A Characterization of Perfect Strategies for Mirror Games

We associate mirror games with the universal game algebra and use the *-representation to describe quantum commuting operator strategies. We provide an algebraic characterization of whether or not a mirror game has perfect commuting operator strategies. This new characterization uses a smaller algebra introduced by Paulsen and others for synchronous games and the noncommutative Nullstellensatz developed by Cimpric, Helton and collaborators. An algorithm based on noncommutative Gröbner basis computation and semidefinite programming is given for certifying that a given mirror game has no perfect commuting operator strategies.


INTRODUCTION
Quantum nonlocal games have been an active area of research for mathematicians, physicists, and computer scientists in past decades.The violation of Bell inequality has verified the non-locality of quantum mechanics [1], which can be explained in the framework of nonlocal games [8,34].A nonlocal game has two or multiple players and a verifier.The verifier sends a question to each player separately, and each player sends an answer back to the verifier without communicating with the others.The verifier determines whether the players win for the given questions and answers.We have a classical strategy if the players can only share classical information.We have a quantum strategy if we allow the players to share quantum information.Bell inequality violations have been proved in the CHSH game [7], where the winning probability using classical strategies is at most 3/4, while a quantum strategy using an entangled state shared by two players can achieve a success probability cos 2 ( /8) ≈ 0.85.Noncommutative Positivstellensätze have been used to study nonlocal games in [11,32].
A synchronous game is a nonlocal game with two players called Alice and Bob, where Alice and Bob are sent the same question and win if and only if they send the same response.Paulsen and his collaborators found a simpler formulation using a smaller algebra and hard zeroes to study synchronous games in [19,35].It has been shown that the success probability of a synchronous game is given by the trace of a bilinear function on a smaller algebra, see Theorem 5.5 in [35], and Theorem 3.2 in [19].In [2,19,42], they give algebraic characterizations of perfect quantum commuting operator strategies for a general game using noncommutative Nullstellensätze [4][5][6] and Positivstellensätze [3,17,18,29].Theorem 8.3 and 8.7 in [2] provide a simplified version of the Nullstellensatz theorem for synchronous games.
In [27], Lupini, etc. introduce a new class of nonlocal games called imitation games, in which another player's answer completely determines each player's answer.Any synchronous game is an imitation game as the players send the same answers for the same questions.Some imitation games are not synchronous, such as mirror games, unique games [37], and variable assignment games [27].Lupini, etc., associates a C*-algebra with any imitation game and characterizes perfect quantum commuting strategies in terms of the properties of this C*-algebra.
As an interesting subclass of imitation games, mirror games include unique games and synchronous games.Theorem 5.5 in [35] for synchronous games has been generalized to Theorem 6.1 in [27] for mirror games, and a representation of perfect quantum commuting strategies for mirror games in terms of traces is also given in the paper.It is natural to ask whether one can obtain similar results as Theorem 8.3 and 8.7 in [2] for mirror games.We answer the question in Theorem 3.1: we provide an algebraic characterization of whether or not a mirror game has perfect commuting operator strategies based on a noncommutative Nullstellensatz and sums of squares.This new characterization uses a smaller algebra introduced by Paulsen and others for synchronous games and the noncommutative Nullstellensatz developed by Cimpric, Helton, and collaborators [4][5][6].An example is given to demonstrate how to use noncommutative Gröbner basis algorithm [30] and semidefinite programming [41] to verify that a given mirror game has no perfect commuting operator strategies.It would be interesting to see how to extend these results to imitation games.
The paper is organized as follows.Section 2 introduces some preliminary results and definitions of nonlocal games.Some background material on classical strategies and quantum strategies of nonlocal games are included.We also introduce the universal game algebra and its *-representation.Section 3 contains our main result on characterizing whether or not a mirror game has perfect commuting operator strategies based on a noncommutative Nullstellensatz and sums of squares.Finally, Section 4 shows how to use noncommutative Gröbner basis and semidefinite programming to verify that a given mirror game has no perfect commuting operator strategies.A running example is given to demonstrate the computations.

