A two-player dimension witness based on embezzlement, and an elementary proof of the non-closure of the set of quantum correlations

We describe a two-player non-local game, with a fixed small number of questions and answers, such that an $\epsilon$-close to optimal strategy requires an entangled state of dimension $2^{\Omega(\epsilon^{-1/8})}$. Our non-local game is inspired by the three-player non-local game of Ji, Leung and Vidick [arXiv:1802.04926]. It reduces the number of players from three to two, as well as the question and answer set sizes. Moreover, it provides an (arguably) elementary proof of the non-closure of the set of quantum correlations, based on embezzlement and self-testing. In contrast, previous proofs involved representation theoretic machinery for finitely-presented groups and $C^*$-algebras.


Introduction
A non-local game consists of a one-round interaction between a trusted referee (who asks questions) and two or more spatially isolated players (who provide answers). The players are cooperating to win the game, but spatial isolation prevents them from communicating. Bell's theorem [Bel64], a landmark result in physics, asserts that there exist games for which players who share entanglement can outperform players who do not, the most famous example being the CHSH game [CHSH69]. The most immediate application of non-local games is to "test quantumness": a referee who observes a winning probability in a non-local game which exceeds what is attainable classically can have high confidence that the players (or devices) she is interacting with were sharing entanglement. A more refined analysis of non-local games allows the referee to obtain more precise characterizations of the devices involved. For example, in some cases, the referee might be able to infer that the devices must share high-dimensional entanglement [BPA + 08]. In special cases, the referee might even be able to completely characterize the quantum state inside the devices and the measurements that they are performing (up to local isometries and some small error) [MY04,MYS12,CGS17]. In light of the above, non-local games have been well-studied both in the context of quantum cryptography and in the context of the complexity of interactive proof systems.
In this work, we focus on the study of non-local games as witnesses of high-dimensional entanglement. Before proceeding further, we clarify that when we use the term non-local game we do not restrict ourselves to games with a binary outcome ("win" or "lose"), but rather we consider games specified by an arbitrary function V taking values in R which determines the players' score as a function of questions and answers.
Certifying high-dimensional entanglement -previous work and state of the art Non-local games with the property that near-optimal strategies require high-dimensional entanglement are referred to as dimension witnesses. The study of games (or correlations) with such a property was initiated by Brunner et al. [BPA + 08], who coined the term.
One of the first examples of a game with such a property was proposed by Leung, Toner and Watrous [LTW13], and is intimately connected to our result. The game that they introduced is not a non-local game in the usual sense, since it involves quantum questions and answers. However, it has the property that in order to succeed with high probability, the players have to perform a coherent state exchange which requires them to share an embezzling state of high dimension. More precisely, the game forces the two players to coherently transform a product state of two qubits into an EPR pair, using only local operations. This task is, of course, impossible to perform exactly, but can be performed to arbitrarily high precision if the two players share an auxiliary entangled state of sufficiently high dimension (referred to as an embezzling state).
Subsequently, several examples of dimension witnesses have been proposed over the years consisting of non-local games with classical questions and answers [BBT11,Slo11,BNV13,MV14,CRSV16,CN16,Col17,CS17a,NV17,CS17b]. However, all of these examples involve families of non-local games whose questions and answers increase as the witnessed dimension increases. For some time, it was an open question to determine whether there exists a non-local game, with a finite number of questions and answers, whose optimal value cannot be attained by any finite-dimensional strategy (in the tensor product model), but which can be attained in the limit of finite-dimensional strategies. This question was answered recently by Slofstra in a sequence of two breakthrough works [Slo16,Slo19]. There, Slofstra developed novel techniques for the study of non-local games based on the representation theory of finitely-presented groups, and proved embedding theorems which relate the study of non-local games with certain properties to the study of finitely presented groups and their representation theory. Using these techniques, he constructed the first example of a game, with question and answer sets of a fixed finite size, with the property that a winning probability of 1 can be attained in the limit of finite-dimensional strategies, but cannot be attained exactly by any finite or infinite-dimensional strategy (in the tensor product model). The existence of such a game implies that the set Cqs of quantum correlations attainable in the tensor-product model (with possibly infinite-dimensional entanglement) is strictly contained in the set Cqa of limits of quantum correlations attainable with finite-dimensional entanglement. Hence, the set of quantum correlations in the tensor product model is not closed.
