Unbounded Entanglement Can Be Needed to Achieve the Optimal Success Probability

Quantum entanglement is known to provide a strong advantage in many two-party distributed tasks. We investigate the question of how much entanglement is needed to reach optimal performance. For the first time we show that there exists a purely classical scenario for which no finite amount of entanglement suffices. To this end we introduce a simple two-party nonlocal game H, inspired by a paradox of Lucien Hardy. In our game each player has only two possible questions and can provide answers in a countable set. We exhibit a sequence of strategies which use entangled states in increasing dimension d and succeed with probability 1 − O(d^(−c)) for some c ≥ 0.13. On the other hand, we show that any strategy using an entangled state of local dimension d has success probability at most 1 − Ω(d⁻²). In addition, we show that any strategy restricted to producing answers in a set of cardinality at most d has success probability at most 1 − Ω(d⁻²).