⚛️ quantum physics

Certifying Randomness or its Lack Thereof for General Network Scenarios

This paper extends device-independent randomness certification to general network scenarios by adapting the inflation technique to certify randomness against beyond-quantum adversaries in bilocality and triangle setups, while also providing methods to certify the absence of randomness and discussing key conceptual challenges in the field.

Original authors: Maria Ciudad Alañón, Daniel Centeno, Andrew Watford, Elie Wolfe

Published 2026-09-30

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Maria Ciudad Alañón, Daniel Centeno, Andrew Watford, Elie Wolfe

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

True randomness is a rare and precious commodity in our universe. In the everyday world, things appear random only because we lack complete information; a coin toss seems unpredictable, but if we knew the exact force of the flip, the air resistance, and the surface it lands on, we could predict the outcome with perfect certainty. This is known as epistemic randomness, or randomness born of ignorance. However, quantum physics offers something far more profound: intrinsic randomness. This is a fundamental unpredictability built into the fabric of reality itself, where no amount of knowledge about the past can ever determine the future. This concept is not just a philosophical curiosity; it is the bedrock of modern secure communication. If we can prove that a number is truly random, we can use it to create encryption keys that are mathematically impossible to crack.

For decades, scientists have tested this intrinsic randomness using a setup known as the Bell scenario, where two distant observers measure particles that are linked in a mysterious way. When their results violate specific mathematical limits, it proves that the universe is not following a hidden, pre-written script, and that the outcomes are genuinely unpredictable. But the universe is more complex than just two people sharing a secret. In recent years, researchers have begun exploring "quantum networks," where multiple independent sources send information to several parties at once, creating intricate webs of connection. The question remained: can we still certify that true randomness exists in these complicated networks, and can we prove when it does not? A team of physicists has now taken a significant step forward, developing new methods to answer these questions in these general network scenarios.

The researchers focused on two specific network shapes: the "bilocality" scenario, which involves three parties connected by two independent sources, and the "triangle" scenario, where three parties are connected in a loop by three independent sources. To tackle the problem, they adapted a powerful computational tool called the inflation technique. Imagine trying to verify if a complex machine is working correctly by building a larger, more detailed version of it to see how the parts interact; the inflation technique works similarly. By mathematically expanding the network into a larger, hypothetical version, the researchers could test whether the observed data could possibly be explained by a predictable, classical system or if it required the fundamental unpredictability of quantum mechanics. They assumed the worst-case scenario for security: an eavesdropper who has access to all the information sources in the network but cannot control them, representing a threat that is even more powerful than what standard quantum theory allows.

Using this method, the team successfully demonstrated that intrinsic randomness can be certified in these network scenarios. They analyzed several specific patterns of data, including those inspired by famous theoretical examples, and found that for certain parties in the network, the outcomes were indeed unpredictable. For instance, in the bilocality scenario, they showed that the middle party, who receives information from two different sources, could still generate certifiable randomness, a phenomenon that does not occur in the simpler two-party setups. In the triangle scenario, they proved that randomness exists even when the parties have no settings to choose from, relying entirely on the structure of the network itself. These findings are significant because they show that the security of random number generation can be extended to more complex, realistic network architectures, provided the sources of information are independent.

However, the work also addressed the flip side of the coin: proving when randomness is absent. It is a common misconception that if a system behaves in a non-classical way, it must be random. The researchers showed that this is not always true. They developed a second set of computational methods to certify that an adversary could perfectly predict the outcomes of a specific party, even if the overall network data looked non-classical. They achieved this by constructing specific causal models where the party in question received only classical, predictable information from their sources, while the rest of the network handled the complex, non-classical parts. In one striking example, they applied this to a specific distribution of data known as RGB3. They found that while the overall pattern was non-classical, a specific party in the triangle network could be predicted with absolute certainty by an eavesdropper. This proves that non-classicality does not automatically guarantee randomness for every individual in the network.

The study highlights a crucial distinction between the randomness of a single person's outcome and the randomness of the group's combined outcome. It is possible for a network to produce data that is unpredictable as a whole, yet for every individual participant to have a predictable result. The researchers found that their methods could certify the lack of randomness for specific individuals by showing that their data could be explained by models where they only received classical inputs. However, they also noted a limitation: if a network is so complex that a party's data cannot be explained by a model where they receive at most one non-classical source, their current tools cannot determine if randomness is present or absent. This leaves an open question for the future: are there networks so complex that they hide their randomness from these detection methods?

Ultimately, this research provides a new toolkit for understanding the flow of information and unpredictability in complex systems. By adapting the inflation technique, the team has shown how to certify the presence of true randomness in networks and, just as importantly, how to certify its absence. These findings refine our understanding of where security can be guaranteed in future quantum networks. While the work is currently theoretical, focusing on the foundational principles of how randomness behaves in these structures, it lays the groundwork for future experiments. As we move toward a world of interconnected quantum devices, knowing exactly where and when true randomness exists—and where it is merely an illusion of complexity—will be essential for building systems that are truly secure.

Technical Summary: Certifying Randomness or its Lack Thereof for General Network Scenarios

Problem Statement
The certification of intrinsic randomness is a foundational element of quantum information theory, crucial for applications like secure communication and quantum key distribution. While device-independent randomness certification based on Bell inequality violations is well-established in the standard bipartite Bell scenario, this work addresses the extension of these concepts to more complex causal structures known as quantum networks.

