An inherently infinite-dimensional quantum correlation

Bell’s theorem, a landmark result in the foundations of physics, establishes that quantum mechanics is a non-local theory. It asserts, in particular, that two spatially separated, but entangled, quantum systems can be correlated in a way that cannot be mimicked by classical systems. A direct operational consequence of Bell’s theorem is the existence of statistical tests which can detect the presence of entanglement. Remarkably, certain correlations not only witness entanglement, but they give quantitative bounds on the minimum dimension of quantum systems attaining them. In this work, we show that there exists a correlation which is not attainable by quantum systems of any arbitrary finite dimension, but is attained exclusively by infinite-dimensional quantum systems (such as infinite-level systems arising from quantum harmonic oscillators). This answers the long-standing open question about the existence of a finite correlation witnessing infinite entanglement.


Supplementary Note 1 -Notation and preliminaries
For a positive integer n, we denote by [n] the set {1, .., n}. δ ij is the Kronecker delta. For a Hilbert space H, L(H) is the space of linear operators on H. Define the Pauli matrices σ z = 1 0 0 −1 , and σ x = 0 1 1 0 . (1) For an operator T ∈ L(H) and a subspace H ⊆ H invariant under T , we denote by T | H ∈ L(H ) the restriction of T to H . Let C N denote the Hilbert space of square-summable sequences, sometimes called 2 (C). We endow it with a standard basis {|i : i ∈ N}. Formally, C N = i a i |i : i a 2 i < ∞ . Correlations and quantum strategies were introduced in the main text. Here, we clarify some additional terminology.
We will sometimes refer to a quantum strategy as a triple |ψ , {Π a Ax } a , {Π b By } b . If we wish to emphasize the underlying Hilbert space, we write |ψ ∈ H A ⊗ H B , {Π a Ax } a , {Π b By } b . Notice that we have chosen our state to be pure and our measurements to be projective. This choice is without loss of generality. The most general measurements are modeled by POVMs, but Naimark's dilation theorem implies that any correlation induced using POVMs can also be induced using projective measurements (possibly of larger dimension). Likewise, any correlation induced by a mixed state can also be induced by using a purification of that state. We sometimes describe a quantum strategy by specifying an observable for each question. The observables in turn specify the projectors through their eigenspaces.
A correlation is said to be quantum if there exists a quantum strategy that induces it. We refine this, and we say that a quantum correlation is finite-dimensional (infinite-dimensional ) if it is induced by a quantum strategy on a finite-dimensional (infinite-dimensional) Hilbert space. We denote by C m,n,r,s q and C m,n,r,s qs respectively the sets of finite and infinite-dimensional quantum correlations on question sets of sizes m, n and answer sets of sizes r, s.
As we will be borrowing inspiration and techniques from the field of device-independent self-testing, we provide a formal definition: Definition 1 (Self-testing). We say that a correlation {p * (a, b|x, y) : a ∈ A, b ∈ B} x∈X ,y∈Y self-tests By } b that induces p * , there exists a local isometry Φ = Φ A ⊗ Φ B and an auxiliary state |aux such that, for all x ∈ X , y ∈ Y, a ∈ A, b ∈ B, Sometimes, we refer to self-testing of the state when we are only concerned with the guarantee of Supplementary Equation (2), and not Supplementary Equation (3).
The tilted CHSH inequality was introduced in the main text. Here, we report the ideal strategy that attains maximal violation of the inequality, as we will employ these ideal measurements as part of of the ideal strategy defining our separating correlation.
The ideal strategy for tilted CHSH with parameter β (i.e. achieving the RHS of Equation (7) from the main text) consists of the joint state |Ψ = cos θ(|00 + α |11 ) and observables A 0 , A 1 and B 0 , B 1 For each observable, we associate the projection onto the +1-eigenspace with answer 0 and the projection onto the −1-eigenspace with answer 1.
We will make use of the following lemma.
). The tilted CHSH correlation for ratio α self-tests the strategy of Definition 2.

Supplementary Note 2 -Direct sums of correlations
In this section, we introduce the notion of a direct sum of correlations. We will later use this to build our desired correlation out of tilted CHSH building blocks. Lemma 5 will allow us to characterize the strategies for the desired correlation from self-testing results about its direct summands. In particular, these strategies also decompose, in a sense made precise below, as a direct sum of strategies corresponding to the direct summands. The proof is somewhat technical, and the ideas in the proof are not necessary to understand the rest of the proof. Some of the ideas in the proof have appeared adhoc in previous works on constructing quantum correlations block-by-block [3], [2]. We have packaged these arguments into a lemma since it may be of independent interest. First, we define formally a direct sum of correlations. For ω ∈ [0, 1] and a correlation table T xy , we write ω · T xy to denote entry-wise multiplication of T xy by ω. We may refer to ω as a weight.
p(a, b|x, y) = δ ij ω i p i (a, b|x, y). (5) Then we say that p is a direct sum of the p i , and we write p = ⊕ l i=1 ω i p i . We sometimes refer to the p i as blocks of p and the ω i as weights of the blocks. We give a visual interpretation of condition (5) in Supplementary Table 1.

