⚛️ quantum physics

Characterising the set of deterministic quantum correlations in prepare-and-measure scenarios

This paper addresses the challenge of identifying non-deterministic quantum correlations in prepare-and-measure scenarios by proposing a certification method based on adversary predictability with classical side-information and developing tailored semidefinite programming relaxations for three specific communication restrictions.

Original authors: Nicola D'Alessandro, Oliver Karlsson, Carles Roch i Carceller

Published 2026-08-11

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Nicola D'Alessandro, Oliver Karlsson, Carles Roch i Carceller

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

The Mystery of the Magic Box

Imagine you are at a carnival, standing in front of a mysterious "Magic Box." You drop a specific colored ball into a slot on the left, and a light flashes on the right. Sometimes the light is red, sometimes blue. In the world of classical physics—the rules that govern our everyday life, like throwing a ball or riding a bike—this box would be a simple machine. If you knew exactly how the ball was made and how the gears inside were set up, you could predict with 100% certainty which light would flash. The outcome would be pre-determined, like a clock ticking.

But quantum physics, the rulebook for the very tiny world of atoms and particles, plays by different rules. Sometimes, even if you know everything about the ball and the box, the light that flashes seems to be a genuine surprise. It's not that we are bad at calculating; it's that the universe itself might not have decided the answer until you looked. This "intrinsic non-determinism" is the secret sauce that makes quantum computers and super-secure codes possible. If we can prove that a system is truly unpredictable, we can use it to generate perfect randomness or prove that a message hasn't been tampered with.

The big question scientists have been asking is: How do we know for sure that a system is truly quantum and not just a clever classical trick? Usually, to prove this, we need to perform very difficult experiments involving two people far apart sharing a special connection called "entanglement." But what if we want to test this in a simpler setup, where one person prepares a particle and sends it to another person to measure? This is called a "prepare-and-measure" scenario. It's like sending a sealed envelope (the preparation) to a friend who opens it and reads a note (the measurement). The challenge is figuring out if the note inside was written in advance (deterministic) or if it was created the moment the envelope was opened (quantum).

The Detective's New Tool

In this new work, Nicola D'Alessandro, Oliver Karlsson, and Carles Roch i Carceller propose a clever new way to solve this mystery. Instead of trying to prove the system is random, they flip the script and ask: "Could a super-smart detective figure out the answer in advance?"

Imagine a villain, let's call her Eve, who is trying to test the system. Eve has a special superpower: she knows a secret code (called a "hidden variable") that controls how the Magic Box works. If the system is truly classical, Eve should be able to look at her code, see what ball you are about to drop, and predict with perfect accuracy which light will flash. If Eve can do this, the system is "classical" and boring. But if Eve tries her best and still can't predict the outcome better than a lucky guess, then the system is "non-classical" and truly quantum.

The authors realized that this "Eve" perspective is a perfect match for existing tools used in a field called Quantum Random Number Generation (QRNG). Usually, QRNG experts try to measure how unpredictable a system is. These researchers turned that idea around: they asked, "Can we prove the system is not predictable?" If the answer is "no, it's not predictable," then we have certified that the system is quantum.

To make this work for different types of experiments, the team developed a set of mathematical "filters" called semidefinite programming (SDP) relaxations. Think of these as different types of security checks for the Magic Box, depending on what rules the box is supposed to follow. They tested three specific scenarios:

The Fixed Ensemble: Imagine the sender always uses a specific, known set of balls. The researchers checked if Eve could predict the outcome if she knew the exact mix of balls being used.
Bounded Overlaps: Here, the sender doesn't have to use a fixed set, but the balls must be "similar" to each other in a specific way (they can't be too different). The team checked if Eve could predict the outcome given this similarity rule.
Restricted Observables: In this case, the sender is limited by how much "energy" or specific properties the balls can have (like a limit on how many photons, or light particles, are in the ball). They checked if Eve could predict the outcome under these energy limits.

What They Found

Using these new tools, the authors ran simulations to see where the line between "predictable" and "quantum" lies. They didn't just guess; they used powerful computer algorithms to find the exact mathematical boundaries.

For the Fixed Ensemble scenario, they derived a general rule (a "witness") that acts like a scorecard. If the score is too high, it proves that no matter how hard Eve tries, she cannot predict the outcome. They showed that for certain setups involving "mutually unbiased bases" (a fancy way of saying the balls are prepared in completely different, incompatible ways), the quantum score is strictly higher than what any predictable, classical model could ever achieve. For example, in a setup with 5 different types of balls, the quantum system could reach a success probability that classical models simply cannot touch.

For the Bounded Overlaps scenario, they looked at how similar the balls need to be to test Eve. They found that if the balls are too similar (high overlap), Eve can predict the outcome. But as the balls become more distinct, there is a tipping point where Eve loses her ability to predict. They even came up with a neat formula to calculate exactly where this tipping point is for specific numbers of inputs.

Finally, for the Restricted Observables scenario, they applied their method to light particles (photons). They simulated a situation where the sender is limited to using only a small number of photons. Their simulations showed that even with these strict limits, there are specific combinations of inputs where the system behaves in a way that Eve cannot predict, proving the quantum nature of the light.

The paper doesn't claim to have solved every mystery in the universe, nor does it say these methods work for every possible experiment. Instead, it offers a versatile toolkit. It shows that by asking "Can Eve predict this?" we can systematically certify quantum behavior in many different situations, from fixed sets of particles to those with strict energy limits. This approach reveals that "classicality" isn't just a simple on/off switch; it's more like a ladder. A system might be predictable for some inputs but quantum for others, creating a rich hierarchy of behaviors that scientists can now explore with greater precision.

