A multi-player, multi-team nonlocal game for the toric code

TL;DR

This paper introduces a multi-player nonlocal game based on the toric code, demonstrating quantum advantage in winning probability and extending to states with M topological order, highlighting the quantumness of complex many-body states.

Contribution

It constructs a novel nonlocal game for the toric code that can be won with certainty using quantum states, unlike classical strategies, and generalizes to M topological order.

Findings

Quantum players win with certainty using the toric code ground state.

Classical players cannot surpass 50% winning probability in the large N limit.

The game framework extends to states with M topological order.

Abstract

Nonlocal games yield an unusual perspective on entangled quantum states. The defining property of such games is that a set of players in joint possession of an entangled state can win the game with higher probability than is allowed by classical physics. Here we construct a nonlocal game that can be won with certainty by $2N$ players if they have access to the ground state of the toric code on as many qubits. By contrast, the game cannot be won by classical players more than half the time in the large $N$ limit. Our game differs from previous examples because it arranges the players on a lattice and allows them to carry out quantum operations in teams, whose composition is dynamically specified. This is natural when seeking to characterize the degree of quantumness of non-trivial many-body states, which potentially include states in much more varied phases of matter than the toric code. We present generalizations of the toric code game to states with $\mathbb{Z}_M$ topological order.