⚛️ quantum physics

Randomness Certification and Trade-offs in the Prepare-and-Broadcast Scenario

This paper investigates the prepare-and-broadcast scenario to derive fundamental trade-offs between quantum witnesses and nonlocality, while establishing a semi-device-independent framework that certifies two bits of joint randomness and remains robust against quantum side information.

Original authors: Tailan S. Sarubi, Moisés Alves, Vinícius F. Alves, Santiago Zamora, Diogo H. G. Duarte, A. de Oliveira Junior, Rafael Chaves

Published 2026-08-11

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CC BY 4.0

Original authors: Tailan S. Sarubi, Moisés Alves, Vinícius F. Alves, Santiago Zamora, Diogo H. G. Duarte, A. de Oliveira Junior, 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 send a secret message using a tiny, fragile marble. In the world of quantum physics, this marble is a "qubit," a particle that holds information in a way that is fundamentally different from a regular bit in your computer. One of the most famous rules of this quantum world is that you cannot perfectly copy an unknown marble. If you try to photocopy it, the copy will always be a little bit blurry or wrong. This rule is actually a superpower for security: if a spy tries to steal your message by copying the marble, they inevitably leave a trace, and you know they were there.

But what happens if you want to send that single, uncopyable marble to two different friends at the same time? You can't just hand it to both, because it's only one object. You have to "broadcast" it, splitting its information between them. This is the tricky puzzle scientists are trying to solve: How much of the secret can Friend A get, and how much can Friend B get, without breaking the laws of physics? This isn't just about sending messages; it's about generating true randomness. In a world where computers can predict almost anything, having a source of pure, unpredictable numbers is like having a magic dice that no one, not even a supercomputer, can guess the outcome of. This paper explores how to split that quantum marble to create the best possible magic dice for two people at once.

The researchers in this paper, led by Tailan S. Sarubi and colleagues, decided to investigate a specific setup they call the "prepare-and-broadcast" scenario. Think of it like a game show where a host (Alice) prepares a special quantum marble and sends it through a magical splitter. This splitter doesn't just copy the marble; it distributes the marble's "quantum essence" to two players, Bob and Charlie. The team wanted to see how well Bob and Charlie could play different games using this shared resource.

First, they looked at the trade-offs. Imagine you have a limited amount of "quantum juice" in your marble. If you squeeze it to give Bob a huge advantage in a guessing game, there is less juice left for Charlie. The paper shows that there is a strict limit to how much advantage both can have simultaneously. If Bob tries to get the maximum possible quantum advantage, Charlie's performance drops to a basic, classical level. It's like trying to share a single slice of pizza between two hungry people; if one person takes the whole slice, the other gets nothing. Interestingly, the team found that the mathematical boundary for this sharing problem looks exactly like the rules for "quantum cloning," a process where scientists try to make imperfect copies of quantum states. This suggests that the reason Bob and Charlie can't both win big is the same reason you can't perfectly copy a quantum state in the first place.

Next, the team tackled the big question: Can we use this setup to certify true randomness? In the world of quantum security, "certifying randomness" means proving that the numbers coming out of the machine are truly unpredictable, even to a super-smart spy. Usually, to get two bits of randomness (which is like flipping two coins and getting four possible outcomes), you need very specific, complex experiments. The authors discovered something surprising: by using their broadcast setup, they could certify two full bits of joint randomness just by observing the results of their experiment. This is a big deal because other famous methods, like the CHSH inequality (a standard test for quantum weirdness), usually cap out at about 1.23 bits of randomness. Their method not only beats that limit but is also much tougher against noise. Even if the experiment is a bit messy or the equipment isn't perfect, they can still prove they have two bits of pure randomness.

The paper also tested how robust this system is against a very sneaky spy. In most scenarios, we assume the spy only has a notebook with classical notes. But what if the spy has a quantum computer and is holding a piece of the quantum system that is entangled with the players? The authors showed that their method still works even against this stronger, more powerful type of spy. They used advanced computer simulations (specifically a technique called semidefinite programming) to prove that even with a quantum spy, the randomness they generate remains secure and unpredictable.

In summary, this paper maps out the rules of sharing a single quantum particle between two people. It shows that while you can't give both people a perfect quantum advantage, you can use this sharing process to generate a massive amount of certified randomness—more than previously thought possible with standard methods. It's a playful yet rigorous demonstration that by splitting a quantum system, we don't just lose information; we can actually unlock new levels of security and unpredictability that were hiding in plain sight.

Technical Summary: Randomness Certification and Trade-offs in the Prepare-and-Broadcast Scenario

Problem Statement
The paper addresses the limitations of standard dimension-constrained prepare-and-measure (PAM) scenarios, where a quantum system is transmitted to a single receiver. While PAM frameworks have successfully characterized dimension witnesses, quantum random access codes (QRACs), and nonclassicality, they do not account for how information encoded in a single quantum system is distributed among multiple observers. The authors investigate the Prepare-and-Broadcast (PAB) scenario, a multipartite extension where a prepared quantum system is distributed via a broadcast channel to multiple receivers (specifically Bob and Charlie). The core problem involves understanding the fundamental trade-offs imposed by quantum mechanics when information is shared among multiple parties and developing a framework for semi-device-independent (SDI) randomness certification within this extended causal structure.

