Trading symmetry for Hilbert-space dimension in Bell-inequality violation

This paper demonstrates that for certain Bell inequalities, achieving maximal quantum violation requires trading party-exchange symmetry for higher-dimensional Hilbert spaces, as symmetric strategies in minimal dimensions can be suboptimal, thereby revealing a complex interplay between symmetry, dimension, and the geometry of quantum correlations.

The Gist

Imagine you are trying to solve a very difficult puzzle. In the world of quantum physics, this puzzle is called a Bell inequality. It's a test designed to prove that the universe operates on "spooky" quantum rules rather than simple, local rules. To win the game (achieve the maximum possible score or "violation" of the inequality), you need to use a specific quantum strategy: a shared state (like a pair of entangled particles) and a set of measurements.

This paper explores a fascinating trade-off between two resources needed to win this game: Symmetry and Size.

The Two Resources

1. Symmetry (The Mirror): Imagine you and your partner are playing the game. A "symmetric" strategy means you both do exactly the same thing. You hold the same type of coin, flip it the same way, and look at it from the same angle. It's like looking in a mirror; your actions are perfectly identical.
2. Hilbert-Space Dimension (The Size of the Toolbox): This is a fancy way of saying "how complex is the quantum system?"
   • A low dimension is like using a simple, small toolbox (e.g., a single coin or a qubit). It's efficient and simple.
   • A high dimension is like having a massive, complex toolbox (e.g., a high-dimensional quantum state). It has more "room" to maneuver.

The Big Question

The researchers asked: Can we always win the game using a simple, small toolbox and a perfectly symmetric strategy?

In other words, if we force the players to be identical (symmetric), do we have to use a bigger, more complex toolbox to get the best score? Or can we get the best score with a small toolbox while staying symmetric?

The Findings: It Depends on the Puzzle

The paper looked at many different "puzzles" (Bell inequalities) and found two very different outcomes:

1. The "No Trade-off" Cases (The Easy Puzzles)

For some famous puzzles, like the CHSH inequality (the simplest test of quantum weirdness) and the CGLMP inequalities (which involve more outcomes), the answer is YES.

• The Analogy: You can win the game with a small, simple toolbox and by having both players do the exact same thing.
• The Result: For these specific puzzles, you don't have to sacrifice symmetry to keep things simple. You can have your cake (symmetry) and eat it too (minimal dimension).

2. The "Trade-off" Cases (The Hard Puzzles)

However, for a specific set of more complex puzzles (involving 3 or 4 different measurement choices), the answer is NO.

• The Analogy: Here, the rules are tricky. If you force the players to be identical (symmetric) and use the smallest possible toolbox, you cannot get the maximum score. You will get a "suboptimal" score (you lose points).
• The Catch: To get the maximum score on these puzzles, you have to choose one of two paths:
  • Path A: Use a symmetric strategy, but you must upgrade to a larger, more complex toolbox (higher dimension).
  • Path B: Keep the small, simple toolbox (minimal dimension), but you must break the symmetry. One player must do something slightly different from the other (an "asymmetric" strategy).
• The Surprise: The paper found that for these specific puzzles, the best way to win with the smallest toolbox is actually to be asymmetric. The players must be different to get the top score.

Why Does This Matter? (The Geometry of the Game)

The paper explains that this trade-off changes the shape of the "winning zone."

• The Flat Spot: Usually, if there is only one way to win a puzzle perfectly, that winning spot is a sharp point. But in these "trade-off" cases, because you can win by being asymmetric (with a small toolbox) OR symmetric (with a big toolbox), the winning area becomes a flat plateau.
• The Self-Testing Problem: In quantum physics, we often try to "self-test" devices. This means we look at the score and say, "Ah, you got the maximum score, so I know exactly what state and measurements you used!"
  • The paper shows that for these specific puzzles, you cannot self-test. Because there are multiple ways to get the maximum score (symmetric vs. asymmetric), seeing the top score doesn't tell you which strategy was used. You can't be sure if the players were identical or different.

