⚛️ quantum physics

Lifting the maximally-entangledness assumption in robust self-testing for synchronous games

This paper establishes that robust self-testing results for synchronous games, previously proven only under the unphysical assumption of symmetric projective maximally entangled strategies, hold for all quantum strategies, thereby enabling the construction of an efficient $n$ -qubit test via the Quantum Low Degree Test.

Original authors: Matthijs Vernooij, Yuming Zhao

Published 2026-09-11

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Matthijs Vernooij, Yuming Zhao

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 strange and powerful world of quantum mechanics, particles can become linked in a way that defies everyday logic. When two particles are linked, or "entangled," measuring one instantly reveals information about the other, no matter how far apart they are. This phenomenon is the engine behind a new generation of technologies, from ultra-secure communication to computers that could solve problems impossible for today's machines. However, because these systems are so delicate and operate on principles that seem to contradict our daily experience, verifying that they are working correctly is a major challenge. If a scientist builds a quantum device, how can they be sure it is truly using the complex entanglement it claims to, rather than just mimicking the results with a simpler trick? This is the problem of "self-testing." It is a method where a classical referee, who has no access to the quantum device's internal workings, can ask questions and check the answers to certify that the device is performing a specific, highly complex quantum strategy.

For years, proving that a device is doing exactly what it should have required a significant shortcut. Researchers had to assume that the quantum particles inside the device were in a perfectly balanced, ideal state of entanglement, known as a maximally entangled state. They also had to assume the measurements were perfectly symmetrical. While these assumptions made the math manageable, they did not reflect reality. In the real world, quantum systems are noisy, imperfect, and rarely exist in that perfect, theoretical state. If a security protocol or a proof relied on these perfect assumptions, a clever adversary could potentially exploit the gap between the theory and the messy reality of actual hardware. The question remained: could the powerful results of self-testing be made to work without these unrealistic assumptions? Could a referee certify a quantum strategy even if the particles were not in a perfect state and the measurements were not perfectly symmetrical?

A team of researchers has now answered this question with a definitive yes. They have proven that if a specific type of quantum game can certify a perfect strategy under the ideal assumptions, it can also certify that same strategy for any real-world, imperfect version of that game. Their work removes the need for the "perfect state" assumption, bridging the gap between elegant mathematical theory and the noisy reality of physical devices. This means that the robustness of these certification methods is not an artifact of idealized math, but a genuine property of the quantum strategies themselves. The researchers showed that the ability to verify a quantum system holds up even when the system is far from perfect, provided the system is playing a "synchronous" game, a type of interaction where the players are asked the same questions and must give consistent answers.

The core of their discovery lies in understanding how close a near-perfect strategy is to the ideal one. In the past, researchers could prove that if a game was a robust self-test for perfect strategies, it was also a robust self-test for strategies that were slightly imperfect, but only if those imperfect strategies still looked a lot like the perfect ones. The new work goes much further. It demonstrates that even if a strategy is completely general—using any kind of quantum state and any kind of measurement—it is still forced to be close to the ideal strategy if it wins the game with high probability. The researchers achieved this by showing that any messy, general strategy can be mathematically broken down into a collection of simpler, perfect strategies. They then proved that the "distance" between the messy strategy and the ideal one is controlled by how well the game itself resists errors. This relationship is not just a vague possibility; they calculated the exact mathematical link, showing that the robustness of the test for general strategies is directly related to the robustness for the perfect ones, with the connection being a simple polynomial relationship.

To make this concrete, the team applied their new theory to a specific, high-stakes test known as the Quantum Low Degree Test. This test is a critical component in the recent breakthrough that proved a massive connection between quantum interactive proofs and the limits of computation. Previously, this test was only known to be a robust self-test if the players were using perfect, maximally entangled states. The researchers used their new lifting technique to show that the test is actually a robust self-test for any strategy. They calculated that the test can verify that players have access to a specific number of qubits and the correct set of quantum operations, even if the entanglement is not perfect. This finding is significant because it means the security and soundness of these complex quantum proofs do not rely on the impossible condition of perfect hardware.

The implications of this work are profound for the future of quantum technology. By removing the requirement for perfect entanglement, the researchers have strengthened the foundation of device-independent cryptography and verifiable quantum computing. In a world where quantum devices will inevitably have noise and imperfections, being able to certify their behavior without assuming perfection is essential. The study confirms that the power of these quantum games is intrinsic to the laws of physics, not just a feature of idealized mathematics. The researchers did not just suggest this was possible; they provided a rigorous proof that holds for all strategies, ensuring that the guarantees offered by these tests are as strong as the laws of quantum mechanics themselves. This removes a major theoretical barrier, allowing these powerful verification tools to be applied to the real, imperfect quantum systems that scientists are building today.

Technical Summary: Lifting the Maximally-Entangledness Assumption in Robust Self-Testing for Synchronous Games

1. Problem Statement

Robust self-testing in non-local games allows a classical referee to certify that two untrustworthy players are executing a specific quantum strategy with high precision, based solely on observed statistics. While significant progress has been made in proving robust self-testing results, particularly for synchronous games, these proofs often rely on restrictive physical assumptions. Specifically, existing results frequently assume that the employed strategies are Projective Maximally Entangled (PME) strategies. In a PME strategy, players share a maximally entangled state and perform projective, symmetric measurements.

This assumption is mathematically convenient because PME strategies admit natural descriptions via tracial von Neumann algebras, a framework heavily utilized in recent breakthroughs like $MIP^*=RE$ . However, from a device-independent perspective, the assumption of a perfectly maximally entangled state is unphysical. Real-world implementations involve noise, and strategies may employ general Positive Operator-Valued Measures (POVMs) or non-maximally entangled states. Consequently, there is a critical gap: results proven under the PME assumption may not hold for general strategies, potentially weakening security guarantees in applications like device-independent cryptography and verifiable quantum delegation.

