⚛️ quantum physics

Beyond the Magic Square Game: Widening the Gap for Two Bell States

This paper demonstrates that the largest gap between the entangled and classical values for a one-round two-player nonlocal game utilizing two Bell states is at least $\frac{4}{35}$ , surpassing the previous $\frac{1}{9}$ record set by the Mermin-Peres magic square game through the explicit construction of a new game with a classical value of $\frac{31}{35}$ based on the 2-qubit Pauli group's symmetry.

Original authors: Tony Lau

Published 2026-03-24

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Tony Lau

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 a detective trying to catch a pair of liars. These liars, let's call them Alice and Bob, claim to share a magical, invisible connection that allows them to coordinate their answers perfectly without ever talking to each other. They say they have "two magic coins" (Bell states) that are linked across space.

Your job is to test them. You ask them questions, and they must give answers. If they are telling the truth (and using quantum magic), they should win 100% of the time. If they are just guessing or using normal tricks (classical physics), they should fail often.

For a long time, the best test we had was the "Magic Square Game."

The Setup: You ask them to fill in a 3x3 grid of numbers.
The Catch: The rules are tricky. The rows must add up to an even number, but the columns must add up to an odd number.
The Result:
- Classical Liars: They can't do it perfectly. The math says the grid is impossible to fill. The best they can do is get it right 8 out of 9 times (88.8%).
- Quantum Heroes: Using their "magic coins," they can fill the grid perfectly 100% of the time.
- The Gap: The difference between the liars (8/9) and the heroes (1) is 1/9. This is the "gap" the paper talks about. It's a decent test, but not perfect. Sometimes, a really clever liar might get lucky and fool you.

The New Game: The "Augmented Magic Square"

The author, Tony Lau, wanted to make the test stricter. He wanted to widen the gap so that the liars have a much harder time fooling the referee.

He looked at the "Magic Square" and realized it was only using a tiny fraction of the available "magic rules." It was like playing a game of chess but only using the pawns and ignoring the knights, bishops, and queens.

So, he built a bigger, more complex game called the Augmented Magic Square (AMS) Game.

The Setup: Instead of a 3x3 grid, imagine a complex 3D shape (a tetrahedron) with 15 different rules and 15 variables. It's like a giant, interconnected web of logic puzzles.
The Result:
- Quantum Heroes: They can still win 100% of the time using their magic coins. The quantum rules still hold up.
- Classical Liars: Surprisingly, they still managed to win 8/9 of the time!
- The Problem: The gap didn't get bigger. The liars were just as good at faking it as before.

The Twist: The "Synchronous" Trap

The author noticed something weird about the liars who were winning the big game. To win, they had to be asymmetric.

Symmetric: Alice and Bob use the exact same strategy.
Asymmetric: Alice and Bob use slightly different strategies to cheat the system.

In the big game, the liars had to be asymmetric to get that 8/9 score. If they tried to be symmetric (use the same strategy), they would only win 13/15 times (which is worse than 8/9).

The Solution: The author created a new game called the p-SAMS Game.
Think of this as a game where, most of the time, you play the big "Augmented" game. But, every now and then (with a specific probability), the referee hits a "Synchronous Button."

The Synchronous Button: The referee asks Alice and Bob the exact same question and demands they give the exact same answer.
The Trap:
- If Alice and Bob are Quantum Heroes, they are perfectly coordinated. They give the same answer every time. They win.
- If Alice and Bob are Classical Liars using the "asymmetric" trick to win the big game, they will disagree on some answers. When the Synchronous Button is hit, they get caught lying!

The Grand Result

By mixing the big game with the "Synchronous Button" just the right amount (specifically, 1 out of every 7 times), the author created the 1/7-SAMS Game.

Quantum Heroes: Still win 100% of the time.
Classical Liars: Now, even their best cheating strategy only works 31 out of 35 times.