A Special Twist: The "Mirror" Strategy

The researchers also discovered a cool way to be asymmetric but still look symmetric.

• Imagine one player is the "mirror image" of the other. If Player A looks left, Player B looks right. If Player A measures a specific way, Player B measures the "conjugate" way.
• Even though they are doing different things (asymmetric), the results they produce look perfectly identical (symmetric).
• The paper proves that for the "trade-off" puzzles, the best strategy with the smallest toolbox is often this kind of "mirror" strategy. It's asymmetric in action but symmetric in result.

Summary

• Symmetry (doing the same thing) is usually helpful, but sometimes it's a burden.
• Dimension (complexity) is a resource.
• For some quantum tests, you can be simple and symmetric.
• For others, you must choose: Be simple but different (asymmetric), OR Be identical (symmetric) but complex. You can't be both simple and identical if you want the perfect score.
• This discovery tells us that the landscape of quantum possibilities has "flat spots" where multiple strategies lead to the same perfect result, making it impossible to know exactly how a device is working just by looking at its score.

Technical Summary

Technical Summary: Trading Symmetry for Hilbert-Space Dimension in Bell-Inequality Violation

Problem Statement
In the study of quantum nonlocality, Bell inequalities serve as witnesses to the incompatibility between quantum theory and local hidden-variable (LHV) models. While it is well-established that higher-dimensional (HD) quantum systems can offer stronger violations and better noise resistance, the minimal Hilbert space dimension (HSD) required to achieve the maximal quantum violation of a specific Bell inequality is often unknown or difficult to determine.

A common simplification in analyzing Bell inequalities with specific symmetries—such as party-permutation invariance (PPI)—is to restrict the search for maximal violations to quantum strategies (QS) that respect the same symmetry. This "symmetry reduction" significantly lowers the computational complexity of optimization problems, particularly when combined with semidefinite programming (SDP) hierarchies. However, this approach raises a fundamental question: Does enforcing symmetry on the quantum strategy come at the cost of requiring a higher HSD? Specifically, is there a trade-off where a symmetric QS in the minimal dimension yields only a suboptimal violation, necessitating either an asymmetric QS of minimal dimension or a symmetric QS of higher dimension to achieve the true quantum maximum?

Methodology
The authors systematically investigate this trade-off across various bipartite Bell scenarios, including those with binary outcomes and up to four measurement settings, as well as scenarios with arbitrary outcomes but only binary measurement choices.

1. Definitions and Framework: The paper formalizes the concepts of symmetric correlations, symmetric Bell inequalities, and symmetric quantum strategies (SQS). An SQS is defined as a strategy where parties share a PPI state and perform identical local measurements. The authors distinguish between the set of all quantum correlations ($Q$), the set of symmetric correlations ($\vec{P}_{\leftrightarrow}$), and the subset of SQSs.
2. Symmetrization Procedures: The authors review and extend existing results (e.g., from Moroder et al.) showing that any symmetric correlation can be realized by an SQS, potentially at the cost of doubling the local HSD via ancillary systems. They also discuss purification procedures to obtain purified symmetric QSs (PSQS).
3. Numerical and Analytical Analysis:
   • For scenarios where the minimal dimension is known or bounded (e.g., CGLMP inequalities), the authors employ numerical optimizations and sum-of-squares decompositions to compare the maximal violation achievable by SQSs against the general quantum bound (often derived from the Navascués-Pironio-Acín (NPA) hierarchy).
   • For scenarios where trade-offs are suspected, they utilize SDP tools (detailed in Appendix A) to compute upper bounds on the violation achievable by symmetric qubit strategies and compare them against the general quantum bounds.
   • They construct explicit counter-examples involving "mirror-symmetric" strategies, where parties use measurements related by a reflection across the Bloch sphere's $x-z$ plane, combined with specific asymmetric states, to generate symmetric correlations.

