The study of quantum correlation sets initiated by Tsirelson in the 1980s and originally motivated by questions in the foundations of quantum mechanics has more recently been tied to questions in quantum cryptography, complexity theory, operator space theory, group theory, and more. Synchronous correlation sets introduced by Paulsen et al. [J. Funct. Anal. 270, 2188–2222 (2016)] are a subclass of correlations that has proven particularly useful to study and arises naturally in applications. We show that any correlation that is almost synchronous, in a natural ℓ1 sense, arises from a state and measurement operators that are well-approximated by a convex combination of projective measurements on a maximally entangled state. This extends a result of Paulsen et al. [J. Funct. Anal. 270, 2188–2222 (2016)] that applies to exactly synchronous correlations. Crucially, the quality of approximation is independent of the dimension of the Hilbert spaces or of the size of the correlation. Our result allows one to reduce the analysis of many classes of nonlocal games, including rigidity properties, to the case of strategies using maximally entangled states that are generally easier to manipulate.
For finite sets and , a quantum correlation is an element of the set
where and range over all finite-dimensional Hilbert spaces and a POVM (positive operator-valued measure) on a Hilbert space is a collection of positive semidefinite operators on that sum to identity. For each , the set is convex, as can be seen by taking direct sums, but there are such that it is not closed.1 We write Cq for the union of over all finite and .
A strategy is a tuple such that is a state (i.e., a unit vector), is a collection of POVM on , and is a collection of POVM on . (The finite-dimensional Hilbert spaces and as well as the index sets and are generally left implicit in the notation.) Given a strategy , we say that induces the correlation .
The study of the set of quantum correlations Cq and its relation to the set of classical correlations C, defined as the convex hull of those correlations that can be induced using a state that is a tensor product , is of importance in the foundations of quantum mechanics. The fact that C ⊊ Cq, as first shown by Bell2 and often termed “quantum nonlocality,” underlies the field of device-independent quantum cryptography and gives rise to the study of entanglement witnesses, protocols for delegated quantum computation, and questions in quantum complexity theory; we refer to Ref. 3 for references. Following the foundational work of Tsirelson,4 multiple variants of the set of quantum correlations have been introduced and their study is connected to a range of problems in mathematics, including operator space theory,5,6 group theory,7 and combinatorics.8
In this paper, we consider a subset of Cq introduced in Ref. 9 and called the synchronous set . It is defined as the union of all , where is the subset of that contains all those correlations C that satisfy Cx,x,a,b = 0 whenever a ≠ b. This set arises naturally in the study of certain classes of nonlocal games. In general, a nonlocal game is specified by a distribution ν on and a function . A nonlocal game gives rise to a linear function on through the quantity
Given a game , one is interested in its quantum value, which is defined as the supremum over all C ∈ Cq of . A game such that , , ν(x, x) > 0 for all x, and D(a, b|x, x) = 0 for all x and a ≠ b is called a synchronous game. Any such game has the property that can only be obtained by a . Synchronous games arise naturally in applications; see, e.g., the classes of graph homomorphism games10 or linear system games.11 (Linear system games are projection games, which can be turned into synchronous games by taking their “square;” see Ref. 12.) The set retains most of the interesting geometric aspects of Cq, and in particular, it is convex and non-closed.12
A key property of synchronous correlations that makes them more amenable to study is the following fact shown in Ref. 9. For every synchronous correlation C, there is a family of strategies and a measure μ on Λ such that for each λ, with
where and is an orthonormal family in , and each measurement and consists entirely of projections, and moreover, for all x, y, a, b, we have
When takes the form in (1), we can express
where Tr(·) is the usual matrix trace and XT denotes the transpose with respect to the basis . The fact that synchronous correlations are “tracial” in the sense given by (2) and (3) contributes largely to their appeal. In contrast, there are correlations C ∈ Cq such that C cannot be induced, even approximately, using a convex combination of strategies using states of the form (1) in any dimension; see Ref. 13 for an example. Such correlations tend to be more difficult to study and their main interest lies in their existence, e.g., they can provide entanglement witnesses for states that are not maximally entangled.
