Maximally entangled states in pseudo-telepathy games

A pseudo-telepathy game is a nonlocal game which can be won with probability one using some finite-dimensional quantum strategy but not using a classical one. Our central question is whether there exist two-party pseudo-telepathy games which cannot be won with probability one using a maximally entangled state. Towards answering this question, we develop conditions under which maximally entangled states suffice. In particular, we show that maximally entangled states suffice for weak projection games which we introduce as a relaxation of projection games. Our results also imply that any pseudo-telepathy weak projection game yields a device-independent certification of a maximally entangled state. In particular, by establishing connections to the setting of communication complexity, we exhibit a class of games $G_n$ for testing maximally entangled states of local dimension $\Omega(n)$. We leave the robustness of these self-tests as an open question.

💡 Research Summary

The paper investigates a fundamental question in quantum non‑locality: whether every two‑party pseudo‑telepathy game that admits a perfect quantum strategy can be won using a maximally entangled state (MES). A pseudo‑telepathy game is a non‑local game that can be won with probability one by some finite‑dimensional quantum strategy but not by any classical strategy. While it is known that in many Bell‑inequality scenarios less‑entangled states can outperform MES, it remained open whether MES are sufficient for perfect strategies.

To address this, the authors introduce the notion of weak projection games. A game is weakly projective for Bob if for each of Bob’s questions (t) there exists an Alice question (s(t)) and a deterministic function (f_{st}:A\to B) such that the verifier accepts exactly when Bob’s answer equals (f_{st}) applied to Alice’s answer. The definition for Alice is symmetric; a game that is weakly projective for either player is called a weak projection game. This class strictly contains the traditional projection games (where the condition must hold for all question pairs) and is tailored to capture the structural property needed for the authors’ arguments.

The technical core consists of three lemmas. Lemma 1 shows that if the reduced density matrix (D) of a shared state (|\psi\rangle) commutes with one party’s measurement operators, then the zero‑probability events of the form (\operatorname{Tr}(E_i\otimes F_j|\psi\rangle\langle\psi|)=0) are preserved when (|\psi\rangle) is replaced by a MES (|\Psi_d\rangle). Lemma 2 proves that if two local measurements on a full‑Schmidt‑rank state produce perfectly correlated outcomes (i.e., (\operatorname{Tr}(E_i\otimes F_j|\psi\rangle\langle\psi|)=0) for all (i\neq j)), then each measurement must be a projective measurement and must commute with (D). Corollary 1 extends this to the case where the two measurements have different numbers of outcomes but are linked by a function (f) that enforces the same zero‑probability condition.

Using these lemmas, Theorem 1 establishes that any weakly projective game that admits a perfect quantum strategy can be won with a MES. The proof proceeds by applying Corollary 1 to the specific pair of measurements associated with each Bob question, thereby showing that Bob’s measurements are projectors that commute with the reduced state. Lemma 1 then allows the substitution of the original shared state by a MES without affecting the winning condition.

A direct consequence is Corollary 2, which provides a device‑independent self‑testing statement: if a weakly projective pseudo‑telepathy game requires at least local dimension (d) of entanglement for a perfect strategy, then any perfect strategy necessarily certifies the presence of a MES of dimension at least (d). This bridges the structural game property with the operational task of entanglement certification.

To illustrate the power of the framework, the authors consider the Hadamard graph coloring games (G_n). The vertices of the Hadamard graph (H_n) are all (n)-bit strings, and two vertices are adjacent if they differ in exactly half of the bits. In the coloring game, Alice and Bob receive vertices and must output colors that agree when the vertices are identical and differ when they are adjacent. Prior work showed that for (4\mid n) and (n\ge12), (G_n) is a pseudo‑telepathy game.

Theorem 2 shows that any perfect strategy for (G_n) must use entanglement of local dimension (\Omega(n)). The proof translates a perfect quantum strategy for (G_n) into a one‑way communication protocol for a related promise equality problem (\mathrm{EQ}_n). By establishing a lower bound on the one‑way communication complexity of (\mathrm{EQ}_n) (Lemma 3), the authors infer a lower bound on the Schmidt rank of the shared state, yielding the (\Omega(n)) dimension requirement. Consequently, Corollary 3 states that (G_n) provides a family of games that self‑test maximally entangled states of growing dimension, a notable advance over earlier self‑tests that were limited to constant dimensions.

The paper further shows that any non‑local game can be turned into a weakly projective game by adding consistency checks. Definition 2 constructs a new game (\tilde G_B) (or (\tilde G_A)) by augmenting the question set with a flag indicating whether the answer should be checked for consistency with the other player’s answer. This modification makes the game automatically weakly projective for the flagged player, while preserving the existence of a perfect strategy: any perfect strategy for the original game yields a perfect strategy for the augmented game, and vice‑versa. Theorem 3 formalizes this equivalence. As a result, any pseudo‑telepathy game can be transformed into a weak projection game, implying that MES are sufficient for perfect success in all such transformed games.

In summary, the authors provide a comprehensive answer to the central question: maximally entangled states are sufficient for perfect quantum strategies in all weak projection games, and by adding simple consistency checks, any pseudo‑telepathy game can be brought into this class. The work not only clarifies the role of MES in perfect non‑local tasks but also yields practical self‑testing protocols for high‑dimensional entanglement, linking game‑theoretic properties with communication‑complexity lower bounds. Open problems include establishing robustness guarantees for the presented self‑tests and extending the analysis to noisy or imperfect implementations.