A central challenge in network scenarios is that the set of correlations compatible with multiple independent sources is non-convex, unlike the standard Bell scenario. Furthermore, the paper distinguishes between two distinct tasks:

Certifying the presence of randomness: Determining if an adversary (Eve) cannot perfectly predict a party's measurement outcomes.
Certifying the absence of randomness: Determining if an adversary can perfectly predict outcomes, even when the observed probability distribution is nonclassical. The authors emphasize that nonclassicality does not guarantee intrinsic randomness; "bound randomness" phenomena exist where correlations violate Bell inequalities yet remain perfectly predictable.

The study operates under specific assumptions:

Adversary Model: Eve is a "beyond-quantum" adversary (limited only by no-signaling or generalized probabilistic theories) who can "listen to" (measure) all sources but cannot "control" them.
Settings: Measurement settings are public and accessible to Eve, modeled as records from bipartite sources.
Laboratory Closure: Eve cannot access information generated strictly inside the private laboratories of the honest parties.
IID Assumption: Events are assumed to be identically and independently distributed, negating coherent attacks.

Methodology
The authors employ two complementary computational approaches to address the certification problem:

Certifying Presence (Outer Approximation):
To certify that randomness exists, the authors adapt the Inflation Technique. This method constructs an outer approximation of the set of probability distributions compatible with a given causal structure (DAG) extended to include an eavesdropper.
- They utilize non-fanout inflation, which is appropriate for certifying randomness against beyond-quantum adversaries.
- The problem is formulated as a linear programming optimization to maximize the adversary's "guessing probability" ( $p_{guess}$ ) subject to the constraint that the marginal distribution of the honest parties matches the observed data.
- If the maximum guessing probability is strictly less than 1, intrinsic randomness is certified.
Certifying Absence (Inner Approximation):
To certify that randomness does not exist (i.e., outcomes are predictable), the Inflation Technique is insufficient because consistency with an inflation does not guarantee the existence of a valid causal model. Instead, the authors propose inner constructions (explicit causal models):
- Classical Parents Method: They construct causal models where the specific party in question receives only classical sources, while other sources in the network may be nonclassical (post-quantum). If such a model reproduces the observed nonclassical distribution, the party's outcomes are predictable.
- Nonclassical Parent Method: Recognizing that some correlations lack randomness even if a party must receive a nonclassical source, they embed a Bell scenario within the network. If the embedded Bell scenario allows for perfect prediction (as shown in prior work by Ramanathan et al.), the network correlation also lacks randomness for that party.
- These methods involve solving linear programs (for bilocality) and bilinear programs (for triangle scenarios) to find feasible causal models.

Key Contributions and Results

Extension to Networks: The paper successfully extends device-independent randomness certification from the standard Bell scenario to general network scenarios, specifically the bilocality and triangle networks.
Certification of Presence: Using non-fanout inflation, the authors certify intrinsic randomness for various distributions:
- In the bilocality scenario, they certify randomness for Alice and Bob in a Fritz-inspired distribution and for the middle party (Charlie) in an entanglement-swapping protocol. Notably, they demonstrate that randomness can be certified for "setting-less" parties in networks, a phenomenon absent in standard Bell scenarios.
- In the triangle scenario, they certify randomness for distributions compatible with both quantum and post-quantum theories.
Certification of Absence: The authors provide computational proofs that certain nonclassical distributions lack single-partite randomness:
- For the bilocality scenario, they show that the Fritz-inspired distribution allows Charlie's outcomes to be predicted by an adversary, as the correlation can be modeled with one classical source between Bob and Charlie.
- For the triangle scenario, they analyze the RGB3 distribution. They construct a causal model where Charlie receives only classical sources (with the nonclassical source between Alice and Bob), proving that Charlie's outcomes are predictable despite the distribution being nonclassical. Analogous models show predictability for Alice and Bob as well.
Distinction of Randomness Types: The work clarifies the distinction between single-partite randomness and multipartite (joint) randomness. A distribution may lack randomness for individual parties while the joint outcomes of multiple parties remain unpredictable.

Significance and Limitations
The paper frames its significance as a foundational step in understanding randomness certification in complex causal structures. It highlights that the assumptions regarding the causal structure (e.g., source independence) are critical; a distribution that appears deterministic in a standard Bell scenario may exhibit certified randomness when analyzed under a network causal structure.

The authors explicitly note the limitations of their current approach:

Scope of Absence Certification: Their methods for certifying the absence of randomness rely on constructing models where a party has at most one nonclassical source. They acknowledge that "Fully Network Nonlocal" correlations (where parties have multiple nonclassical parents) may resist this explanation, and it remains an open question whether such correlations necessarily yield certifiable randomness.
Quantification: The work focuses on the binary presence or absence of randomness, not the quantitative degree of randomness.
Exogenous Structures: The analysis is restricted to exogenous causal structures (where latent nodes have no parents). The authors note that non-exogenous scenarios may alter the results and leave this for future study.
Applications: While the theoretical framework is established, the paper does not propose specific experimental implementations or immediate cryptographic applications, leaving the practical utility of network-based randomness certification as an open area for future research.

Technical Summary: Certifying Randomness or its Lack Thereof for General Network Scenarios

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