⚛️ quantum physics

Post-selection games

This paper introduces post-selection games as a generalization of nonlocal games that allow rounds to be discarded, providing algorithms to compute their bounds and demonstrating their superior statistical power for analyzing Bell tests with low detection efficiency.

Original author: Víctor Calleja Rodríguez, Ivan A. Bocanegra-Garay, Mateus Araújo

Published 2026-11-04

📖 5 min read🧠 Deep dive

CC BY 4.0

Original author: Víctor Calleja Rodríguez, Ivan A. Bocanegra-Garay, Mateus Araújo

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 author. For technical accuracy, refer to the original paper. Read full disclaimer

In the early days of quantum physics, a profound puzzle emerged: could the strange, interconnected behavior of particles be explained by hidden rules we simply hadn't found yet? For decades, physicists tested this idea by playing a cooperative game with two separated players. In this game, a referee sends each player a random question, and they must answer without communicating. If their answers match a specific pattern more often than classical logic allows, it proves that the universe is fundamentally nonlocal—meaning particles can influence each other instantly across vast distances, defying our everyday intuition. These tests usually rely on statistics, looking for a slight statistical edge that classical physics cannot replicate. However, this approach has a weakness: real-world experiments are messy. Detectors miss particles, and noise creeps in, often making it hard to distinguish a genuine quantum signal from a fluke.

A team of researchers in Spain has now introduced a new way to play this game, one that turns the weakness of missing data into a strength. They developed what they call "post-selection games." In a standard test, every round counts, whether the players succeed or fail. In this new version, the referee has the power to discard a round entirely if the outcome doesn't fit a specific, rare pattern. By ignoring the messy, unhelpful rounds and focusing only on the specific events that matter, the players can win the game with absolute certainty if they are using quantum mechanics, while a classical player can never do the same. This method transforms a subtle statistical proof into a clear, binary victory, offering a powerful new tool for testing the foundations of reality, especially in difficult experimental conditions where detectors are not perfect.

The researchers, Víctor Calleja Rodríguez, Ivan A. Bocanegra-Garay, and Mateus Araújo, built their work on a famous logical puzzle known as Hardy's paradox. For years, this paradox was seen as a beautiful but impractical proof of quantum weirdness because it relied on the idea that certain events are strictly impossible. In the real world, you can never prove something is impossible; you can only show that it hasn't happened yet. If an experiment shows a zero frequency for an event, it might just be bad luck or a faulty detector. The new approach solves this by redefining the game. Instead of trying to prove an event never happens, the players agree to throw away any round where that event does happen. They only keep the rounds where the event is absent. In this filtered set of data, the quantum players can win every single time, while a classical player, bound by local rules, will inevitably lose some rounds. This turns a theoretical impossibility into a practical, winnable game.

To make this work, the team had to invent new mathematical tools to calculate the limits of these games. They needed to know the best possible score a classical player could achieve (the local bound) and the best score a quantum player could achieve (the Tsirelson bound). They developed algorithms to compute these limits, proving that for their new games, the quantum advantage is real and calculable. They showed that even with the noise and imperfections of real life, these post-selection games remain robust. In fact, they found something surprising: while the new games are weaker than traditional tests in a perfect, noise-free world, they become incredibly powerful when the experiment is noisy. As the quality of the detectors drops, the advantage of the new method grows without limit.

The team tested this idea using data from a famous, high-profile experiment that had previously been analyzed using standard methods. When they re-analyzed the same raw numbers using their new post-selection game, the results were striking. The standard analysis of that data was inconclusive, unable to definitively rule out classical explanations. However, when viewed through the lens of the new game, the same data provided a decisive rejection of classical physics. The new method was able to reach a conclusion with the same statistical strength using far fewer rounds of the game. In the noisy conditions of that specific experiment, the new approach was roughly sixteen times more efficient than the traditional one.

The researchers also explored whether this advantage holds up as the experiment gets even worse, simulating scenarios where detectors are very inefficient. They found that as the detection efficiency drops toward a critical threshold, the statistical power of their new game grows infinitely larger than that of any traditional game. This suggests that for experiments where detectors are poor or where data is scarce, this new method is not just an alternative but a necessity. They also looked at more complex versions of the game with more questions and answers, finding that the statistical power can be further tuned, though the basic advantage remains rooted in the ability to discard the unhelpful data.

This work does not just offer a new way to calculate numbers; it changes how we think about proving the strange nature of our universe. By accepting that some data is useless and discarding it, the researchers have found a way to make the quantum world speak more clearly. They have shown that what was once a theoretical curiosity, a paradox that could not be tested, can be turned into a rigorous, high-powered experiment. The result is a tool that is ideally suited for the messy reality of modern physics, where detectors fail and noise is inevitable, allowing scientists to see the quantum signature with a clarity that was previously out of reach.

Technical Summary of "Post-selection games"

Problem Statement
The paper addresses the limitations of traditional nonlocal games and possibilistic proofs of nonlocality, such as Hardy's paradox. While proofs like GHZ-Mermin allow for "pseudo-telepathy" (winning with certainty), Hardy-like proofs rely on possibilistic nonlocality—demonstrating that certain events are possible or impossible rather than relying on statistical inequalities. However, standard experimental tests of Hardy's paradox face two major issues:

Impossibility of Proof: One cannot experimentally prove an event is strictly impossible; observing a frequency of zero does not guarantee impossibility.
Suboptimal Statistical Power: When experimentalists attempt to test Hardy's paradox using standard Bell inequalities (like the CH inequality), the resulting games often do not maximize the violation for correlations satisfying Hardy's conditions, leading to weak statistical evidence against local hidden variable (LHV) models.

