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Post-selection games

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

TL;DR

This work extends nonlocal games by introducing post-selection games, where rounds can be discarded, to capture possibilistic nonlocality such as Hardy within a post-selected framework. It develops a rigorous formalism with $S,V,\mu$ and derives the win probability $\omega(P) = \frac{\langle V_\mu, P\rangle}{\langle S_\mu, P\rangle}$, establishing that the local bound $\omega_\ell$ can be found at extreme points and the Tsirelson bound $\omega_q$ can be computed via conic optimization and NPA-type hierarchies. Through both analytic and numerical analysis, the paper finds that ideal post-selection can yield modest statistical power gains over CHSH, but in the presence of detection inefficiency the ratio of statistical powers can become unbounded, giving a dramatic advantage for Hardy-type post-selection tests in noisy Bell experiments. The authors also generalize Hardy to more inputs/outputs, deriving scaling laws for statistical power and demonstrating practical implications for low-detection Bell tests and experimental design.

Abstract

In this paper, we introduce post-selection games, a generalization of nonlocal games where each round can be not only won or lost by the players, but also discarded by the referee. Such games naturally formalize possibilistic proofs of nonlocality, such as Hardy's paradox. We develop algorithms for computing the local and Tsirelson bounds of post-selection games. Furthermore, we show that they have an unbounded advantage in statistical power over traditional nonlocal games, making them ideally suited for analysing Bell tests with low detection efficiency.

Post-selection games

TL;DR

This work extends nonlocal games by introducing post-selection games, where rounds can be discarded, to capture possibilistic nonlocality such as Hardy within a post-selected framework. It develops a rigorous formalism with and derives the win probability , establishing that the local bound can be found at extreme points and the Tsirelson bound can be computed via conic optimization and NPA-type hierarchies. Through both analytic and numerical analysis, the paper finds that ideal post-selection can yield modest statistical power gains over CHSH, but in the presence of detection inefficiency the ratio of statistical powers can become unbounded, giving a dramatic advantage for Hardy-type post-selection tests in noisy Bell experiments. The authors also generalize Hardy to more inputs/outputs, deriving scaling laws for statistical power and demonstrating practical implications for low-detection Bell tests and experimental design.

Abstract

In this paper, we introduce post-selection games, a generalization of nonlocal games where each round can be not only won or lost by the players, but also discarded by the referee. Such games naturally formalize possibilistic proofs of nonlocality, such as Hardy's paradox. We develop algorithms for computing the local and Tsirelson bounds of post-selection games. Furthermore, we show that they have an unbounded advantage in statistical power over traditional nonlocal games, making them ideally suited for analysing Bell tests with low detection efficiency.
Paper Structure (11 sections, 4 theorems, 58 equations, 2 figures, 1 table)

This paper contains 11 sections, 4 theorems, 58 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Hardy's paradox
  4. Post-selection games
  5. Local bound
  6. Tsirelson bound
  7. Statistical power
  8. Examples
  9. Unbounded advantage
  10. Generalizations of the Hardy game
  11. Acknowledgements

Key Result

Lemma 1

For all behaviours $P$ with post-selection probablity $\gamma(P) > 0$, there exists a behaviour $P'$ such that $\omega(P') = \omega(P)$ and $P'$ is a convex combination only of extreme points $E_i$ with post-selection probability $\gamma(E_i) > 0$.

Figures (2)

  • Figure 1: Maximal statistical power of CHSH and Hardy games as a function of detection efficiency, Equations (\ref{['eq:chshpower']})-(\ref{['eq:hardypower']}).
  • Figure 2: Ratio between maximal statistical powers of CHSH and Hardy games as a function of detection efficiency, Equation \ref{['eq:ratio']}.

Theorems & Definitions (8)

  • Lemma 1
  • proof
  • Definition 2
  • Lemma 3
  • Lemma 4
  • proof
  • Theorem 5
  • proof