PRELIMINARIES
A nonlocal game G involves a verifier and two players, Alice and Bob.For fixed non-empty finite sets , and , , there exists a distribution on × .After choosing a pair ( , ) ∈ × randomly according to ( , ), the verifier sends elements to Alice and to Bob as questions.Alice and Bob send the verifier corresponding answers ∈ and ∈ .After receiving an answer from each player, the verifier evaluates the scoring function If ( , , , ) = 1, we say Alice and Bob win; otherwise, they lose the game.Alice and Bob know the sets , , , and the scoring function , but they can't communicate during the game.Alice and Bob can make some arrangements before the game starts.
A deterministic strategy for the players consists of two functions: : ) and Alice sends ( ) to the verifier if she receives , and Bob sends ( ) to the verifier if he receives .Given a deterministic strategy, the players win the game G with an expectation , ( , ) ( , , ( ), ( )). ( We can also give a probabilistic strategy for G as follows: for each pair ( , ) ∈ × , let Alice and Bob have mutually independent distributions , , , for ∈ , ∈ .When the players receive the questions ( , ), Alice sends the answer to the verifier with probability , and Bob sends the answer to the verifier with probability , .The winning expectation is , , , ( , ) , , ( , , , ). (2.4) All deterministic strategies and probabilistic strategies are collectively referred to as classical strategies.We record the set of all classical strategies as , which is a closed set.Notice that any probabilistic strategy can be expressed as a convex combination of deterministic strategies so that the maximal winning expectation of a nonlocal game G with classical strategies is always obtained by some deterministic strategy.The classical value of G is defined as the maximal winning expectation ( , ) ( , , ( ), ( )). (2.5) We use the Dirac notation in quantum information to represent the unit vector (a state) in Hilbert space.If Alice and Bob are allowed to share a quantum entangled state | ∈ H ⊗ H , where both H and H are finite-dimensional Hilbert space, and then they can have a quantum strategy described as follows: • If Alice receives , she performs the projection-valued measure (PVM) , on H part of | and sends the measurement result to the verifier.• If Bob receives , he performs the PVM , on H part of | and sends the measurement result to the verifier.
If we replace PVM by POVM (positive operator-valued measure), the results below will also hold [14,36].
We record the set of all finite-dimensional quantum strategies as .If we drop the requirement of finite dimension, i.e., H , H can be infinite-dimensional Hilbert spaces, then we get a set of quantum strategies denoted as .Slofstra [38,39] has proved that neither nor is a closed set.We denote the closure of as .It is evident that Each of the above " ⊆ " is strictly inclusive.The first strict inclusion comes from Bell's inequality, and the last two strict inclusions come from results in [9,12,38,39].The winning expectation for the given quantum strategy is If we take all of the quantum strategies, the supremum of winning expectations is ) which is called the quantum value of G.The quantum value can certainly be attained in , but not necessarily in or .Now we give a quantum commuting operator strategy for G as follows.Let H be a (perhaps infinite-dimensional) Hilbert space, | ∈ H , and for every ( , ) ∈ × , Alice and Bob have PVMs { (1) , ∈ } and { (2) , ∈ }, respectively.Those two sets of PVMs satisfy the following conditions: When Alice receives an input , she performs { (1) , ∈ } on | and sends the result to the verifier; Similarly, when Bob receives an input , he performs { (2) , ∈ } on | and sends the result to the verifier.
We denote the set of all the quantum commuting operator strategies as .We know that is closed [13].Given a quantum commuting operator strategy of G, the winning expectation is (2.9) Then the supremum of winning expectation (note that it can certainly be obtained) is It is easy to see that ⊆ [13], so that we have (G) ≤ (G) ≤ (G).If we restrict the Hilbert space H to be finitedimensional in the commuting operator strategies, then (G) = (G) (see [38,40]).There exist games G for which (G) < (G) in the infinite-dimensional case, see [13,39].The problem of whether = is the famous Tsirelson's problem, and it is true if and only if the Connes' embedding conjecture is true [10].Kirchberg shows that Connes' conjecture has several equivalent reformulations in operator algebras and Banach space theory [22].In [25], Klep and Schweighofer show that Connes' embedding conjecture on von Neumann algebras is equivalent to the tracial version of the Positivstellensatz.In 2020, Ji and his collaborators proved * = , which implies that Connes' embedding conjecture is false [20].But we still don't know an explicit counterexample.See [13,16,33] for recent results on the Connes' embedding problem.This is the main motivation for us to study quantum nonlocal games.
We say a strategy is perfect if and only if the players can certainly win the game with this strategy.A natural problem is to ask whether there exists a perfect strategy in (or , , , ) for a given game G.
In [27], the authors introduce a new class of nonlocal games called imitation games, and they provide an algebraic characterization of perfect commuting operator strategies for these games.In this paper, we mainly discuss the mirror game, which is a special subclass of imitation games.