An alternative proof of the latter result was given subsequently by Dykema, Paulsen and Prakash [DPP17], and more recently by Musat and Rørdam [MR18], using techniques based on the representation theory of C * -algebras. The games constructed in [DPP17] and [MR18] have significantly smaller question and answer set sizes, namely 5 and 2.
In contrast, a more recent result by Coladangelo and Stark [CS18] gives an example of a point in the set of quantum correlations on question sets of size 5 and answer sets of size 3 which cannot be attained using finite-dimensional entanglement but can be attained exactly using infinitedimensional entanglement, in the tensor product model. This asserts that the the set Cq of quantum correlations attainable with finite-dimensional entanglement is strictly contained in the set Cqs of correlations attainable with possibly infinite-dimensional entanglement.
All of the above results are not explicit or quantitative about the tradeoff between winning probability (or expected score in the game) and the dimension required to attain it. What we desire from a dimension witness is a quantitative statement of the following form: if the players' score is -close to optimal, then their strategy has dimension at least f ( ), where f ( ) is a function that tends to infinity as tends to zero. In [SV18], Slofstra and Vidick analyze such a tradeoff for the machinery introduced by Slofstra in [Slo16], and they relate such tradeoff to a quantity called the hyperlinear profile of a group. In a subsequent work [Slo18], Slofstra provides a finitelypresented group whose hyperlinear profile is at least subexponential. As a corollary, this yields a two-player non-local game, with question and answer sets of finite size, with the property that a 1 − winning probability requires dimension at least 2 Ω( −c ) to attain for some constant 0 < c < 1. The caveat of such a non-local game is that its description is quite involved and the size of question and answer sets is large. Moreover, it is not clear whether a winning probability of 1 in the game can be attained in the limit of finite-dimensional strategies or not (although it can be attained in the commuting-operator model). A much simpler game with a similar exponential tradeoff between optimality and dimension, and without this caveat, but involving three players, was proposed recently by Ji, Leung and Vidick [JLV18]. Their work constitutes, in some sense, a return to the original ideas of Leung, Toner and Watrous [LTW13], which are cleverly translated to a setting in which all questions and answers are classical. The crux of the non-local game of Ji, Leung and Vidick is that by combining different kinds of tests, the verifier is still able to enforce that any players succeeding with high probability in the game must be performing a coherent state exchange which involves a high-dimensional embezzling state as a resource.
Our result Our main result is the following: Theorem 1. (informal) There exists a two-player non-local game on question sets of size 5 and 6, and answer sets of size 3, with the property that: • (completeness) For any > 0, there exists a strategy of dimension 2 O( −1 ) that is -close to optimal.
Our game is inspired by the three-player game of [JLV18], but it involves only two players and is considerably simpler. For a comparison, the question and answer sets are of size 12 and 8 respectively in [JLV18]. As mentioned earlier, the only other known two-player non-local game with an exponential tradeoff between optimality and dimension is from [Slo18]. Our game is thus the smallest game (in terms of question and answer set size) with an exponential tradeoff. The structure of our game is markedly similar to the game of Leung, Toner and Watrous (which involved quantum questions and answers) [LTW13]. However, by combining different self-tests we are able to circumvent the quantum messages, and obtain a non-local game with classical questions and answers.
Our game provides yet another proof of the non-closure of the set of quantum correlations. However, strikingly, compared to the proofs in [Slo19], [DPP17] and [MR18], our proof is arguably elementary, and does not involve any representation-theoretic machinery. We point out, additionally, that an exponential tradeoff between optimality and dimension does not hold for the game in [DPP17], where a strategy of dimension 1/poly( ) can be -close to optimal (and we suspect that this is also the case for the game in [MR18]).
Outline Section 2 covers some preliminaries: 2.1 introduces some notation, 2.2 introduces nonlocal games, 2.3 gives two important examples of non-local games which are used as sub-tests in our non-local game, and 2.4 briefly introduces embezzlement. Section 3 describes our non-local game. Section 4 covers completeness: we give a family of strategies that approximates arbitrarily well the optimal value in our non-local game. Section 5 covers soundness: we show that any close to optimal strategy requires a lot of entanglement. Section 6 briefly discusses how our non-local game implies the non-closure of the set of quantum correlations.