Supplementary
such that: Proof. For the remainder of the proof, when an operator acts only on one tensor factor we omit writing the identity on the other factors. Our first goal is to construct the subspaces H i A , H i B . We first study the action of the projectors corresponding to answers in A i and B i on the state |ψ . We will use these properties to define the states |ψ i . Then from these, we will construct The second equality follows from the fact that Π b By forms a complete measurement. The third equality comes from the block structure of the correlation. More specifically, suppose that a ∈ A i but b ∈ B i . Then the block structure demands that p(a, b|x, y) = 0 for all x, y. So we conclude that Π a Ax ⊗ Π b By |ψ 2 = p(a, b|x, y) = 0. This forces the appropriate terms of the sum in Equation (9) to vanish. The same argument with the roles of A and B reversed gives Combined with Equation (11), this implies that, for any i, x, y, In particular, the action of Π Ai Ax on |ψ is the same for all x, and similarly for the B operators. This lets us define where the choice of x does not matter. Now we compute the norm of |ψ i . The block structure p = ⊕ i ω i p i of the correlation gives us that for any fixed x and y, where the last line follows from Supplementary Equation (13). This establishes condition (ii). Now where λ j are the eigenvalues and |j the eigenvectors of ρ i A . These are guaranteed to exist even if |ψ i is infinite-dimensional, because the existence of a Schmidt decomposition for any bipartite state holds also in infinite-dimensional Hilbert spaces. Notice that We wish to compute the action of Π Ai Ax on the eigenstates of ρ i A . We calculate Since ω i = j λ j , we must have Π Ai Ax |j 2 = 1 for each j. In other words, Π Ai Ax |j = |j . This motivates us to define the space H i A as the span of the nontrivial eigenvectors of ρ i A . Define also P i as the projection onto subspace H i A . It follows from the definition of the |ψ i and the H i A that Furthermore, notice that Π Ai Ax P i = P i . Thus the H i A are suitable spaces for the new strategies to be defined on. In particular, the restricted operators Π a Ax | H i A are projectors. To see this, notice that they are orthogonal for distinct a and that they sum to identity.
Let H null Thus, we have established condition (i) of the lemma. It follows straightforwardly from the Definition of |ψ i and (24) that for Ax |ψ , and similarly for B. This establishes condition (iv). Finally, we show condition (iii), that the strategies in each block induce the appropriate correlations. We fix arbitrary a ∈ A i , b ∈ B i , x ∈ X , y ∈ Y, and calculate = p i (a, b|x, y).
In the above, the first quantity is the correlation induced by the strategy defined in Equation (6), and the last quantity is the desired correlation p i . Thus, we have shown condition (iii).

Supplementary Note 3 -The separating correlation
In this section, we describe in detail the correlation p * that separates C q and C qs , which was described only at a high-level in the main text.
Recall the definition of C N from Supplementary Note 1. For each m ≥ 0, we define two isometries V even m , V odd m : C 2 → C N as follows: We use these isometries to define observables on C N . By abuse of notation, for an isometry V : C 2 → C N and an operator O on C 2 , we write V (O) to refer to the pushforward V OV † of O along V . For example, V even m (σ z ) = |2m 2m| − |2m + 1 2m + 1|. 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. One can check that with this notation O = O + − O − . We use the notation A i to denote the direct sum of observables A i . We will make use of the α-tilted Paulis σ z α , σ x α from Definition 2. The following is the ideal strategy in detail.
Supplementary Table 2: Alice's ideal measurements. The entry in cell x, a is the projector Π a Ax .
See the main text, for an informal intuition about the ideal measurements above. The ideal state and measurements defining p * specify correlation tables T xy for all pairs of questions x ∈ {0, 1, 2, 3}, y ∈ {0, 1, 2, 3, 4}. We report some of them here for convenience (they also appear in the main text), as we will later analyse the constraints that these impose on the measurement projectors. Let C = 1 1−α 2 in the tables below (note C > 1). Recall from the main text that we denote by CHSH α x,y the correlation table on question x, y ∈ {0, 1} for the ideal tilted CHSH correlation for ratio α. Supplementary

Supplementary Note 4 -Proof of separation
In this section, we give a detailed proof of Theorem 1 from the main text. We start from a strategy that induces p * : in Supplementary Note 4.1, we prove properties of the state and the measurement operators, and in Supplementary Note 4.2, we characterize the non-zero Schmidt coefficients of the state, concluding that they must be infinitely many.

