We give an explicit family of XOR games with O(n)-bit questions requiring 2n ebits to play near-optimally. More generally, we introduce a new technique for proving lower bounds on the amount of entanglement required by an XOR game: we show that near-optimal strategies for an XOR game G correspond to approximate representations of a certain C*-algebra associated to G. Our results extend an earlier theorem of Tsirelson characterising the set of quantum strategies which implement extremal quantum correlations.
We study optimal and nearly-optimal quantum strategies for non-local XOR games. First, we prove the following general result: for every non-local XOR game, there exists a set of relations with the properties: (1) a quantum strategy is optimal for the game if and only if it satisfies the relations, and (2) a quantum strategy is nearly optimal for the game if and only if it approximately satisfies the relations. Next, we focus attention on a specific infinite family of XOR games: the CHSH(n) games. This family generalizes the well-known CHSH game. We describe the general form of CHSH(n) optimal strategies. Then, we adapt the concept of intertwining operator from representation theory and use that to characterize nearly-optimal CHSH(n) strategies.
Entangled games are a quantum analog of constraint satisfaction problems and have had important applications to quantum complexity theory, quantum cryptography, and the foundations of quantum mechanics. Given a game, the basic computational problem is to compute its entangled value: the supremum success probability attainable by a quantum strategy. We study the complexity of computing the (commuting-operator) entangled value omega^* of entangled XOR games with any number of players. Based on a duality theory for systems of operator equations, we introduce necessary and sufficient criteria for an XOR game to have omega^* = 1, and use these criteria to derive the following results: 1) An algorithm for symmetric games that decides in polynomial time whether omega^* = 1 or omega^* < 1, a task that was not previously known to be decidable, together with a simple tensor-product strategy that achieves value 1 in the former case. The only previous candidate algorithm for this problem was the Navascués-Pironio-Acín (also known as noncommutative Sum of Squares or ncSoS) hierarchy, but no convergence bounds were known. 2) A family of games with three players and with omega^* < 1, where it takes doubly exponential time for the ncSoS algorithm to witness this. By contrast, our algorithm runs in polynomial time. 3) Existence of an unsatisfiable phase for random (non-symmetric) XOR games. We show that there exists a constant C_k^{unsat} depending only on the number k of players, such that a random k-XOR game over an alphabet of size n has omega^* < 1 with high probability when the number of clauses is above C_k^{unsat} n. 4) A lower bound of Omega(n log(n)/log log(n)) on the number of levels in the ncSoS hierarchy required to detect unsatisfiability for most random 3-XOR games. This is in contrast with the classical case where the (3n)^{th} level of the sum-of-squares hierarchy is equivalent to brute-force enumeration of all possible solutions.
We study the complexity of computing the commuting-operator value
ω∗ of entangled XOR games with any number of players. We introduce necessary and sufficient criteria for an XOR game to have
ω∗=1, and use these criteria to derive the following results: 1. An algorithm for symmetric games that decides in polynomial time whether
ω∗=1 or
ω∗<1, a task that was not previously known to be decidable, together with a simple tensor-product strategy that achieves value 1 in the former case. The only previous candidate algorithm for this problem was the Navascués-Pironio-Acín (also known as noncommutative Sum of Squares or ncSoS) hierarchy, but no convergence bounds were known. 2. A family of games with three players and with
ω∗<1, where it takes doubly exponential time for the ncSoS algorithm to witness this (in contrast with our algorithm which runs in polynomial time). 3. A family of games achieving a bias difference
2(ω∗−ω) arbitrarily close to the maximum possible value of
1 (and as a consequence, achieving an unbounded bias ratio), answering an open question of Briët and Vidick. 4. Existence of an unsatisfiable phase for random (non-symmetric) XOR games: that is, we show that there exists a constant
Ckunsat depending only on the number
k of players, such that a random
k-XOR game over an alphabet of size
n has
ω∗<1 with high probability when the number of clauses is above
Ckunsatn. 5. A lower bound of
Ω(nlog(n)/loglog(n)) on the number of levels in the ncSoS hierarchy required to detect unsatisfiability for most random 3-XOR games. This is in contrast with the classical case where the
n-th level of the sum-of-squares hierarchy is equivalent to brute-force enumeration of all possible solutions.