Key Contributions and Results

• No Trade-off in Specific Families: For the family of symmetric Collins-Gisin-Linden-Massar-Popescu (CGLMP) inequalities (specifically the $I_{22dd}$ form) and the Salavrakos-Augusiak-Tura-Wittek-Acín-Pironio (SATWAP) inequalities in $(2, 2, n)$ scenarios, the authors provide evidence that no trade-off exists. They demonstrate that for dimensions $d \in [2, 19]$ (and conjectured for all $d$), there exists a symmetric quantum strategy (SQS) in the minimal dimension $d_{min}$ that achieves the maximal quantum violation.
• Existence of Trade-offs in Other Scenarios: Conversely, the authors identify several Bell inequalities where a trade-off is unavoidable:
  • In the $(2, 3, 2)$ scenario, a family of symmetric inequalities $I_S(\alpha)$ is shown to have a maximal quantum violation achievable only by an asymmetric two-qubit strategy. Symmetric qubit strategies (SQS) fail to violate these inequalities at all for certain parameter ranges, or yield strictly suboptimal values.
  • In the $(2, 4, 2)$ scenario, out of 52 non-trivial symmetric facet-defining inequalities, 9 exhibit a trade-off. For these, the maximal violation achievable by a symmetric qubit strategy is strictly lower than the general quantum maximum. In some cases (e.g., $J_{42}^{4422}$), even the maximal violation by any symmetric correlation (including those from asymmetric strategies) is lower than the general quantum maximum, though the gap between symmetric correlations and asymmetric correlations is smaller than the gap between symmetric correlations and the general bound.
• Mirror-Symmetric Strategies: The paper introduces a class of "mirror-symmetric" strategies where parties perform measurements related by a reflection ($M_B = M_A^*$) on an asymmetric state. The authors prove that such strategies can generate symmetric correlations even when the underlying state and measurements are not symmetric. They further prove that for non-coplanar measurement directions, these strategies cannot be symmetrized via local unitary transformations, confirming the necessity of asymmetry in the minimal dimension.

Significance and Implications

The paper claims several significant implications for the geometry of the set of quantum correlations and device-independent protocols:

1. Geometry of the Quantum Set: The existence of asymmetric maximizers for symmetric Bell inequalities implies that the set of quantum maximizers for these inequalities forms a flat region (a one-dimensional boundary) in the space of correlations. This is because the convex combination of an asymmetric maximizer and its permuted version also yields the maximal value.
2. Limitations on Self-Testing: A direct consequence of the flat boundary is that symmetric Bell inequalities admitting asymmetric maximizers cannot be used for self-testing based solely on the observed maximal violation. Self-testing requires a unique quantum correlation (up to local isometries) to certify the underlying state and measurements. Since multiple distinct strategies (symmetric and asymmetric) yield the same maximal value, uniqueness is lost.
3. Asymmetry as a Resource: The findings highlight that asymmetry is a resource for Bell-inequality violation. In specific cases, achieving the maximal violation with minimal resources (HSD) requires sacrificing symmetry.
4. Semi-Device-Independent Witnesses: The paper suggests that inequalities like $J_{42}^{4422}$ can serve as semi-device-independent witnesses for asymmetry. If one assumes the underlying strategy is a qubit strategy, observing a violation above the symmetric qubit bound certifies that the strategy must be asymmetric, even if the observed correlation appears symmetric.

Conclusion
The authors conclude that the relationship between symmetry and dimension is not uniform across all Bell inequalities. While symmetric strategies in minimal dimensions suffice for families like CGLMP and SATWAP, they are strictly suboptimal for others, such as certain inequalities in $(2, 3, 2)$ and $(2, 4, 2)$ scenarios. This trade-off reveals a complex structure in the quantum set of correlations, where enforcing symmetry can restrict the achievable violation unless higher dimensions are employed, and conversely, minimal dimensionality may require the abandonment of symmetry. The paper leaves open the question of whether such trade-offs exist in simpler scenarios or more complex multipartite settings, noting that analytic criteria for predicting these trade-offs remain an open challenge.