⚛️ quantum physics

Noise robustness of three outcome Bell certified quantum randomness

This paper demonstrates that device-independent certification of global randomness in bipartite scenarios with three outcomes per party is generically robust against realistic noise, with many newly identified Bell expressions achieving near-maximal min-entropy using fewer measurement settings than previously possible.

Original authors: Raffaele D'Avino, Ignacio Perito, Piotr Mironowicz, Antonio Acín, Remigiusz Augusiak

Published 2026-06-23

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Raffaele D'Avino, Ignacio Perito, Piotr Mironowicz, Antonio Acín, Remigiusz Augusiak

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 a perfect roll of a die, using a mysterious black box. In the old days, scientists had to trust that the box was built exactly as they thought it was. But what if the box was a trick? What if a sneaky hacker inside was secretly rigging the rolls?

This paper is about a way to prove the numbers are truly random without trusting the box at all. It uses a concept called "Bell inequalities," which are like a special test to see if two distant boxes are "talking" to each other in a way that only quantum physics allows. If they pass the test, we know the numbers are genuinely unpredictable.

Here is the breakdown of what the researchers did, using some everyday analogies:

1. The Problem: The "Coin Flip" Limit

Most previous experiments used "binary" outcomes, like flipping a coin (Heads or Tails). Even if you win the quantum test, a coin flip can only give you one bit of randomness. It's like trying to fill a swimming pool with a teaspoon; you can get some water, but it's slow and limited.

The researchers asked: What if we use a six-sided die instead of a coin? Or, in their case, a three-sided die. This allows for more possibilities (more "outcomes") and potentially much more randomness.

2. The Challenge: The "Noise" in the Room

In the real world, nothing is perfect. There is always "noise"—static on a radio, a wobbly table, or a slightly broken machine. In quantum experiments, this noise can ruin the test. If the noise is too high, the "randomness" disappears, and the test fails.

The big question was: Do these fancy "three-sided die" experiments break easily when there is noise, or are they tough enough to handle real-world imperfections?

3. The Experiment: A Massive Digital Search

Instead of just building one or two perfect "dice" experiments by hand, the researchers used a computer to generate thousands of random Bell inequalities (different rules for the test).

Think of it like a chef trying to find the best recipe for a cake. Instead of writing one recipe from scratch, they randomly mixed ingredients (coefficients and settings) to create 50,000 different recipes. Then, they tested each one to see:

How much randomness does it produce?
How much "noise" (bad ingredients) can it handle before the cake collapses?

4. The Surprising Results

The team found two major things:

More Randomness is Possible: They confirmed that using three outcomes (like a 3-sided die) allows them to certify significantly more randomness than the old "coin flip" (binary) methods. In the best cases, they could get close to 3.17 bits of randomness, which is a huge jump from the 2-bit limit of the best coin-flip methods.
Robustness is Common, Not Rare: They expected that only very carefully engineered, complex recipes would work well in noisy conditions. Instead, they found that many of the randomly generated recipes were surprisingly tough. Even when they added a lot of noise, a large number of these random inequalities still produced valid, certified randomness.

5. The "Golden Ticket" Inequality

Among the thousands of random recipes, they found one specific inequality (a specific set of rules) that was a "golden ticket."

It was surprisingly simple.
It required fewer settings (inputs) than other complex methods.
It produced almost the maximum amount of randomness possible.
It was very resistant to noise.

It was like finding a simple, homemade cookie recipe that tasted just as good as a Michelin-star dessert and didn't crumble if you dropped a little flour on it.

The Bottom Line

The paper concludes that you don't need to be a genius engineer to build a perfect quantum randomness generator. By using higher-dimensional systems (more than just "heads or tails") and looking at families of rules, you can naturally find methods that are:

Stronger: They generate more randomness.
Tougher: They survive real-world noise better than we thought.
Simpler: Some of the best ones are actually quite easy to set up.

This proves that high-quality, secure randomness isn't just a fragile theoretical idea; it's a robust resource that can be found generically in these quantum systems.