In short, these researchers gave us a new magnifying glass. By imagining a villain trying to test the system, they found a way to prove that the universe is genuinely playing hide-and-seek, and that sometimes, even the best detective can't find the answer until the box is opened.

Technical Summary: Characterising the Set of Deterministic Quantum Correlations in Prepare-and-Measure Scenarios

Problem Statement
Quantum theory predicts correlations that cannot be explained by deterministic models, where measurement outcomes are not merely revelations of pre-existing properties. While this non-determinism is a central resource for quantum information processing, identifying and certifying such correlations remains a fundamental challenge. Previous investigations have largely focused on Bell scenarios, which require demanding resources like entanglement distribution and loophole-free implementations. Alternatively, prepare-and-measure scenarios offer a simpler framework where a system is prepared and subsequently measured. However, existing methods for certifying non-classicality in these scenarios are often tailored to specific constraints (e.g., fixed dimension) and lack a systematic general framework applicable to a wide range of communication restrictions. The core problem addressed is how to certify whether observed correlations in a prepare-and-measure scenario admit a deterministic explanation under arbitrary communication constraints.

Methodology
The authors propose an adversarial approach to certify non-classicality. Instead of quantifying the unpredictability of outcomes (as in Quantum Random Number Generation, QRNG), the work asks whether observed correlations admit a model where an adversary, equipped with classical side-information (the shared random variable $\lambda$ ), can perfectly predict the measurement outcomes.

Adversarial Framework: An adversary, Eve, is introduced who has complete knowledge of the hidden variable $\lambda$ governing the preparation and measurement processes. The scenario is modeled by purifying the quantum states and extending the Hilbert space to include Eve's system.
Definition of Classicality: A correlation is deemed "classical" for a specific set of inputs if the adversary's optimal guessing probability ( $p_g$ ) equals 1. This implies that for every realization of $\lambda$ , the measurement outcome is deterministic ( $D_\lambda(b|x,y) \in \{0,1\}$ ).
SDP Relaxations: To test for the existence of such deterministic models, the authors formulate Semidefinite Programming (SDP) feasibility problems tailored to three representative communication restrictions:
- Fixed Ensemble (SE): The average ensemble of prepared states is known ( $E = \{\rho_x\}$ ). The method uses a Block Moment Matrix (BMM) approach to search for a decomposition into hidden-variable states that reproduce the statistics with deterministic outcomes.
- Bounded Overlaps (S $\beta$ ): Only lower-bounds on the pairwise overlaps of pure state preparations are known ( $\sum_\lambda q(\lambda)|\langle \psi^\lambda_x | \psi^\lambda_{x'} \rangle| \ge \beta_{xx'}$ ). This is addressed using a Gram matrix SDP hierarchy.
- Restricted Observables (SO): Constraints are placed on the expectation values of specific observables ( $\sum_\lambda q(\lambda) \text{tr}(\rho^\lambda_x O_{ix}) \le \omega_{ix}$ ). This utilizes a tracial moment-matrix SDP hierarchy.

In all cases, if the SDP is infeasible, the observed correlations cannot be explained by a deterministic model compatible with the given restriction, thereby certifying non-classicality.

Key Contributions and Results

General Certification Framework: The paper establishes a systematic route to study determinism beyond standard dimension restrictions by adapting QRNG techniques to feasibility tests for deterministic explanations.
Analytical Bounds: The authors derive analytical bounds for generic linear witnesses under the assumption of deterministic models. Specifically, for a fixed ensemble of pure states where the non-orthogonality graph is connected, they prove that any deterministic correlation satisfies a specific bound on the witness value $W$ .
Numerical and Analytical Examples:
- Fixed Ensemble: Using mutually unbiased bases (MUBs), the authors show that the classical bound for a specific witness is $W_{Cl} = n$ , whereas the quantum maximum is $W_Q = nd$ . This allows for the certification of non-classicality whenever the observed value exceeds $n$ . Critical visibilities for qubit ensembles are numerically determined.
- Bounded Overlaps: For state discrimination tasks with symmetric overlap bounds, the authors derive a closed-form expression for the maximum success probability compatible with a deterministic model when only one input contributes deterministically ( $n^*=1$ ). They note that for $n=2$ , this bound coincides with the maximum success probability of a non-contextual model.
- Restricted Observables: Applying the framework to photon-number restrictions (Poissonian statistics), the authors compute maximal witness values for state discrimination tasks where the number of photon-number components is restricted.

Significance and Claims
The paper claims to advance the problem of certifying non-classical correlations in prepare-and-measure scenarios by shifting the perspective from randomness generation to the exclusion of deterministic explanations. The authors highlight two conceptual advantages:

Versatility: Unlike previous approaches limited to dimension-restricted frameworks, this method naturally inherits existing QRNG methodologies to certify non-classicality under a wide range of communication constraints (energy, information content, overlaps, etc.).
Nested Structure of Classicality: The approach reveals that classicality is not merely a binary property (classical vs. non-classical for all settings). Instead, it introduces a hierarchy where correlations may be classical for specific subsets of inputs but non-classical for others. This "nested classicality" allows for a more granular characterization of non-classical behaviors.

The work demonstrates the utility of this approach through three representative communication restrictions, providing both analytical bounds and numerical results that define the boundaries between deterministic and quantum correlations in these scenarios.

The Mystery of the Magic Box

The Detective's New Tool

What They Found

Technical Summary: Characterising the Set of Deterministic Quantum Correlations in Prepare-and-Measure Scenarios

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