Non-local games
X and Y are referred to as question sets, and A and B as answer sets. V is referred to as the scoring function.
We denote by D(x, y) the probability of outcome x, y according to distribution D. Note that we use the term non-local game to refer to games in which the scoring function V can take any real value, not just values in {0, 1} like is sometimes the case in the literature. With this nomenclature, non-local games and Bell inequalities are equivalent.
Definition 2 (Quantum strategy for a non-local game). A quantum strategy for a non-local game where HA, HB are Hilbert spaces, {P a x : a ∈ A}x∈X is a set of projective measurements on HA, and {Q b y : b ∈ B}y∈Y on HB. Definition 3 (Value of a quantum strategy in a game). Let G = (X , Y, A, B, D, V ) be a non-local game, and S = (|Ψ ∈ HA ⊗ HB, Note that the value ω(S, G) corresponds to the expected score of strategy S in game G, assuming that questions are distributed according to D, and that the score is determined by the function V .
Definition 4 (Quantum value of a game). The quantum value ω * (G) of a game G = (X , Y, A, B, D, V ) is defined as follows: where the supremum is taken over all quantum strategies for G.
Since the closure of the set of finite-dimensional quantum correlations contains the set of infinite-dimensional quantum correlations [SW08], it does not matter whether the supremum in the definition of ω * is taken over finite or infinite-dimensional strategies (i.e. whether HA and HB are finite or infinite-dimensional).
Finally, we introduce some terminology which we will primarily employ in section 6.
Definition 5 (Correlation). Given sets X ,Y,A,B, a (bipartite) correlation is a collection {p(a, b|x, y) : a ∈ A, b ∈ B} (x,y)∈X ×Y , where each p(·, ·|x, y) is a probability distribution over A × B.
Definition 6 (Correlation induced by a quantum strategy). Given a quantum strategy S = (|Ψ ∈ HA ⊗ HB, {P a x : a ∈ A}x∈X , Definition 7 (Value of a correlation in a game). Let G = (X , Y, A, B, D, V ) be a non-local game, Clearly, if p is the correlation induced by a quantum strategy S for game G, then ω(p, G) = ω(S, G).
We note that quantum strategies can also be considered separately from non-local games (i.e. without specifying any distribution D or scoring function V ). We then denote by Cq (resp. Cqs) the set of correlations induced by a finite-dimensional (resp. possibly infinite-dimensional) quantum strategy. We denote by Cqa the closure of Cq, which by [SW08] is known to also be the closure of Cqs.

Useful examples of non-local games
In this section, we describe two families of non-local games which we will employ as sub-tests in our non-local game.
Tilted CHSH We introduce the tilted CHSH inequality [AMP12], which is a building block for the non-local game in this work. First, we recall the CHSH inequality. It states that for binary observables A0, A1 on Hilbert space HA and binary observables B0, B1 on Hilbert space HB together with a product state |φ = |φA ⊗ |φB , we have where the maximum is achieved (for example setting all observables to identity). However, if instead of the product state |φ we allow an entangled state |ψ , then the right-hand side of the inequality increases to 2 √ 2. This maximum requires a maximally entangled pair of qubits to achieve. In this work, we would like to use an inequality that requires a non-maximally entangled state to achieve the maximum; this is the tilted CHSH inequality. Given a real parameter β ∈ [0, 2), for a product state |φ = |φA ⊗ |φB , (2) For entangled |ψ , we have instead that The maximum in the tilted CHSH inequality is attained by the following strategy: Definition 8 (Ideal strategy for tilted CHSH). Given parameter β ∈ [0, 2), let θ ∈ (0, π 4 ] be such that sin 2θ = 4−β 2 4+β 2 , µ = arctan sin 2θ, and α = tan θ. Define the α-tilted Pauli operators as σ z α := cos µσ z + sin µσ x , and σ x α := cos µσ z − sin µσ x . (4) The ideal strategy for tilted CHSH with parameter β (i.e. achieving maximal violation of (3)) consists of the joint state |Ψ = cos θ(|00 + α |11 ) and observables A0, A1 and B0, B1 with A0 = σ z , A1 = σ x , B0 = σ z α and B1 = σ x α . β and α are related by an invertible function, and α is typically the parameter of interest, so we choose to denote by tCHSH(α) the tilted CHSH game whose ideal state is |Ψ = cos θ(|00 +α |11 ).