Methodology
The authors employ a combination of geometric analysis, optimization techniques, and semidefinite programming (SDP) relaxations:

Scenario Definition: The PAB scenario is defined where Alice prepares a $d$ -dimensional system (specifically a qubit, $d=2$ ) based on input $x$ . This system passes through a broadcast channel $T$ (classical, quantum, or nonsignaling) to produce outputs for Bob ( $y, b$ ) and Charlie ( $z, c$ ). The observed statistics are $p(b, c|x, y, z)$ .
Trade-off Analysis: The authors analyze the coexistence of PAM witnesses (marginal behaviors) and Bell nonlocality (joint conditional correlations). They utilize the support-function method to derive lower bounds on trade-off curves and SDP relaxations (specifically the $(1, 2)$ level hierarchy from Ref. [29]) to establish upper bounds.
Randomness Certification: A semi-device-independent framework is developed to bound an adversary's (Eve's) guessing probability.
- Classical Side Information: Eve is modeled as having access to a classical hidden variable $\lambda$ that correlates with the devices. The guessing probability is bounded by decomposing the observed behavior into convex branches compatible with the observed witness value.
- Quantum Side Information: The adversarial model is strengthened to allow Eve to hold a quantum system correlated with the measurement device via an isometry, representing a more powerful adversary in the SDI regime.
Optimization: The certification process involves solving SDPs to maximize Eve's guessing probability subject to constraints imposed by the observed witness values (e.g., $S_3$ , $S_4$ , or $W_{PAB}$ ) and the dimension constraint.

Key Contributions and Results

Fundamental Trade-offs:
- PAM vs. Bell: The authors demonstrate that while PAM violations (quantum communication advantage) and Bell violations (nonlocality) can coexist in the PAB scenario, they are subject to quantitative trade-offs. Increasing the PAM witness value reduces the maximum achievable Bell violation.
- Receiver-Receiver Trade-off: A stronger trade-off exists between the marginal PAM witnesses of the two receivers. If one receiver violates a PAM witness (e.g., $S_4$ ), the other cannot violate the corresponding witness simultaneously.
- Connection to Cloning: The trade-off curve for the $2 \to 1$ QRAC success probabilities ( $P_B, P_C$ ) in the PAB scenario is shown to coincide exactly with the boundary for optimal asymmetric phase-covariant quantum cloning. Specifically, $(P_B - 1/2)^2 + (P_C - 1/2)^2 = 1/8$ . This establishes that the quantum advantage of the QRAC cannot be shared; if one receiver exceeds the classical limit ( $3/4$ ), the other must fall below it.
Randomness Certification in PAM:
- The authors analyze the $S_3$ (three-preparation) and $S_4$ (four-preparation) witnesses.
- They find that the $S_3$ witness is more effective for certifying randomness at specific generation inputs, capable of certifying one full bit of randomness at maximal violation.
- Conversely, the $S_4$ witness, while optimal for QRAC tasks, is biased at the optimal generation input, limiting certifiable randomness to approximately 0.228 bits in that specific configuration.
Randomness Certification in PAB:
- Using the PAB witness $W_{PAB}$ (Eq. 11), the authors show that the maximal quantum violation ( $W_Q = 8 + 2\sqrt{2}$ ) certifies two bits of joint randomness for the output pair $(B, C)$ .
- This result exceeds the limit of approximately 1.23 bits achievable from the maximal violation of the CHSH inequality in standard Bell scenarios.
- The certification is robust, tolerating significantly higher noise levels than protocols based on tilted Bell inequalities. Notably, the certification of two bits occurs even when the conditional behavior at the generation input is Bell-local, relying instead on the dimension constraint and correlations across different preparation inputs.
Quantum Side Information:
- The paper extends the analysis to an adversary holding quantum side information. By modeling the broadcast channel as an isometry where Eve retains the complementary output, the authors derive bounds on the guessing probability.
- The results indicate that while quantum side information reduces the certifiable randomness compared to the classical side information model, the PAB framework remains a viable setting for SDI certification against quantum adversaries.

Significance and Claims
The paper claims that the Prepare-and-Broadcast scenario provides a natural and powerful framework for studying the distribution of quantum information and nonclassicality. Its primary significance lies in:

Unifying Nonclassicality: It reveals quantitative trade-offs between communication advantages (PAM) and nonlocality (Bell), linking them to fundamental limits in quantum cloning.
Enhanced Randomness: It demonstrates that the PAB scenario can certify more randomness (two bits) than standard Bell-based approaches (CHSH) while remaining robust to noise, without requiring full device independence.
Adversarial Modeling: It offers a natural setting for SDI randomness certification against stronger adversarial models where the eavesdropper retains quantum side information, bridging the gap between PAM and fully quantum network scenarios.

The authors conclude that these findings open avenues for developing semi-device-independent cryptographic protocols and information-processing tasks that have no counterpart in standard prepare-and-measure frameworks.

Technical Summary: Randomness Certification and Trade-offs in the Prepare-and-Broadcast Scenario

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