A. Our results
We consider strategies that are almost synchronous, where the default to synchronicity is measured by the quantity
where ν is some distribution on . This averaged ℓ1 distance is motivated by applications to nonlocal games, which we describe below. Informally, our main result is that any strategy that induces a correlation C is well-approximated by a convex combination of strategies each using a maximally entangled state, where the approximation is controlled by δsync(C; ν) for any ν (ν also enters in the measure of approximation between and the ) and, crucially for applications, does not depend on the dimension of or the size of the sets and . In particular, each gives rise to a synchronous correlation Cλ such that ∫λCλ ≈ C in a suitable ℓ1 sense. Moreover, and crucially for the applications that we describe next, specific structural properties of the , such as algebraic relations between some of the measurement operators, can be transferred to the strategy . A simplified version of our theorem specialized to the case of a single measurement can be stated as follows:
For the complete statement and additional remarks, see Theorem 3.1. The first part of the theorem (5) is very simple to obtain; it is the second part that is meaningful. In particular, since is a maximally entangled state, the approximation on the left-hand side can be seen as a form of weighted approximation over certain (overlapping) diagonal blocks of A. The fact that the spaces and the states depend on only allows us to apply the theorem repeatedly for different measurements in order to decompose an arbitrary strategy as a convex combination of projective maximally entangled strategies, with the right-hand side in (6) replaced by Cδsync(C;ν)c for a ν of one’s choice (which naturally will also appear on the left-hand side).
A consequence of the theorem is that any that is also synchronous can be approximated by elements of ; this is because any sequence of approximations to C taken from Cq must, by definition, be almost synchronous and so Theorem 3.1 can be applied. (For this observation, it is crucial that the approximation provided in Theorem 3.1 does not depend on the dimension of the Hilbert spaces; however, it could depend on the size of C.) This particular application was already shown in Ref. 12 (Theorem 3.6).
Our result and its formulation are motivated by the study of nonlocal games. For a strategy , we write for , where C is the correlation induced by . Recall that the game value is the supremum over all strategies of . The fact that the supremum is taken over Cq and not is motivated by applications to entanglement tests, cryptography, and complexity theory, as in those contexts, there is no a priori reason to enforce hard constraints of the form Cx,x,a,b = 0; indeed, such a constraint cannot be verified with absolute confidence in any statistical test.
Given a game and a strategy , it is possible to obtain statistical confidence that for finite ɛ > 0 by playing the game many times. For this reason, the characterization of nearly optimal strategies plays a central role in applications of nonlocality. Recall that a synchronous game has the property that D(x, x, a, b) = 0 whenever a ≠ b. Given a synchronous game such that furthermore , it follows that any strategy for such that must satisfy , where νdiag(x) = ν(x, x)/(∑x′ν(x′, x′)) and the constant implicit in the O(·) notation will, in general, depend on the weight that ν places on the diagonal. (In particular, a better bound on δsync will be obtained in cases when the distribution ν is not a uniform distribution, as the uniform distribution places weight on the diagonal, which can be quite small.) Thus, nearly optimal strategies in synchronous games give rise to almost synchronous correlations. This conclusion may also hold for games that are not necessarily synchronous, for example, because the sets and are disjoint; an example is the class of projection games that we consider in Sec. IV B. Examples of projection games include linear system games11 and games such as the low-degree test14 that play an important role in complexity theory.
Given the importance of studying nearly optimal strategies, the fact that for many games any nearly optimal strategy is almost synchronous ought to be useful. Our work allows one to reduce the analysis of almost synchronous strategies to that of exactly synchronous strategies in a broad variety of settings. The most direct application of our results is to the study of the phenomenon of rigidity, which seeks to extract necessary conditions of any strategy that is nearly optimal for a certain game. Informally, our results imply that a general rigidity result for a synchronous game can be obtained in an automatic manner from a rigidity result that applies only to perfectly synchronous strategies. In order for the implication to not lose factors depending on the size of the game in the approximation quality for the rigidity statement, it is sufficient that a high success probability in the game implies a low for ν being the marginal distribution on either player’s questions in the game; see Corollary 4.1 and the remarks following it for further discussion. To give just one example, the entire analysis carried out in the recent work15 could be simplified by making all calculations with the maximally entangled state only, making manipulations of the “state-dependent distance” far easier to carry out. We refer to Sec. IV A for a precise formulation of how our main result can be used in this context as well as another application showing algebraic relations between measurement operators.