D 2.1 (
).Let G be a nonlocal game with a question set × , an answer set × and a scoring function : × × × → {0, 1}.The distribution on × is the uniform distribution.We say is a mirror game if there exist functions : → and : → such that: and the scoring function be given as follows: We use the universal game algebra and representation defined in [2] to describe the relations between the PVMs in the commuting operator strategy below.and C e be the noncommutative free algebra generated by the tuple e.Let I be the two-sided ideal generated by the following polynomials: (2.14) Then we define U = C e /I and equip U with the involution induced by ( ( 1) where the " * " of a complex number is its conjugate.We call U the universal game algebra of G.
For the universal game algebra U, we can use *-representation to describe a commuting operator strategy.A *-representation of U is a unital *-homomorphism where B (H ) denotes the set of bounded linear operators on a Hilbert space H and satisfies ( * ) = ( ) * , ∀ ∈ .It is obvious that any commutative PVMs { (1) , ∈ } and { (2) , ∈ } can be obtained by the unital *-homomorphism and given an arbitrary unital *-homomorphism, the image of U's generators is commutative PVMs.Therefore, each commuting operator strategy corresponds to a pair ( , | ), where : U → B (H ) is a *-representation and | ∈ H is a state (a unit vector).We can use the language of representation to rewrite (G) as follows: ( where and the supremum is taken over all *-representations of U into bounded operators on a Hilbert space H and state | ∈ H . Since we assume that is a uniform distribution, Φ G can be simplified to is a determining set.We call it the invalid determining set. C 2.2.The left ideal L(N ) generated by N is also a determining set.
For a mirror game G, suppose its universal game algebra is U, and we define:  (2.30)

P
. By the definition of regularity and the universal game algebra.
In the following sections, we'll only consider regular mirror games.

MAIN RESULT
Given a universal game algebra U of a nonlocal game G, a general noncommutative Nullstellensatz developed by Cimpric, Helton, and their collaborators [5,6] which is also equivalent to where L(N ) is the left ideal generated by the invalid determining set N .For synchronous games, the authors use a smaller algebra U (1) which is the subalgebra of U generated by (1) , and prove that a synchronous game has a perfect commuting operator strategy if and only if there exists a *-representation ′ : U (1) → B (H ) and a tracial state | ∈ H satisfying ′ (J (synchB (1)))| = {0}, (3.4) where J (synchB (1)) is a two-sided ideal in U (1), see Theorem 8.3 and 8.7 in [2].In Theorem 3.1, we generalize Theorem 8.3 and 8.7 in [2] for mirror games and provide a characterization of whether or not a mirror game has perfect commuting operator strategies using smaller algebras U (1) and U (2), where U (1) is the subalgebra of U generated by (1) only, and U (2) is the subalgebra of U generated by (2) only.
Let e = ( (1) ) ∈ , ∈ ∪ ( (2) ) ∈ , ∈(2.13) mir2) be the two-sided ideal of U (2) generated by continued).Let's continue the computation in Example 2.1.The two-sided ideal J (mir1) is generated by the following elements: Given a Hilbert space H and an operator algebra A acting on H , a state | ∈ H is called a tracial state if the linear mapping it induces is tracial, i.e.Let G be a nonlocal game; its universal game algebra is U.A set F ⊆ U is denoted as a determining set of G if it satisfies that a pair ( , | ) is a perfect commuting operator strategy if and only if (F )| = {0}. )