⚛️ quantum physics

Sequential Semi-Device-Independent Quantum Randomness Certification

This paper presents a general framework and numerical methods, including a semidefinite programming approach for min-tradeoff functions, to certify randomness in sequential semi-device-independent scenarios using maximum confidence measurements, thereby demonstrating their capability to distribute and certify randomness against adaptive attacks.

Original authors: Carles Roch I Carceller, Hanwool Lee, Jonatan Bohr Brask, Kieran Flatt, Joonwoo Bae

Published 2026-09-07

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Carles Roch I Carceller, Hanwool Lee, Jonatan Bohr Brask, Kieran Flatt, Joonwoo Bae

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

In the world of physics, the most reliable source of true randomness is the quantum realm. Unlike a coin toss, which is unpredictable only because we cannot measure the wind or the force of the thumb perfectly, a quantum event is fundamentally unpredictable. This inherent uncertainty is not a flaw in our tools but a feature of nature itself. For decades, scientists have sought to harness this unpredictability to create unbreakable codes and secure communications. However, verifying that a device is truly producing this kind of randomness is difficult. If we trust the device completely, we might be fooled by a clever trickster who has pre-programmed the machine to look random while actually following a secret script. If we demand proof that the device is behaving in a way that defies all classical logic, the experimental requirements become so strict that they are nearly impossible to achieve in a real laboratory. This leaves researchers searching for a middle ground: a way to certify randomness with fewer assumptions about the internal workings of the machines, making the process practical for everyday technology.

A team of physicists has now developed a new method to solve this problem by looking at how quantum information flows through a chain of observers. Imagine a single quantum particle being passed from one person to the next, where each person measures it and then passes the remaining state of the particle to the following person. In this new framework, the researchers showed that it is possible to certify that each person in the chain is generating their own unique, certified randomness, even if the devices they use are not fully trusted. The key to this discovery lies in a specific type of measurement called a "maximum confidence" measurement. This is a strategy where an observer tries to identify a quantum state with the highest possible certainty, accepting that sometimes they will have to admit they cannot tell the difference between two options. The researchers found that by using this approach, the "randomness" inherent in the inability to perfectly distinguish between quantum states can be shared and verified across multiple people in a sequence, rather than being consumed by the first person to look at it.

The researchers built a general mathematical framework to test this idea, focusing on a scenario where a sender prepares a quantum state and sends it to a first receiver, who measures it and passes the result to a second receiver. They considered two levels of trust. In the first, more idealized scenario, they assumed the states of the particles after the first measurement were known and fixed. In this case, they could prove that both receivers could generate certified randomness. However, to make the method robust for real-world use, they moved to a second, more challenging scenario where they assumed nothing about what happened to the particle after the first measurement. They only trusted the initial preparation of the particles. Even with this much less information, they proved that randomness could still be certified. They achieved this by treating the entire chain of operations as a single, complex process and using advanced computer algorithms to calculate the limits of how much an eavesdropper could possibly guess about the outcomes.

One of the most significant findings of the study is the existence of a delicate balance required for this sharing of randomness to work. The researchers discovered that if the initial quantum states are too similar to each other, or if the first receiver is too aggressive in trying to extract information, the chain breaks. Specifically, they found that there is a critical threshold for how often a receiver must say "I don't know" to the measurement. If the first receiver tries to guess the state too often without admitting uncertainty, they consume all the available randomness, leaving nothing for the next person. The team calculated precise limits for this behavior, showing that for a chain of measurements to all produce certified randomness simultaneously, the initial states must be distinguishable enough, and the rate of "inconclusive" results must stay within a specific range. If these conditions are met, the randomness is not a finite resource that runs out after the first use; it is a property that can be distributed and verified across the entire chain.

The study also addressed the threat of adaptive attacks, where a malicious observer might try to learn from previous rounds to improve their guesses in future ones. By using a technique that treats the sequence of measurements as a unified whole, the researchers showed that their method remains secure even against such sophisticated strategies. They demonstrated that the amount of certified randomness that can be accumulated over many rounds is bounded by a specific mathematical function, ensuring that the security holds up even as the chain grows longer. Their simulations revealed that while it is possible to certify randomness for a few people in a sequence, the requirements become increasingly strict as the number of people increases. Eventually, the conditions become so demanding that it becomes practically impossible to certify randomness for a very long chain of observers, suggesting a natural limit to how far this specific type of distribution can be extended.

This work represents a significant step forward in making quantum randomness certification practical. By moving away from the need for perfect knowledge of every device and instead relying on the fundamental limits of how well quantum states can be distinguished, the researchers have provided a toolkit that can be applied to real-world systems. The methods they developed, which rely on solving complex optimization problems on a computer, are designed to handle the noise and imperfections found in actual experiments. This means that future devices, perhaps built on integrated photonic chips or optical systems, could use these techniques to generate and verify secure random numbers without needing to trust every component of their hardware. The results confirm that the unpredictability of the quantum world is a shared resource, capable of being distributed among multiple parties, provided the flow of information is managed with the right balance of certainty and uncertainty.

Technical Summary: Sequential Semi-Device-Independent Quantum Randomness Certification

Problem Statement
Quantum systems are natural sources of randomness due to the inherent unpredictability of quantum measurements. While device-independent protocols based on Bell nonlocality offer strong security guarantees, they impose stringent experimental demands. Conversely, semi-device-independent (SDI) prepare-and-measure scenarios lower these demands by adding assumptions about device internal workings, yet they often struggle to certify randomness in realistic conditions where measurements may yield inconclusive or undetected events.

