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The Grothendieck Game and Self-Testing Continuous Groups

This paper establishes the first robust self-test of the full continuous family of Majorana operators and the spin representation of the continuous pin group by demonstrating that optimal performance in the Grothendieck game certifies these quantum measurements solely from observed correlations.

Original authors: Alexander Kulpe, Giulio Malavolta, Simon Schmidt, Michael Walter

Published 2026-10-06

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Alexander Kulpe, Giulio Malavolta, Simon Schmidt, Michael Walter

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 trying to verify that a mysterious machine is doing exactly what it claims to do, without ever being allowed to open its casing or see its internal gears. You can only watch the lights it flashes and the sounds it makes in response to your inputs. In the strange world of quantum physics, this is the central challenge of certifying quantum devices. Scientists have long known how to prove that a device is behaving in a genuinely quantum way, distinct from any classical machine, by checking the correlations between the answers it gives to different questions. This process, known as self-testing, allows researchers to confirm that a device is holding a specific quantum state and performing specific measurements, even if they have no idea how the device was built. Until now, however, these proofs have been limited to checking a fixed, finite set of measurements, like verifying a few specific buttons on a control panel.

The question that has remained unanswered is whether one can certify an entire continuous family of measurements using a single test. In the physical world, many properties change smoothly, like the direction a compass needle points or the angle of a spinning top. If a quantum device is designed to measure along any possible direction on a sphere, can a single game prove that it is doing so correctly for every single direction, not just a few chosen ones? This is the fundamental gap that a new study by Alexander Kulpe, Giulio Malavolta, Simon Schmidt, and Michael Walter sets out to fill. They tackle this by designing a specific game that forces the players to reveal the behavior of their quantum devices across a continuous range of possibilities, proving that near-perfect performance in the game guarantees the device is working as intended for the entire spectrum of measurements.

The researchers turned to a mathematical concept known as the Grothendieck game to solve this problem. In this game, two players, who cannot communicate with each other, receive questions in the form of directions on a sphere. They must respond with a simple "yes" or "no" answer. The rules of the game dictate that their answers should agree if the angle between their two directions is acute, and disagree if the angle is obtuse. The challenge is to win as often as possible. While classical strategies, which rely on pre-agreed plans or shared randomness, have a strict limit on how often they can win, quantum players can do better by sharing a special entangled state and measuring it in specific ways. The researchers showed that the optimal quantum strategy for this game involves a continuous family of measurements known as Majorana operators. These are mathematical objects that behave like fermions, a type of fundamental particle, and they can be oriented in any direction on the sphere.

The core discovery of the paper is that the Grothendieck game acts as a rigorous self-test for this entire continuous family. The authors proved that if a quantum device achieves a winning rate that is even slightly below the theoretical maximum, it must be using a strategy that is extremely close to the ideal one. This means that by simply observing the game's outcomes, one can certify that the device is effectively measuring along every possible direction on the sphere, not just a few. This is a significant leap forward because previous methods could only certify a finite number of specific measurements. The researchers demonstrated that the device's behavior is rigid; it cannot deviate from the ideal strategy without losing its winning advantage. This rigidity holds true even when the device is imperfect, providing a robust guarantee that the continuous family of measurements is being implemented correctly.

Beyond just the individual measurements, the study extends this certification to the complex structures built from them. The Majorana operators can be combined in various ways to form a representation of a continuous group known as the pin group, which describes symmetries in space. The researchers showed that the same game also self-tests this larger, more complex structure. They proved that the near-optimal strategy for the game effectively implements the spin representation of this group, which is a fundamental concept in physics describing how particles with spin behave under rotation. This means the game does not just verify a list of measurements; it verifies the underlying mathematical symmetry that connects them all. The proof relies on showing that the device's responses are consistent with the rules of this group, even when the questions are drawn from a continuous distribution.

To make these abstract concepts concrete, the researchers used a technique involving random sampling. Since it is impossible to check every single direction on a sphere, they showed that checking a random selection of directions is sufficient. If the device performs well on a random set of questions, the mathematics guarantees it is performing well on the whole sphere. They also developed a method to handle the fact that the device's internal state might be slightly different from the ideal one. By using a mathematical tool called an isometry, they showed how to map the device's actual state and measurements onto the ideal ones, proving that the difference between them is small and controlled. This mapping acts as a bridge, allowing the researchers to say with certainty that the unknown device is functionally equivalent to the known ideal device.