Proof. Notice that for any strategy S, the value ω(S, GtCHSH(α)) takes precisely the form of the LHS of (3) (upon associating, for each observable in (3), the projection onto the +1-eigenspace with answer 0 and the projection onto the −1-eigenspace with answer 1, and up to a factor of 1 4 from sampling the questions uniformly).
In other words, the LHS of the tilted CHSH inequality and the value of the tilted CHSH game are equivalent reformulations of one another. The following theorem asserts a robust self-testing result for tilted CHSH, i.e. that any strategy that attains a value close to the quantum value of the game, must be close to the ideal strategy of Definition 8 (in the following statement we only write down the conditions that we make use of later).
For clarity of notation and exposition in later sections, it is convenient for us to define the game G ∼tCHSH(α) , for α ∈ (0, 1]. This is an equivalent version of G tCHSH(α) with the only difference that the scoring function is V ∼tCHSH(α) := (−1) a⊕b−xy − δ {x=y=0} · β · (−1) a (notice the minus sign). It is easy to see that this game is equivalent to the original tilted CHSH up to a flip of the answer labels (so in particular ω * tCHSH(α) = ω * ∼tCHSH(α) ). The corresponding version of Theorem 2 for G ∼tCHSH(α) is as follows: Theorem 3. Let α ∈ (0, 1]. Maximal value in G ∼tCHSH(α) self-tests the ideal strategy of Definition 8 with robustness O( √ ), i.e. for any strategy S = (|Ψ ∈ HA ⊗ HB, {P a x }, {Q b y }) with value ω(S, G ∼tCHSH(α) ) > ω * ∼tCHSH(α) − there exists a local unitary U and an auxiliary state |aux such that: Generalization of CHSH self-testing states of local dimension d There is a family of non-local games, parametrized by d ≥ 2 ∈ N, which generalizes the CHSH game [Col18]. The games in this family have the property that, for the game with parameter d, maximal score in the game self-tests the maximally entangled state of local dimension d. Each of the games in this family is a 2-player game in which question sets are of size 2 + 1 d>2 and 2 + 2 · 1 d>2 , and answer sets are of size d. When d = 2, the game coincides with the usual CHSH game. We denote by G d-CHSH the game in the family with parameter d. We do not describe this family of games in full detail here (for details we refer to [Col18]). We will just recall the self-testing properties of the game that we need in the following theorem, and describe the ideal strategy for the case of d = 3 (we will use G3-CHSH later as a sub-test in our non-local game).
Theorem 4 ( [Col18]). There exists a family of non-local games {G d-CHSH } d≥2∈N with the following properties: • Question sets are: Answer sets are A = B = {0, 1, .., d−1}. For all d, the distribution over questions is uniform. Denote by V d-CHSH the scoring function for G d-CHSH .
• (Self-testing) Let ω * d-CHSH be the value of the game with parameter d. There exists a constant C > 0 such that the following holds. Any strategy S = (|Ψ , {P a x }, {Q b y }) with value ω(S, G d-CHSH ) ≥ ω * d-CHSH − , for some 0 < < C d 3 , is such that there exists a local unitary U and an auxiliary state |aux such that: Again, the last condition means that the first player's measurement on question "0" is equivalent (up to a change of basis) to a computational basis measurement.
