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Commuting Embeddings for Parallel Strategies in Non-local Games

Sarah Chehade, Andrea Delgado, Elaine Wong

TL;DR

The paper tackles the qubit inefficiency of parallel non-local games by introducing a commuting-embedding framework and leveraging Cartan decompositions from Lie theory. It proves two compression results: (i) a random-referee single-game compression using a maximally entangled state of dimension equal to the largest game, and (ii) a parallel-compression scheme whereby multiple games can be played on a shared, smaller Hilbert space if their algebras embed into commuting subalgebras. The authors provide a constructive approach, including a Common Winning Sector analysis, and illustrate the ideas with a two-game MSG/MRG example that reduces the qubit count from $N= obreak extstyle\sum_i n_i$ to $n< N$. This framework reframes NLGs as algebraic primitives for distributed and resource-constrained quantum computation and points to device-independent dimension witnessing as a potential experimental probe of such embeddings.

Abstract

Non-local games (NLGs) provide a versatile framework for probing quantum correlations and for benchmarking the power of entanglement. In finite dimensions, the standard method for playing several games in parallel requires a tensor product of the local Hilbert spaces, which scales additively in the number of qubits. In this work, we show that this additive cost can be reduced by exploiting algebraic embeddings. We introduce two forms of compressions. First, when a referee selects one game from a finite collection of games at random, the game quantum strategy can be implemented using a maximally entangled state of dimension equal to the largest individual game, thereby eliminating the need for repeated state preparations. Second, we establish conditions under which several games can be played simultaneously in parallel on fewer qubits than the tensor product baseline. These conditions are expressed in terms of commuting embeddings of the game algebras. Moreover, we provide a constructive framework for building such embeddings. Using tools from Lie theory, we show that aligning the various game algebras into a common Cartan decomposition enables such a qubit reduction. Beyond the theoretical contribution, our framework casts NLGs as algebraic primitives for distributed and resource constrained quantum computations and suggested NLGs as a comparable device independent dimension witness.

Commuting Embeddings for Parallel Strategies in Non-local Games

TL;DR

The paper tackles the qubit inefficiency of parallel non-local games by introducing a commuting-embedding framework and leveraging Cartan decompositions from Lie theory. It proves two compression results: (i) a random-referee single-game compression using a maximally entangled state of dimension equal to the largest game, and (ii) a parallel-compression scheme whereby multiple games can be played on a shared, smaller Hilbert space if their algebras embed into commuting subalgebras. The authors provide a constructive approach, including a Common Winning Sector analysis, and illustrate the ideas with a two-game MSG/MRG example that reduces the qubit count from to . This framework reframes NLGs as algebraic primitives for distributed and resource-constrained quantum computation and points to device-independent dimension witnessing as a potential experimental probe of such embeddings.

Abstract

Non-local games (NLGs) provide a versatile framework for probing quantum correlations and for benchmarking the power of entanglement. In finite dimensions, the standard method for playing several games in parallel requires a tensor product of the local Hilbert spaces, which scales additively in the number of qubits. In this work, we show that this additive cost can be reduced by exploiting algebraic embeddings. We introduce two forms of compressions. First, when a referee selects one game from a finite collection of games at random, the game quantum strategy can be implemented using a maximally entangled state of dimension equal to the largest individual game, thereby eliminating the need for repeated state preparations. Second, we establish conditions under which several games can be played simultaneously in parallel on fewer qubits than the tensor product baseline. These conditions are expressed in terms of commuting embeddings of the game algebras. Moreover, we provide a constructive framework for building such embeddings. Using tools from Lie theory, we show that aligning the various game algebras into a common Cartan decomposition enables such a qubit reduction. Beyond the theoretical contribution, our framework casts NLGs as algebraic primitives for distributed and resource constrained quantum computations and suggested NLGs as a comparable device independent dimension witness.
Paper Structure (9 sections, 7 theorems, 80 equations, 2 figures, 1 table, 1 algorithm)

This paper contains 9 sections, 7 theorems, 80 equations, 2 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Baseline Games
  4. Cartan Extrapolation
  5. Cartan preliminaries
  6. Game algebras and Cartan sectors
  7. Compressed Games
  8. Conclusion
  9. Acknowledgments

Key Result

Theorem 1

Let $\{\mathcal{G}_i\}_{i=1}^K$ be $K$ non-local games, each admitting a perfect quantum strategy on a fixed number of qubits $n_i$ for each player. Let $\mathcal{A}_i$ be the $C^*$-algebra generated by the measurement operators of $\mathcal{G}_i$ for Alice, and $\mathcal{B}_i$ the corresponding alg and measurement operators $\{M_{x_i,a_i}\}_a \subset \mathcal{A}_i$ and $\{N_{y_i,b_i}\}_b \subset

Figures (2)

  • Figure 1: For $K=4$, this graphic depicts 4 parallel and independent games, where each game requires 2 entangled states. For game $i$ and qubit $j$, the $q_{i,j}$ denotes qubits while the dashed line represents the entanglement between the qubits (players).
  • Figure 2: Two MSGs from \ref{['ex: compressed']} compressed to 3 qubits per player.

Theorems & Definitions (23)

  • Theorem 1
  • proof
  • Example 1
  • Theorem 2
  • proof
  • Remark 1
  • Example 2
  • Example 3
  • Definition 1
  • Theorem 3
  • ...and 13 more