The implications of this work are profound for the future of quantum technology. As quantum computers and sensors become more complex, the ability to verify their operation without trusting the manufacturer becomes increasingly critical. This new method provides a way to certify that a device is capable of performing a continuous range of operations, which is essential for many advanced quantum protocols. It moves the field from checking a few discrete points to verifying a smooth, continuous landscape of quantum behavior. The researchers have established that the Grothendieck game is not just a theoretical curiosity but a powerful tool for certifying the fundamental operations of quantum devices. Their results offer a robust framework for ensuring that the quantum machines of the future are doing exactly what they are supposed to do, across the full spectrum of their capabilities.

In the end, the paper demonstrates that the boundaries of what can be certified in quantum mechanics are broader than previously thought. By linking a simple game with continuous questions to the deep mathematical structures of quantum mechanics, the authors have opened a new door. They have shown that a single, well-designed test can verify an infinite number of possibilities, providing a level of assurance that was previously out of reach. This work stands as a testament to the power of combining game theory, group theory, and quantum physics to solve practical problems in verification. It confirms that even in the most abstract realms of mathematics, there are concrete ways to ensure that the physical world behaves as expected, providing a solid foundation for the next generation of quantum technologies.

Technical Summary: The Grothendieck Game and Self-Testing Continuous Groups

Problem Statement

Self-testing is a protocol in quantum information that certifies quantum states and measurements based solely on observed correlations, without requiring knowledge of the internal workings of the devices. While robust self-testing has been established for finite sets of observables (e.g., Pauli matrices in CHSH games, Majorana operators in CHSH( $n$ ) games), a fundamental open question remains: Is robust self-testing possible for a continuous family of measurements using a single nonlocal game?

Existing self-tests typically involve finite question sets, allowing every designated measurement to occur with positive probability. In contrast, continuous distributions of questions imply that any single question has probability zero. Consequently, an optimal strategy could be altered arbitrarily on a finite set of questions without affecting the game's value, making it impossible to certify a fixed finite set of generators in the traditional sense. The challenge is to establish a probabilistic guarantee: that a near-optimal strategy approximates an ideal implementation of a continuous family of observables on average (or with high probability) over the choice of questions, under a single pair of local isometries.

Methodology

The authors address this problem by analyzing the Grothendieck game ( $G_n$ ), a two-player nonlocal game derived from the "little Grothendieck inequality."

The Grothendieck Game ( $G_n$ ):
- Setup: The referee samples a pair of vectors $(x, y)$ from the unit sphere $S^{n-1}$ with a probability density proportional to $|\langle x, y \rangle|$ .
- Winning Condition: Alice and Bob return binary answers $a, b \in \{\pm 1\}$ such that $ab = \text{sgn}(\langle x, y \rangle)$ .
- Optimal Strategy: The optimal quantum strategy involves sharing a maximally entangled state $|\psi^+_{2m}\rangle$ (where $n=2m$ ) and measuring Majorana observables defined as $\gamma(x) = \sum_{i=1}^n x_i \gamma_i$ , where $\gamma_i$ are the standard Majorana operators satisfying the Clifford algebra relations $\{\gamma_i, \gamma_j\} = 2\delta_{ij}I$ .
Technical Approach:
- Classical and Vector Rigidity: The authors first establish rigidity for classical and vector strategies. They prove that any near-optimal classical strategy is close to a strategy defined by a fixed vector $c$ (i.e., $a(x) = \text{sgn}\langle c, x \rangle$ ), and any near-optimal vector strategy is close to the identity map (up to an isometry).
- From Generators to Continuous Families: A key difficulty is that near-optimality only implies approximate relations for specific pairs of questions. To extend this to the entire continuous family, the authors employ a Haar-random orthonormal basis $Q = (q_1, \dots, q_n)$ . They show that if the observables $A(q_i)$ and $B(q_i)$ for the basis vectors are close to the ideal Majorana operators, then the linear extensions $\tilde{A}_Q(x) = \sum \langle q_i, x \rangle A(q_i)$ approximate $A(x)$ for all $x$ on average.
- Approximate Representations: The authors construct approximate representations of the finite Clifford group $C_n$ using the observables from the random basis. They utilize the Gowers–Hatami theorem (adapted for state-dependent seminorms) to "round" these approximate representations into exact representations, thereby extracting the local isometries that map the physical observables to the ideal Majorana operators.
- Extension to the Pin Group: To certify the continuous group structure, the authors extend the self-test from the generators (Majorana operators) to the spin representation of the Pin group $\text{Pin}(n)$ . Since group elements can be decomposed into products of generators in non-unique ways, they introduce a symmetrization trick: they average over Haar-random rotations $R \in SO(n)$ applied to the decomposition of group elements. This ensures that the unit vectors appearing in the product decompositions are uniformly distributed on the sphere, allowing the application of the Majorana self-test to each factor in the product.