Next, we describe the ideal strategy for G3-CHSH. First, we fix some notation. We define an isometry V : (C 2 )A → (C 3 )Ã as follows: For an operator O on C 2 , we write V (O) to refer to the pushforward V OV † of O along V . For example, V (σ z ) = |1 1| − |2 2|. If O has +1, 0, −1 eigenvalues, we write O + for the projection onto the +1 eigenspace and O − for the projection onto the −1 eigenspace. One can check that with this notation O = O + − O − . We use the notation Ai to denote the direct sum of observables Ai. If HA ≈ C 3 , we still write σ z A to mean σ z A = |0 0| A − |1 1| A . On the other hand, in accordance with the notation above, we write V (σ z )A to mean V (σ z )A = |1 1| − |2 2|. We adopt an analogous notation for all other Paulis and tilted Paulis, and projections onto their eigenspaces. (We will make use of the α-tilted Paulis σ z α , σ x α from Definition 8). Definition 10 (Ideal strategy for G3-CHSH [Col18]). The ideal strategy for G3-CHSH is , and the ideal measurements are described in Tables 1 and 2.   Table 1: Alice's ideal measurements for G 3-CHSH . The entry in cell x, a is the projector P a x .
x a 0 1 2 Table 2: Bob's ideal measurements for G 3-CHSH . The entry in cell y, b is the projector P b y .
We emphasize, as it will be important later, that both the ideal strategies for G tCHSH(α) and G d-CHSH include a computational basis measurement for the first player on question "0".

Embezzlement
The phenomenon of embezzlement was first discovered by van Dam and Hayden [vDH03]. A family of embezzling states can be used to coherently transform a product state into an EPR pair (or viceversa). The fidelity of this transformation increases with the dimension of the embezzling state.
, where N d is a normalizing constant.
Then, the family of states {|Γ d } is an embezzling family. The unitaries W AA and W BB are the "left-shift" unitaries, which act on C 2 A ⊗ (C 2 ) ⊗d A and C 2 B ⊗ (C 2 ) ⊗d B respectively, by shifting by one to the left each of the d + 1 qubit registers. It is easy to check that the family of states {|Γ d } d∈N satisfies Definition 11.

Our non-local game
The following is our non-local game. We describe it informally first, and then we give a precise description in Fig. 1. We refer to Alice and Bob as the two players in our non-local game.
The non-local game consists of three tests, run with equal probability.
(a) In the first test, the verifier sends both players questions from the game G3-CHSH, and they obtain a score according to its scoring function. The intuition behind this game is the following. If Alice and Bob's strategy attains an -close to optimal expected score overall (where optimally here means playing perfectly in all three tests), then it must attain a 3 -close to optimal expected score in each of the three tests. By the self-testing result of Theorem 4, in order to play 3 -close to optimally in (a), the players need to be sharing a state close to a maximally entangled state of qutrits, up to a local isometry, and moreover one of Alice's measurements is a "computational basis" measurement. By Theorem 3, in order to play 3 -close to optimally in (b), Alice and Bob must be measuring a state close to a tilted EPR pair with ratio 1 √ 2 , up to a local isometry. Moreover one of Bob's measurements must be a "computational basis" measurement. Crucially, Alice cannot distinguish her question in (c) from a "computational basis" question in (a), while Bob cannot distinguish his question in (c) from a "computational basis" question in (b). In order to play close to optimally in (c), Alice and Bob's computational basis measurements need to satisfy a consistency condition. It is this consistency condition that forces the two players to "agree" on a computational basis element |00 ∈ C 3 A ⊗ C 3 B , and to perform a coherent state exchange such that: with |00 AB → |00 AB and 1 √ 2 (|11 + |22 )AB → |11 AB . The LHS of (7) is the state that the players need in order to play part (a) perfectly, while the RHS is the state that they need to play part (b) perfectly. Part (c) ensures that players have to "agree" on the term |00 , and this enforces that they must perform coherently the exchange in (7) to high accuracy if they are to perform well in all three parts.
All three of the above are clearly impossible.
On the other hand, we will construct in the next section a sequence of strategies whose value in G gets arbitrarily close to 1 3 (ω * 3-CHSH + ω * ∼tCHSH( 1 √ 2 ) + 1). This completes the proof.

Completeness
In this section, we describe a family of strategies whose value in our non-local game G emb gets arbitrarily close to 1 3 (ω * 3-CHSH + ω * ∼tCHSH( 1 √ 2 ) + 1) (which also completes the proof of Proposition 2). A strategy in the family is parametrized by d ∈ N. The provers start with the state where |Γ d is an embezzling state. We give first an informal description of the ideal measurements, and we follow this by a formal description.
• Upon receiving a question with prefix "3-CHSH", Alice and Bob perform the corresponding ideal measurement for 3-CHSH. In particular on question ("3-CHSH", 0), Alice measures her half of the state in (8) in the computational basis.
