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Transfer of quantum game strategies

Gage Hoefer

TL;DR

The paper develops a framework for transferring perfect strategies between two-player, one-round quantum non-local games with quantum inputs/outputs by embedding games into a unified 'game-of-games' and using strongly quantum no-signalling (SQNS) correlations as simulators. It establishes that such transfers preserve strategy class (local, quantum, QA, QC, NS) under mild conditions, via the simulation paradigm and Kraus-space characterizations, and extends these ideas to quantum graphs and concurrent games. Central to the approach is a detailed operator-system/ tensor-product analysis that represents SQNS correlations as states on canonical operator-system coproducts and tensor products, yielding a robust algebraic handle on strategy transport. The work further connects these transport results to concurrent (synchronous) quantum games through jointly tracial SQNS correlations and traces on associated ${ m C}^{*}$-algebras, highlighting the deep links between non-local games, operator algebras, and quantum information theory. Overall, it provides a coherent, mathematically rigorous pathway for transferring perfect strategies across a broad class of quantum games and lays groundwork for concurrent-game analyses via operator-algebraic invariants.

Abstract

We develop a method for the transfer of perfect strategies between various classes of two-player, one round cooperative non-local games with quantum inputs and outputs using the simulation paradigm in quantum information theory. We show that such a transfer is possible when canonically associated operator spaces for each game are quantum homomorphic or isomorphic, as defined in the joint work of H. and Todorov (2024). We examine a new class of QNS correlations, needed for the transfer of strategies between games, and characterize them in terms of states on tensor products of canonical operator systems. We define jointly tracial correlations and show they correspond to traces acting on tensor products of canonical ${\rm C}^{*}$-algebras associated with individual game parties. We then make an inquiry into the initial application of such results to the study of concurrent quantum games.

Transfer of quantum game strategies

TL;DR

The paper develops a framework for transferring perfect strategies between two-player, one-round quantum non-local games with quantum inputs/outputs by embedding games into a unified 'game-of-games' and using strongly quantum no-signalling (SQNS) correlations as simulators. It establishes that such transfers preserve strategy class (local, quantum, QA, QC, NS) under mild conditions, via the simulation paradigm and Kraus-space characterizations, and extends these ideas to quantum graphs and concurrent games. Central to the approach is a detailed operator-system/ tensor-product analysis that represents SQNS correlations as states on canonical operator-system coproducts and tensor products, yielding a robust algebraic handle on strategy transport. The work further connects these transport results to concurrent (synchronous) quantum games through jointly tracial SQNS correlations and traces on associated -algebras, highlighting the deep links between non-local games, operator algebras, and quantum information theory. Overall, it provides a coherent, mathematically rigorous pathway for transferring perfect strategies across a broad class of quantum games and lays groundwork for concurrent-game analyses via operator-algebraic invariants.

Abstract

We develop a method for the transfer of perfect strategies between various classes of two-player, one round cooperative non-local games with quantum inputs and outputs using the simulation paradigm in quantum information theory. We show that such a transfer is possible when canonically associated operator spaces for each game are quantum homomorphic or isomorphic, as defined in the joint work of H. and Todorov (2024). We examine a new class of QNS correlations, needed for the transfer of strategies between games, and characterize them in terms of states on tensor products of canonical operator systems. We define jointly tracial correlations and show they correspond to traces acting on tensor products of canonical -algebras associated with individual game parties. We then make an inquiry into the initial application of such results to the study of concurrent quantum games.

Paper Structure

This paper contains 12 sections, 23 theorems, 192 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Strongly stochastic operator matrices
  4. SQNS correlations
  5. Strategy transport
  6. Perfect strategies for various quantum games
  7. Perfect strategies for quantum implication games
  8. Classical-to-quantum non-local games
  9. Quantum graph games
  10. Characterization of SQNS correlations and applications to concurrent games
  11. Representations of SQNS correlations via operator systems
  12. Concurrent strategies

Key Result

Proposition 3.3

A QNS correlation $\Gamma$ over $(X, Y, A, B)$ belongs to $\mathcal{Q}_{\rm loc}$ if and only if there exists a Hilbert space $H$, a locally dilatable strongly stochastic operator matrix $P = (P_{xx', yy'}^{aa', bb'})_{xx', yy', aa', bb'}$ acting on $H$ and a unit vector $\xi \in H$ such that for $x, x' \in X, y, y' \in Y, a, a' \in A, b, b' \in B$.

Theorems & Definitions (66)

  • Remark 3.1
  • Remark 3.2
  • Proposition 3.3
  • proof
  • Remark 3.4
  • Remark 3.5
  • Proposition 3.6
  • proof
  • Definition 3.7
  • Remark 3.8
  • ...and 56 more