Key Contributions and Results

1. Robust Self-Test for Majorana Observables

The paper establishes the first robust self-test for a full continuous family of measurements.

Theorem 1.1 (Majorana Self-Testing): For an $\epsilon$ -optimal strategy in $G_n$ , there exist local isometries $\Phi_A, \Phi_B$ such that the shared state is close to a maximally entangled state, and the observables $A(x), B(y)$ are close to the ideal Majorana observables $\gamma(x), \gamma(y)$ on average over $x, y \in S^{n-1}$ .
Corollary 5.10: For a Haar-random orthonormal basis, all $2n$ observables corresponding to the basis vectors are simultaneously certified with high probability.

2. Robust Self-Test for the Spin Representation

The authors extend the certification to the continuous group generated by the Majorana operators.

Theorem 1.2 (Spin Representation Self-Test): The Grothendieck game is a self-test for the spin representation of the Pin group. Specifically, the unitaries constructed by players from their observables (via products of generators) approximate the ideal spin representation $\Gamma(g)$ for any $g \in \text{Pin}(n)$ , averaged over random group elements and random rotations.

3. Classical and Vector Rigidity

Theorem 3.1: Proves robust classical rigidity, showing that any $\epsilon$ -optimal deterministic strategy is close to a strategy defined by a single vector $c$ .
Theorem 4.8: Proves robust vector rigidity, showing that any $\epsilon$ -optimal vector strategy is close to the identity strategy $a(x)=x, b(y)=y$ (up to a common isometry).

4. Effective Gowers–Hatami Theorem

Section 6: The authors provide a simplified, effective proof of the Gowers–Hatami stability theorem applicable to both finite and compact continuous groups. This version clarifies the conditions under which the resulting isometry can be implemented efficiently (requiring efficient preparation of uniform superpositions over the group and controlled approximate representations).

Significance and Claims

The paper claims to provide the first robust self-test of the full continuous family of measurements generated by Majorana operators. Prior work had established self-tests for finite sets of Majorana operators (e.g., in CHSH( $n$ ) games), but certifying the entire continuous parameterization simultaneously had not been achieved.

The significance lies in:

Bridging Finite and Continuous: It demonstrates that robust self-testing is possible for continuous families of observables, overcoming the probabilistic nature of continuous question distributions.
Group Representation Certification: It extends self-testing from individual observables to the spin representation of the Pin group, a continuous group structure, which is a non-trivial generalization of previous results on finite groups (like the Pauli group or Weyl-Heisenberg group).
Methodological Advancement: The use of Haar-random bases and symmetrization over rotations provides a general technique for lifting self-tests from generators to continuous families and groups, potentially applicable to other continuous groups.
Efficiency: The paper clarifies that the isometries required for the self-test can be implemented efficiently if the players have unitary access to their observables and can prepare uniform superpositions over the group.

The authors note that the dependence of the error bounds on the dimension $n$ (e.g., $O(n^2 \epsilon)$ for the state and $O(n \epsilon)$ for observables) is optimal for uniform bounds. The work does not propose specific experimental implementations or future cryptographic protocols but establishes the theoretical foundation for certifying continuous quantum devices.