• Upon receiving a question with prefix "∼tCHSH( 1 √ 2 )", Alice and Bob first apply embezzling unitaries WÃ A and WB B respectively, such that (approximately) 1 √ 2 (|11 + |22 ) → |11 and |00 → |00 . So the resulting state is 2 3 They then perform the corresponding ideal measurements for ∼tCHSH( 1 √ 2 ) on registers A,B (where Alice takes the role of the second player, and Bob takes the role of the first player). In particular, on question ("∼tCHSH( 1 √ 2 )", 0), Bob measures his half of the state in (9) in the computational basis.
A key observation is that when Alice and Bob are asked questions ("3-CHSH", 0) and (∼ "tCHSH( 1 √ 2 )", 0) respectively, then it is straightforward to see that, if they follow the above strategy, they reply with answers (a, b) which attain a score of 1 in part (c) of Fig. 1, i.e.  (a, b) Next, we define the players' ideal measurements precisely. Recall the isometry V : C 2 → C 3 defined in subsection 2.3 as follows: (10) Recall also the notation introduced in subsection 2.3 along with V . In particular, we write V (O) to refer to the pushforward V OV † of O along V . For O an operator with +1, 0, −1 eigenvalues, we write O + for the projection onto the +1 eigenspace and O − for the projection onto the −1 eigenspace. If HA ≈ C 3 , we still write σ z A to mean σ z On the other hand, in accordance with the notation above, we write Let {|Γ d ABA B } be the embezzling family from Example 1, and W AA : and defineWB B analogously.
The following is the family of ideal strategies for G emb achieving a value arbitrarily close to 1 3 (ω * 3-CHSH + ω * ∼tCHSH( 1 √ 2 ) + 1). Definition 12 (Ideal strategy for G emb ). The family of ideal strategies is and the ideal measurements are described in Tables 3 and 4. Table 3: Alice's ideal measurements for G emb . The entry in cell x, a is the projector P a x (tensored identities are implied where omitted, and P rest completes the set of orthogonal projections in a row).
x a 0 1 2 Table 4: Bob's ideal measurements for G emb . The entry in cell y, b is the projector P b y (tensored identities are implied where omitted, and P rest completes the set of orthogonal projections in a row).
Proposition 3 (Completeness). Let {S d } d∈N be the family of strategies from Definition 12, and G emb the non-local game from Fig. 1.
Proof. The value of strategy S d in part (a) is exactly ω * 3-CHSH . This is because the starting state is the ideal state for ω * 3-CHSH and measurements are the ideal ones from Definition 10. The value . This is because the joint state resulting from the embezzling transformation has fidelity 1 − O( 1 d ) with the ideal state for ∼tCHSH( 1 √ 2 ) (from Theorem 3), and the measurements for part (b) are also ideal. The value in part (c) is easily seen to be exactly 1.
Together with the upper bound in the proof of Proposition 2, this completes the proof of Proposition 2 (i.e. ω * (G emb ) =
The proof of Theorem 5 can be broken down into two parts: (i) First, we will show that performing well in parts (a), (b) and (c) of the game imposes a certain structure on the strategy of the provers.
(ii) Second, we show that such a structured strategy can be used to play well also in the "coherent state exchange" game of Leung, Toner and Watrous [LTW13]. This reduction allows us to translate the lower bounds on the dimension of an approximately optimal strategy in the "'coherent state exchange" game to lower bounds on the dimension of an approximately optimal strategy for our game.