Why This Matters

Let's look at the "Gap" again:

Old Game (Magic Square): Liars win 8/9 (0.888). Gap = 1/9 (0.111).
New Game (1/7-SAMS): Liars win 31/35 (0.885). Gap = 4/35 (0.114).

Wait, 4/35 is bigger than 1/9!

1/9 is roughly 0.1111.
4/35 is roughly 0.1142.

It seems like a tiny difference, but in the world of quantum physics, this is a massive improvement. It means we can now distinguish between "real quantum magic" and "clever classical cheating" with much higher confidence.

The Analogy Summary

Imagine you are trying to tell the difference between a Master Magician and a Sleight-of-Hand Artist.

The Old Test: You ask them to pull a rabbit out of a hat. The Artist can do it 88% of the time. The Magician does it 100% of the time. It's hard to tell them apart.
The New Test: You ask them to pull a rabbit out of a hat, but you also add a rule: "If I clap my hands, you must both pull out the exact same rabbit."
The Result: The Artist, who relies on secret signals and different tricks for each person, gets confused by the clap and fails more often. The Magician, who is truly connected, doesn't care.

The paper proves that by adding this specific "clap" (the synchronous question) to a complex puzzle, we can catch the liars more often than ever before. This helps us build better quantum computers and more secure encryption, because we can be more sure that the "magic" we are seeing is real.

1. Problem Statement

The paper addresses the problem of device-independent self-testing in quantum information theory. Specifically, it seeks to distinguish between two scenarios:

Two parties (Alice and Bob) share two Bell states (maximally entangled qubits).
The parties share no entanglement (classical correlation only).

The standard method for this distinction is the Mermin-Peres Magic Square (MS) game.

Entangled Value: 1 (Players win with certainty using two Bell states).
Classical Value: $8/9$ (The optimal classical strategy wins with probability $8/9$ ).
The Gap: The difference is $1 - 8/9 = 1/9$ .

The central question is: Can this "false positive" rate (the classical winning probability) be reduced below $8/9$ while maintaining a perfect entangled strategy for two Bell states? The paper aims to construct a nonlocal game that widens the gap between the classical and entangled values.

2. Methodology

The author employs a constructive approach based on the algebraic structure of the two-qubit Pauli group ( $\tilde{P}_2$ ) and its quotient group ( $P_2$ ).

A. The Augmented Magic Square (AMS) Game

The MS game utilizes only 6 commuting triples of Pauli operators out of the 15 available non-identity elements in the Pauli quotient group $P_2 \cong \mathbb{Z}_2^4$ .

Construction: The author constructs a new game, the Augmented Magic Square (AMS), which utilizes all 15 commuting triples of $P_2$ .
Structure: The game involves 15 variables ( $V_0$ to $V_{14}$ ) and 15 equations ( $E_0$ to $E_{14}$ ). The referee selects two intersecting equations (sharing exactly one variable) and sends them to Alice and Bob.
Entangled Strategy: Just like the MS game, the AMS game admits a perfect entangled strategy using two Bell states, where players measure observables corresponding to the commuting triples.
Classical Result: Surprisingly, the AMS game alone does not improve the gap; its classical value remains $8/9$ . However, the author identifies a crucial structural property: all optimal classical strategies for the AMS game are asymmetric (Alice's strategy $\neq$ Bob's strategy). Symmetric strategies perform strictly worse ( $13/15$ ).

B. The $p$ -Synchronous AMS ( $p$ -SAMS) Game

To exploit the asymmetry of optimal classical strategies, the author introduces a probabilistic mixture game called the $p$ -SAMS game.

Mechanism: With probability $1-p$ , the game proceeds as the standard AMS game. With probability $p$ , the referee sends the same equation to both Alice and Bob (a "synchronous" question).
Constraint: In the synchronous round, Alice and Bob must output identical answers (synchronous condition) while satisfying the parity constraint.
Effect:
- Entangled Players: Can still win with probability 1 because the quantum strategy is symmetric (or can be made symmetric without loss of optimality for the specific operator solution).
- Classical Players: Since optimal classical strategies for the AMS game are asymmetric, they will inevitably disagree on at least one equation. If the referee picks that specific equation for the synchronous round, the players lose. This penalizes asymmetric strategies.

