and an analytical bound for the tilted witness $S_3(w)$ .
Robustness Modeling:
- Losses: Modeled via an amplitude-damping channel (photon loss) affecting displacement measurements.
- Reference Frames: The study investigates scenarios where Alice and Bob do not share a reference frame (random relative phases), testing if violations persist.

3. Key Contributions

First CV SDI Randomness Certification: Demonstrates that dimension-witness violations and certified randomness can be achieved in CV platforms using only standard linear-optics measurements (homodyne and displacement).
Analytical Witness-to-Randomness Map: Derives a closed-form analytical lower bound for the min-entropy based on the violation of the tilted witness $S_3(w)$ , providing a device-independent guarantee without needing numerical optimization for every run.
New Quantum Bounds: For the $(3, 3, 2, 2)$ $(3, 3, 2, 2)$ scenario, the authors analytically derive previously unknown quantum maxima for two witnesses:
- $S_{33,1}^{Q} = 3\sqrt{3} \approx 5.196$
- $S_{33,2}^{Q} = 6$
Reference Frame Independence: Proves that shared reference frames are not necessary for SDI randomness certification in PAM scenarios. By increasing the pool of available measurement settings, a violating pair can always be found regardless of phase misalignment.
Optimization of Configurations: Systematically compares Homodyne-only (HH), Displacement-only (DD), and Hybrid (HD/DH) setups, identifying optimal strategies for maximizing violations and robustness.

4. Key Results

Violation of Dimension Witnesses:
- Displacement (DD) schemes achieve violations closest to the theoretical quantum maximum (e.g., $S_3 \approx 3.78$ vs. theoretical $3.82$).
- Homodyne (HH) schemes, while yielding lower violations ( $S_3 \approx 3.11$ ), are significant because they are loophole-free regarding detection efficiency (near-unity efficiency).
- Hybrid schemes offer a balance, often outperforming pure homodyne setups in terms of violation magnitude while maintaining better robustness to losses than pure displacement setups.
Certified Randomness (Min-Entropy):
- For the optimized tilted witness $S_3(w)$ , the scheme certifies up to $H_\infty \approx 0.382$ bits per trial.
- For the $(3, 3, 2, 2)$ scenario, certifiable randomness reaches $0.342$ bits ( $S_{33,1}$ ) and $0.263$ bits ( $S_{33,2}$ ).
- These values exceed previous best-known results for simpler schemes requiring fewer state preparations.
Robustness to Imperfections:
- Detection Efficiency: The critical detection efficiency ( $\eta_{crit}$ ) required to observe a violation is approximately 45.8% for displacement-based measurements in the $(4, 2, 2, 2)$ scenario. Homodyne measurements remain robust even at lower efficiencies due to their high intrinsic efficiency.
- Reference Frames: In the absence of shared reference frames, less than 0.1% of random phase rotations fail to produce a violation. With a pool of just 4 displacement settings, a violation is guaranteed for any relative phase.
Analytical Bound: The derived analytical bound for $S_3(w)$ closely tracks the numerically optimized min-entropy, validating its use for real-time certification.

5. Significance and Outlook

Practicality: This work bridges the gap between theoretical SDI protocols and practical optical implementations. It shows that high-quality randomness can be certified using standard, scalable optical hardware (homodyne detectors and on/off photodiodes) without the extreme complexity of single-photon DI setups.
Scalability: The ability to certify randomness without shared reference frames and with high tolerance to loss makes this approach highly suitable for real-world Quantum Random Number Generators (QRNGs).
Future Directions: The authors suggest extending the framework to include dark counts, mode mismatch, and non-Gaussian resources. They also highlight the potential for "energy-constrained" SDI protocols as a complementary approach to dimension-bounded ones.

In summary, the paper establishes a robust, scalable, and experimentally feasible framework for certifying quantum randomness in continuous-variable systems, overcoming previous limitations related to detection efficiency, reference frame alignment, and the lack of analytical certification tools.