A specific challenge arises in sequential scenarios, where a quantum state is measured by a first observer (Bob), who then passes the post-measurement state to a second observer (Charlie), and potentially further parties. In such chains, quantum properties like the monogamy of nonlocal correlations and the no-cloning theorem limit the extension of randomness to multiple parties. Furthermore, existing literature on sequential state discrimination (e.g., unambiguous or maximum-confidence discrimination) has not fully addressed how to certify randomness when measurements are imperfect, inconclusive events occur, and the internal workings of devices are uncharacterized. The core problem is to establish a framework that can bound and certify the amount of randomness extractable by multiple sequential observers under realistic, semi-device-independent assumptions.

Methodology
The authors develop a general framework and numerical tools to bound certifiable randomness in sequential prepare-and-measure scenarios using Maximum Confidence Measurements (MCM). The work proceeds through two distinct approaches:

Trusted State Preparations (Measurement-Device-Independent):
- Assumptions: All state preparations, including the post-measurement states passed from Bob to Charlie, are fully known and trusted. Measurements are completely uncharacterized.
- Technique: The authors formulate the problem of bounding Eve's guessing probability ( $p_g$ ) as a Semidefinite Program (SDP). By utilizing SDP duality, they derive analytical bounds on the min-entropy ( $H_{min} = -\log_2 p_g$ ). This dual formulation is crucial for experimental robustness, ensuring a solution exists even if observed frequencies do not correspond to a perfect physical realization.
Untrusted State Preparations (Semi-Device-Independent):
- Assumptions: Post-measurement states and intermediate operations are completely uncharacterized. The only assumption is that Alice's initial pure state preparations have a bounded pairwise overlap ( $|\langle \psi_x | \psi_{x'} \rangle| \ge \delta$ ).
- Technique: To handle the lack of knowledge about intermediate states, the authors define a compact operator $G_{b,c} = K_b N_c K_b^\dagger$ that encapsulates all sequential operations (Bob's measurement and state update, followed by Charlie's measurement) into a single effective POVM.
- Entropy Accumulation: This compact representation allows the framework to address non-independent and identically distributed (non-i.i.d.) attacks. The authors derive min-tradeoff functions via SDP duality to bound the Shannon entropy. This enables the application of the Entropy Accumulation Theorem (EAT), which is necessary for security proofs against adaptive attacking strategies where Eve gathers information from previous rounds to improve future guesses.

Key Contributions and Results

General Framework for Sequential Randomness: The paper establishes a rigorous method to certify randomness distributed across a chain of sequential measurements. It demonstrates that randomness, originating from the indistinguishability of non-orthogonal states, can be shared among multiple parties.
Dual SDP Formulations: The authors provide versatile numerical methods to compute bounds on both min-entropy (for single-shot randomness) and Shannon entropy (for non-i.i.d. scenarios). The use of dual SDPs ensures the framework is applicable to real-world experimental data where constraints might not be perfectly satisfied.
Certification Conditions for Two Parties: In a two-party scenario (Bob and Charlie), the authors analyze the trade-off between the rate of inconclusive events ( $Q$ $Q$ ) and certifiable randomness. They identify critical thresholds:
- If the inconclusive rate is too high, the observed statistics can be reproduced by deterministic strategies, yielding zero certifiable randomness.
- There exists a specific range of inconclusive rates where randomness can be simultaneously certified for both Bob and Charlie.
Extension to $N$ Sequential Measurements: The authors generalize the two-party analysis to an arbitrary number $N$ $N$ of sequential measurements. They derive a sufficient condition (Eq. 26) for certifying randomness in all $N$ $N$ devices simultaneously.
- Numerical Bounds: Through numerical search, they determine a bound on the product of the initial state purity ( $r$ ) and overlap ( $\delta$ ), denoted as $r\delta \le \Delta$ . As $N$ increases, $\Delta$ decreases rapidly, indicating that certifying randomness in long sequential chains becomes practically impossible unless the initial states are highly distinguishable.
Adaptive Attack Resilience: By computing min-tradeoff functions, the framework successfully bounds randomness against adaptive eavesdropping strategies, a significant advancement over previous i.i.d.-only sequential models.

Significance and Claims
The paper claims to provide a sufficient criterion showing that Maximum Confidence Measurements enable the distribution and certification of randomness across a sequential measurement chain. The significance of the work lies in:

Relaxing Assumptions: It bridges the gap between fully device-independent and fully trusted models, offering a practical SDI framework where only minimal assumptions (bounded overlap of initial states) are required.
Handling Realism: The framework explicitly accounts for inconclusive events and experimental imperfections, making it suitable for realistic implementations where perfect state discrimination is impossible.
Foundational Insight: It clarifies the connection between sequential state discrimination and randomness distribution, showing that quantum unpredictability is not an isolated property of a single measurement but can be sequentially shared.
Practical Applicability: The authors note that sequential prepare-and-measure scenarios can be realized with modest experimental resources (e.g., integrated photonic platforms) and that their methods are immediately applicable to scenarios with energy constraints or other semi-device-independent frameworks.

The authors conclude that while the ability to certify randomness diminishes as the number of sequential parties increases, their framework provides the necessary tools to quantify and certify this distributed unpredictability, potentially paving the way for applications in semi-device-independent quantum key distribution and multi-party cryptographic schemes.

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