Proof of Theorem 5. Let |ψ ∈ HA ⊗ HB, {P a x }, {Q b y } be a strategy whose value in G emb isclose to ω * (G emb ) = 1 3 (w * 3CHSH + w * 2CHSH + 1). This implies that, for each part of the game, the strategy's expected score is 3 -close to optimal. From each part we deduce the following: and an auxiliary state |aux ∈ H A ⊗ H B such that and moreover (11) and (12) immediately imply that We claim that the local unitaryŨ can be used to approximately win the "coherent state exchange" game of Leung, Toner and Watrous [LTW13]. More precisely, since Equation (13) is O( 1/8 )approximate (with respect to Euclidean norm), we claim that one can construct a strategy which employsŨ , and in which the provers' initial state is |aux , which wins the game of [LTW13] with probability 1−O( 1/4 ). Assuming this claim is true, the rest of the proof is straightforward: it was shown in [LTW13] that the winning probability of any strategy in the "coherent state exchange game" is upper bounded by 1 − 1 32 log 2 (3d) , where d is the dimension of the states used; this implies that it must be 1 32 log 2 (3d) To conclude the proof of Theorem 5, we prove the above claim. The "coherent state exchange" game of [LTW13] between a quantum referee and two noncommunicating provers, proceeds as follows: • The referee initializes a qubit register R and qutrit registers S and T in the state where |φ + = 1 √ 2 (|00 + |11 ). The referee sends registers S and T to Alice and Bob respectively.
• The referee receives single-qubit registers A and B from Alice and Bob respectively. The triple (R, A, B) is measured with projective measurement {Π0, Π1}, where Π0 = I − |γ γ| and Π1 = |γ γ|, and |γ = 1 √ 2 (|000 + |111 ). Consider the following strategy of the provers for this game. They start by sharing the state |aux ∈ H A ⊗ H B . Upon receiving the qutrit registers S and T of the state (14), they applyŨ to registers (C 3 ) S ⊗ (C 3 ) T ⊗ H A ⊗ H B (up to relabelling registers A1 and B1 as S and T), obtaining a state in (C 2 )A 2 ⊗ (C 2 )B 2 ⊗ H A ⊗ H B . Equations (12) and (13) imply that the resulting state on registers R, A2, B2, A , B is O( 1/8 )-close to 1 √ 2 (|000 + |111 ) ⊗ |aux . And hence the state on R, A2, B2 is O( 1/8 )-close to the desired state (in Euclidean norm). Qubit registers A2 and B2 are then sent back to the referee as A and B. Converting the O( 1/8 )-closeness to a probability of winning in the game, gives a lower bound of 1 − O( 1/4 ), and thus concludes the proof.
6 Non-closure of the set of quantum correlations A corollary of Proposition 3 and Theorem 5 (completeness and soundness for our game) is that the set Cqs of quantum correlations induced by, possibly infinite-dimensional, quantum strategies (in the tensor product model) is not closed, i.e. Cqs = Cqa, where the latter is the closure. For precise definitions of these sets see [CS18]. We use superscripts to denote question and answer set sizes. For instance C m,n,r,s qs is on question sets of size m, n and answer sets of size r, s.
Proof. In the proof of Theorem 5, we argued that any strategy with value ω * (G emb ) − in our game G emb can be used to construct a strategy that embezzles an EPR pair into a product state, up to O( 1/8 ) error in Euclidean norm. This implies that no strategy has value exactly ω * (G emb ). Suppose otherwise for a contradiction. Then, by the reduction in the proof of Theorem 5, we can construct a strategy that wins the game of [LTW13] with probability 1. From [LTW13], this is known to imply existence of a strategy that embezzles perfectly (the argument that shows this implication in [LTW13] is phrased for finite-dimensional strategies, but it holds also for infinitedimensional ones). A perfect embezzling strategy consists of a state |Ψ ∈ H A ⊗ H B and a local unitary U = U AA ⊗ U BB such that U |φ + AB ⊗ |Ψ A B = |00 AB ⊗ |Ψ A B . Since Schmidt coefficients are preserved under local unitaries, it is clear that, whatever the Schmidt coefficients of |Ψ are, the Schmidt coefficients of the LHS and RHS are different. This gives a contradiction.
On the other hand, Proposition 3 gives a sequence of strategies whose value tends to ω * (G emb ). If one considers the sequence of correlations induced by such strategies, it is clear that such a sequence has a limit, and that the limiting correlation has value ω * (G emb ). Such a limiting correlation is thus in C 5,6,3,3 qa but not in C 5,6,3,3 qs . We emphasize that strictly stronger separations (for question sets of size 5 and answer sets of size 2) are known [DPP17,MR18]. The latter appeared after the original breakthrough proof of Slofstra, for much larger question and answer sets [Slo19]. What stands out about our proof is that, unlike all previous proofs, it does not involve any representation theoretic machinery.