The central question addressed by this work is: If a synchronous game robustly self-tests a perfect strategy when restricted to PME strategies, does it follow that it robustly self-tests the same strategy for general POVM strategies?

2. Methodology

The authors develop a framework to lift the PME assumption by bridging the gap between general strategies and the algebraic structures of PME strategies. Their approach proceeds in three main technical steps:

A. Algebraic Reformulation and Distance Metrics

The authors first formulate PME strategies using tracial von Neumann algebras. They define a "von Neumann distance" between strategies based on the closeness of measurement operators within the algebra. A key technical contribution is establishing that this algebraic distance is equivalent (up to constant factors) to the standard Hilbert-space distance defined via local dilations. This equivalence allows the authors to reformulate robust self-testing for PME strategies entirely within the language of von Neumann algebras, leveraging the symmetry of these strategies to work with the algebra generated by a single player's measurements.

B. Decomposition of Almost Synchronous Strategies

To handle general strategies that are only "almost" synchronous (due to noise), the authors utilize a decomposition theorem by Vidick [Vid22]. They show that any almost synchronous strategy $S$ can be approximately decomposed into a convex mixture of orthogonal symmetric maximally entangled (ME) strategies $S_\lambda$ .

Challenge: Standard decomposition results guarantee closeness of measurements with respect to the specific state $|\psi_\lambda\rangle$ of each block, but robust self-testing requires closeness with respect to the global state $|\psi\rangle$ .
Solution: The authors construct a single pair of local isometries that works for the entire strategy by carefully combining the isometries derived for each block $S_\lambda$ . They prove that if the game is a robust self-test for PME strategies, the "spectral gap" of the game polynomial for the ideal strategy controls the robustness of the general case.

C. Spectral Gap Analysis

The core of the proof relies on the spectral gap of the game polynomial $T_{G, \tilde{S}}$ associated with the ideal strategy $\tilde{S}$ . The spectral gap $\Delta$ is the difference between the largest and second-largest eigenvalues of this operator.

The authors prove that if a synchronous game is a $\kappa$ -robust self-test for PME strategies, the spectral gap of the ideal strategy is lower-bounded by a function of $\kappa$ (specifically, polynomially related).
This lower bound on the spectral gap is crucial because it ensures that any strategy achieving a near-optimal winning probability must be close to the ideal strategy, even if it does not start with a maximally entangled state.

3. Key Contributions and Results

Main Theorem (Theorem 1.2)

The paper establishes that for any synchronous game $G$ , if $G$ $\kappa$ -robustly self-tests a perfect quantum strategy $\tilde{S}$ for the class of PME strategies, then $G$ $\kappa'$ -robustly self-tests $\tilde{S}$ for the class of general POVM strategies.

Quantitative Relationship: The new robustness function $\kappa'$ is polynomially related to the original robustness $\kappa$ .
Independence: This relationship is independent of the size of the game (number of questions and answers) and depends only on the synchronicity of the game.
Generalization: The result holds even when the distribution used for the game ( $\nu$ ) differs from the distribution used to measure the distance between strategies ( $\hat{\nu}$ ), a generalization necessary for analyzing specific tests like the Quantum Low Degree Test.

Application to Quantum Low Degree Test (Theorem 1.3)

The authors apply their main result to the Quantum Low Degree Test (QLDT), a key component of the $MIP^*=RE$ proof.

They show that the QLDT, when combined with a synchronicity test, $\kappa'$ -robustly self-tests the maximally entangled state on $n$ qubits and a generating set of Pauli operators.
Crucially, they explicitly compute the spectral gap of the QLDT game polynomial to be $d/2$ , where $d$ is the relative distance of the underlying linear code. This explicit calculation confirms that the QLDT satisfies the conditions required for the lifting theorem, providing an efficient $n$ -qubit test robust against general strategies.

Technical Lemmas

Lemma 3.2: Establishes the equivalence between local dilations in the Hilbert space formalism and local dilations in the von Neumann algebra formalism for ME strategies.
Theorem 5.4: Proves that for a $\kappa$ -PME-robust self-test, the spectral gap of the game polynomial is bounded below by a function of $\kappa$ and the synchronicity parameter $\beta$ . This closes the loop, ensuring that the robustness of the PME case implies a non-trivial spectral gap, which in turn guarantees robustness for the general case.

4. Significance and Claims

The paper claims to resolve a fundamental limitation in the theory of robust self-testing for synchronous games. By removing the unphysical assumption of maximally entangled states, the authors upgrade existing self-testing results to be physically relevant and applicable to general device-independent scenarios.

Theoretical Impact: The work demonstrates that the algebraic tractability of PME strategies does not come at the cost of generality. It proves that the "rigidity" of synchronous games is strong enough to enforce the correct structure even when players use arbitrary POVMs and non-maximally entangled states.
Practical Relevance: The results strengthen the soundness guarantees for quantum interactive proof systems (like $MIP^*=RE$ ) and device-independent cryptographic protocols. By showing that the Quantum Low Degree Test is robust against general strategies, the paper ensures that the verification of quantum proofs remains valid in the presence of noise and imperfect devices.
Modesty: The authors note that while their results lift the PME assumption, the quantitative relationship between the robustness parameters ( $\kappa$ and $\kappa'$ ) involves polynomial factors. They do not claim to eliminate all assumptions (e.g., finite dimensionality is still assumed) but rather to remove the specific constraint on the entanglement structure and measurement type.

In summary, this work provides the necessary theoretical machinery to transition robust self-testing results from the idealized world of PME strategies to the realistic world of general quantum strategies, ensuring that the powerful certification tools developed for synchronous games are applicable to real-world quantum devices.