The authors propose that these issues stem from the inability of traditional nonlocal games to naturally formalize possibilistic proofs where specific events are discarded rather than counted as wins or losses.

Methodology
The authors introduce post-selection games, a generalization of nonlocal games defined by three functions: input distribution $\mu(x,y)$ , win function $V(a,b,x,y)$ , and a post-selection function $S(a,b,x,y)$ .

Mechanism: In each round, the referee can declare the round a win, a loss, or discarded (post-selected out). The probability of winning is defined as the conditional probability of winning given that the round was not discarded:
$\omega_G(P) = \frac{\langle V_\mu, P \rangle}{\langle S_\mu, P \rangle}$
where $P$ is the behavior (conditional probability distribution), and the denominator represents the probability of successful post-selection.
Formalization of Hardy's Paradox: The authors map Hardy's paradox to a post-selection game where post-selection occurs only on the specific events involved in the paradox ( $E_{Hardy}$ ), and the win condition is the specific event that quantum mechanics predicts to be non-zero while local theories predict to be zero.
Computational Framework:
- Local Bound: The authors prove that the local bound of a post-selection game is a quasiconvex function maximized at the extreme points (vertices) of the local polytope with non-zero post-selection probability. This allows the local bound to be computed by testing a finite set of deterministic behaviors.
- Tsirelson Bound: They formulate the computation of the quantum (Tsirelson) bound as a conic optimization problem. By utilizing the homogeneity of the ratio, they transform the problem into maximizing a linear function over the cone of unnormalized quantum behaviors subject to a normalization constraint, solvable via the NPA hierarchy and semidefinite programming.
Statistical Power Analysis: The paper adopts a hypothesis testing framework (Bayes factor and p-value) to define the statistical power ( $W_G$ ) of a game. This metric incorporates both the winning probability gap between quantum and local models and the probability of post-selection (efficiency):
$W_G(Q) = \gamma_G(Q) D(\omega_G(Q) \parallel \omega_\ell(G))$
where $\gamma_G$ is the post-selection probability and $D$ is the Kullback-Leibler divergence.

Key Contributions and Results

Definition and Formalization: The paper rigorously defines post-selection games, showing they naturally formalize possibilistic proofs like Hardy's paradox, turning them into games that can be won with certainty (Tsirelson bound = 1) by quantum strategies.
Algorithms: The authors provide algorithms to compute the local and Tsirelson bounds for these games, proving that the local bound is attained at deterministic vertices and formulating the quantum bound as a conic optimization problem.
Statistical Power Comparison (Ideal vs. Noisy):
- Ideal Case: In a noiseless scenario, the Hardy post-selection game has a lower statistical power than the standard CHSH game. This is because the low probability of post-selection (discarding most rounds) outweighs the large gap between the local and quantum winning probabilities.
- Noisy Case (Low Detection Efficiency): The paper demonstrates a counterintuitive result: as detection efficiency ( $\eta$ ) decreases, the statistical power of the Hardy post-selection game surpasses that of the CHSH game.
- Unbounded Advantage: The authors analytically prove that as detection efficiency approaches the critical threshold of $\eta \to 2/3$ , the ratio of the statistical power of the Hardy game to the CHSH game diverges to infinity. This suggests that post-selection games are ideally suited for Bell tests with low detection efficiency, where traditional games fail to provide conclusive evidence.
Generalizations: The framework is extended to generalized Hardy games with $s$ inputs and $k$ outputs. The authors show that while statistical power varies with parameters, it generally increases with the number of outputs ( $k$ ).
Non-Possibilistic Examples: The authors identify other non-trivial post-selection games not derived from possibilistic proofs (e.g., game $G_{\neg}$ ), though these were found to have lower statistical power than the Hardy game in the tested scenarios.

Significance and Claims
The paper claims that post-selection games offer a "natural generalization" of nonlocal games that bridges the gap between possibilistic proofs and experimental reality.

Robustness to Noise: Unlike traditional possibilistic proofs which are fragile to noise, post-selection games naturally tolerate noise and can still demonstrate nonlocality even when correlations are close to the local bound.
Superiority in Low-Efficiency Regimes: The primary significance lies in the finding that post-selection games possess an unbounded advantage in statistical power over traditional nonlocal games (specifically CHSH) as detection efficiency drops. This makes them theoretically optimal for analyzing Bell tests with low detection efficiency, potentially allowing for loophole-free demonstrations of nonlocality in regimes where standard inequalities fail.
Conceptual Clarity: The framework simplifies the analysis of Bell tests by treating the post-selection process as an intrinsic part of the game definition rather than an external statistical correction, thereby avoiding loopholes (like the fair-sampling loophole) by explicitly computing bounds that account for post-selection.

The authors modestly conjecture that this unbounded advantage holds against any nonlocal game without post-selection, though they only provide numerical evidence for a subset of games and a rigorous proof against CHSH.

Technical Summary of "Post-selection games"

More like this