Supplementary Note 4.1 -Characterizing the state and the projectors
The following lemma establishes the existence of two local isometries which decompose any state achieving p * into two different ways (as anticipated in the proof overview of Section 2 from the main text).
Lemma 7 (Characterizing the state and projectors).
By }) be a strategy inducing the ideal correlation p * from Definition 6. Let C = 1 1−α 2 . Then there exist two local isometries Φ and Φ and (normalized) states |aux , |aux and |aux such that Proof. (i): Let p be the restriction of p * to questions x, y ∈ {0, 1}. From Table 4, we know that p is the ideal tilted CHSH correlation for ratio α (except that it has an extra answer "2" which has zero probability mass). Applying the block decomposition lemma (Lemma 5) with ω 1 = 1 and ω 2 = 0, we have that there exist subspaces H Moreover, by Lemma 3, it is also the case that Since (I + Z)/2 = |0 0| and (I − Z)/2 = |1 1|, we deduce by linearity that (30) Letting Φ be any isometric extension of Φ 1 to H A ⊗ H B and applying condition (iv) of Lemma 5 gives (i).
(ii): Let p be the restriction of p * to questions x, y ∈ {2, 3}. Then from table 5 we have that p = ω 1 p 1 ⊕ ω 2 p 2 where p 1 is the ideal tilted CHSH correlation (for ratio α) and p 2 is the correlation in which answer (2, 2) has probability 1 on all question pairs, and ω 1 = C−1 C , ω 2 = 1 C .
,aux be a local isometry, and |aux ∈H 2 A,aux ⊗H 2 B,aux a (normalized) state such that Such Φ 2 and |aux trivially exist. Define Then (a), (b), (c) and (d), together with condition (iv) of Lemma 5, imply that Φ satisfies condition (ii) of Lemma 7, as desired.
We also need the following properties, obtained using the y = 4 question on Bob's side.

Supplementary Note 4.2 -Characterizing the Schmidt coefficients
From now onwards, let (|ψ , {Π a Ax }, {Π b By }) be any strategy inducing p * . In the previous subsection, we gave a partial characterization of the operators and state . In this subsection, we make use of these properties to show that |ψ has infinitely many non-zero Schmidt coefficients. For a bipartite state |φ AB , we denote by Sch (|φ AB ) the multiset of non-zero Schmidt coefficients of |φ AB (here by multiset we mean a set with multiplicity, sometimes called an unordered list; for example, the multiset of Schmidt coefficients of the EPR pair is ( 1 √ 2 , 1 √ 2 )). Recall that the Schmidt coefficients {λ i } are the unique nonnegative real numbers so that |φ AB = i λ i |i A ⊗ |i B for some bases of the A and B registers. Any such pair of bases is called a pair of Schmidt bases with respect to |φ . Usually the tensor product decomposition of the Hilbert space will be clear, in which case we'll simply write Sch(|φ ) without the subscripts. We will use the following basic fact about Schmidt coefficients; we provide a proof for completeness.
Proof. A Schmidt basis for H A with respect to |ψ is the same as an eigenbasis for the reduced density operator Tr B |ψ ψ|. Using the orthogonality of |φ and |η , one can check that the three densities ρ A , σ A , τ A commute. Therefore, the densities have a common eigenbasis. This is also a common Schmidt basis. After repeating the argument to find a common Schmidt basis on H B , we can write the states as with a i + b i = λ i . By the orthogonality of |η and |φ , we have a i b i = 0 for each i. This implies that for each i, exactly one of the following two equalities holds: λ i = a i or λ i = b i . This yields the lemma.
Lemma 10. Let Φ, Φ and |aux , |aux , |aux be the local isometries and auxiliary states from Lemma 7. Let S = Sch (|ψ ), and let S 2 = Sch 1 √ C |22 ⊗ |aux . Then there exists a partition S = S 0 S 1 such that: Notice that these two equalities give us two different correspondences between the Schmidt coefficients of |aux and |aux , where one involves multiplying by α and the other involves dividing by α.
Composing the maps f and g yields a bijection between S 0 and S 0 \ S 2 . Since S 2 is nonempty, this implies that S 0 must be infinite.