3. Key Contributions and Theoretical Results

1. Analysis of Symmetric vs. Asymmetric Strategies

The paper rigorously proves that for the AMS game:

Theorem 3.5: The classical value is $8/9$ .
Theorem 3.7: The optimal symmetric classical strategy wins with probability $13/15$ (which is $\approx 0.866$ , strictly less than $8/9 \approx 0.888$ ).
Theorem 3.11: All optimal classical strategies ( $8/9$ ) must be asymmetric.

2. Bounding Asymmetric Strategies in Synchronous Rounds

Lemma 4.3: An optimal asymmetric strategy for the AMS game can win on at most 13 out of the 15 synchronous questions. (It cannot win on 14 or 15).
- Proof Logic: The proof uses a contradiction argument involving "symmetrizing" an asymmetric strategy and analyzing the change in winning probability based on variable consistency (1-consistent vs. 2/3-consistent assignments).

3. Optimization of the Gap

The author calculates the optimal mixing parameter $p$ to minimize the classical winning probability while keeping the entangled value at 1.

Let $S_1$ $S_{1}$ be a symmetric strategy: Wins synchronous rounds (prob $p$ $p$ ) with prob 1, and AMS rounds (prob $1-p$ $1 - p$ ) with max prob $13/15$ $13/15$ .
- $P(S_1) = p + (1-p)\frac{13}{15}$
Let $S_2$ $S_{2}$ be an asymmetric strategy: Wins synchronous rounds with max prob $13/15$ $13/15$ (due to Lemma 4.3), and AMS rounds with max prob $8/9$ $8/9$ .
- $P(S_2) = p\frac{13}{15} + (1-p)\frac{8}{9}$
Equating the bounds: Setting $P(S_1) = P(S_2)$ yields $p = 1/7$ .
Resulting Classical Value: At $p = 1/7$ , the classical value is $31/35$ .

4. Results

The paper presents the $1/7$ -SAMS game as the primary result:

Entangled Value: 1 (Perfect win with two Bell states).
Classical Value: $31/35$ .
The Gap: $1 - 31/35 = \mathbf{4/35}$ .

Comparison:

Magic Square (MS) Game: Gap = $1/9 \approx 0.111$ .
$1/7$ -SAMS Game: Gap = $4/35 \approx 0.114$ .
Improvement: The gap is widened from $1/9$ to $4/35$ . This represents a reduction in the classical false positive rate from $8/9$ ($0.888...$) to $31/35$ ($0.8857...$).

5. Significance

Tightening Self-Testing: The result provides a more robust test for verifying the presence of two Bell states. A lower classical winning probability means that if parties claim to share two Bell states and win with high probability, it is statistically harder for them to be cheating using only classical resources.
Structural Insight: The work highlights the deep connection between the geometry of the Pauli group (specifically the projective space $PG(3,2)$) and nonlocal games. It demonstrates that "augmenting" a game to use the full symmetry of the underlying algebraic structure, combined with synchronous constraints, can reveal limitations in classical strategies that simpler games miss.
Future Directions: The author suggests that further improvements might be possible by:
1. Proving that asymmetric strategies can win on even fewer synchronous questions (e.g., 12 instead of 13), which could lower the classical value to $22/25$ .
2. Generalizing these constructions to higher dimensions ( $d > 2$ ).
3. Investigating equivalence classes for the AMS game similar to those for the Magic Square game.

In summary, this paper successfully constructs a nonlocal game that strictly improves the distinguishability between two Bell states and classical correlations, pushing the theoretical limit of the entangled-classical gap for this specific resource.