Technical Summary: Noise Robustness of Three-Outcome Bell Certified Quantum Randomness

Problem Statement
Device-independent (DI) randomness generation relies on the violation of Bell inequalities to certify that measurement outcomes are intrinsically unpredictable, without requiring assumptions about the internal workings of the devices. While standard protocols based on binary-outcome measurements (e.g., CHSH) are well-established, they are fundamentally limited in the amount of certifiable randomness per experimental round due to the constraints of qubit systems. Although high-dimensional systems and multi-outcome measurements theoretically offer richer nonlocal correlations and higher entropy, their practical utility remains unclear, particularly regarding their robustness against realistic noise. A critical gap exists in understanding whether multi-outcome scenarios can sustain significant global randomness (the joint unpredictability of all parties' outputs) in noisy environments, and whether such advantages are restricted to carefully engineered inequalities or arise generically.

Methodology
The authors investigate DI randomness certification in bipartite scenarios where each party has three measurement outcomes ( $d=3$ ). The study employs two primary methodological approaches:

Evaluation of Known Inequalities: The authors analyze several established Bell expressions, including SATWAP, CGLMP, Buhrman–Massar (BM), Kaniewski et al., Meyer et al., and a new inequality by Perito et al. To enable global randomness certification, these inequalities are extended by adding terms that require an additional measurement setting on Bob's side, ensuring the "spot setting" (the specific input pair used for generation) certifies uniformly random outcomes.
Systematic Random Search: The authors introduce a method for generating new Bell expressions within two structured families:
- Family 1: Randomized selection of modulo- $d$ probability terms for each setting pair (excluding the spot setting).
- Family 2: Randomized assignment of independent coefficients to each modulo-3 probability term.
  Extensive numerical searches (50,000 expressions per scenario) are conducted across various measurement settings configurations (e.g., 3-2, 3-3, 4-2, 4-3).

Noise Models and Certification Metrics
The robustness of these expressions is evaluated under two noise models:

Werner-type noise: A mixture of the ideal quantum state and the maximally mixed state.
Bell-value noise: Direct degradation of the observed Bell value.

Randomness is quantified via the min-entropy ( $H_\infty = -\log_2 P_{guess}$ ), calculated using the semi-definite programming (SDP) method based on the NPA hierarchy. The study evaluates both the randomness certified at a specific "spot setting" and the average randomness over all possible input settings.

Key Results

Superiority of Multi-Outcome Scenarios: The study confirms that three-outcome scenarios can certify significantly more global randomness than binary-outcome (qubit) counterparts, even under comparable noise levels. Specifically, certain inequalities can certify close to the theoretical maximum of $\approx 3.17$ bits of min-entropy.
Noise Robustness of Known Inequalities: Among the known inequalities, the one introduced by Perito et al. [21] demonstrates the highest noise tolerance, maintaining certification of more than two bits of global randomness at noise levels where other inequalities fail. This surpasses the limit achievable with optimal projective qubit strategies.
Generic Robustness in Random Families: The large-scale numerical study reveals that strong randomness certificates are not rare artifacts of specific engineering.
- In Family 1, a non-negligible fraction of randomly generated expressions certifies substantial randomness (often exceeding 2 bits).
- Crucially, the vast majority of non-trivial certificates exhibit robustness against realistic noise (e.g., $v=0.99$ ), maintaining positive certified randomness well away from the ideal regime.
- Family 2 produces fewer near-optimal certificates compared to Family 1, suggesting that the structural constraints of Family 1 are more conducive to finding high-performing inequalities.
Identification of a Near-Optimal Simple Inequality: The authors identify a specific inequality from Family 1 (Scenario 4-2, denoted as "Case F") that achieves near-maximal global randomness ( $\approx 3.165$ bits under Bell-value noise) while requiring only two measurement settings for one party. This inequality is structurally simple and equivalent to a member of the $(2, 4, 3)$ family from [21] under relabeling.

Significance and Claims
The paper claims that the potential for strong, device-independent randomness expansion in multi-outcome scenarios is not restricted to a few carefully constructed inequalities. Instead, the authors demonstrate that:

High-dimensional correlations are a practical resource: They provide a genuine advantage in certifiable randomness that persists under realistic noise conditions.
Robustness is generic: The ability to certify significant randomness in noisy environments is a common feature within suitably constructed families of Bell expressions, rather than an exceptional property.
Efficiency in measurement settings: It is possible to achieve near-optimal randomness certification with minimal measurement complexity (e.g., using only two settings for one party), which improves the balance between output randomness and the input randomness required for protocol execution.

The authors conclude that structured random generation is a powerful tool for discovering competitive Bell inequalities, offering a systematic path to identify expressions that balance maximal entropy, noise robustness, and experimental simplicity.