Given an almost synchronous strategy , it is not hard to show that the state and operators that underlie the strategy behave in an “approximately” cyclic manner, e.g., letting ρA denote the reduced density of on , it holds that for all x, a, where ‖·‖1 denotes the Schatten-1 norm; see, e.g., Ref. 16 (Lemma 3.7) for a precise statement. The strength of our result lies in showing that such relations imply an approximate decomposition in terms of maximally entangled strategies, where crucially the approximation quality does not depend on the dimension of the Hilbert space nor on the size of the sets or . A similar decomposition implicitly appears in Ref. 7, where it is used to reduce the analysis of nearly optimal strategies for a specific linear system game to the case of maximally entangled strategies; in the context of that paper, the reduction is motivated by a connection with the study of approximate representations of a certain finitely presented group. The main technical ingredient that enables the reduction in Ref. 7 is also the main ingredient in the present paper, which can be seen as a direct generalization of the work done there. Informally, the key idea is to write any density matrix ρ as a convex combination of projections , where λ is any non-negative real and is the indicator of the interval ; see Lemma 2.11. The main additional observation needed is a calculation that originally appears in Ref. 17 and is restated as Lemma 2.12; informally, the calculation allows us to transfer approximate commutation conditions such as those obtained in Ref. 16 (Lemma 3.7) for any almost synchronous strategy to the same conditions, evaluated on the matrix . The latter is a scaled multiple of the identity and is thus directly related to a maximally entangled state.
We use to denote finite sets. We use to denote a finite-dimensional Hilbert space, which we generally endow with a canonical orthonormal basis with . We use ‖·‖ to denote the operator norm (largest singular value) on . Tr(·) is the trace on and ‖·‖F is the Frobenius norm for any operator X on , where X† is the conjugate-transpose. A positive operator-valued measure (POVM), or measurement for short, on is a finite collection of positive semidefinite operators such that ∑aAa = Id. A measurement is projective if each Aa is a projection.
We use poly(δ) to denote any real-valued function f such that there exists constants C, c > 0 with |f(δ)| ≤ Cδc for all non-negative real δ. The precise function f as well as the constants c, C may differ each time the notation is used. For a distribution ν on a finite set , we write Ex∼ν for the expectation with respect to x with distribution ν.
B. Strategies, correlations, and games
(synchronous correlations). For finite sets and , a correlation is called synchronous if Cx,x,a,b = 0 for all and such that a ≠ b. Given a distribution ν on , recall the definition of δsync(C; ν) in (4). Given a strategy , we also write for δsync(C; ν), where C is the correlation induced by .
Observe that a correlation defined from a PME strategy is synchronous. [A converse to this statement is shown in Ref. 9, i.e., every synchronous strategy arises from (a convex combination of) PME strategies.] To see this, first recall Ando’s formula: for any X, Y, and of the form (7) with reduced density ρ, it holds that
where the transpose is taken with respect to the basis as in (7). Now, for a PME strategy and any ν, write
where the first equality is by definition, the second uses (8) together with the fact that for a PME strategy, the reduced density of on either system is proportional to the identity and hence commutes with any operator, the third equality uses that all are projections, and the last uses that they sum to identity.
We reproduce a definition from Ref. 16.
In Ref. 16, the second condition is required for all x, y, a, b. We require it to hold in an averaged sense only because this is more natural when seeking approximations that are independent of the size of the sets , as is the case in the context of this paper.
The following lemma implies that for any correlation C ∈ Cq, there is a projective (but not necessarily maximally entangled) strategy that realizes it.
A nonlocal game (or game for short) is specified by a tuple of finite question sets and , finite answer sets and , a distribution ν on , and a decision predicate that we conventionally write as D(a, b|x, y) for and . The game is symmetric if , , ν(x, y) = ν(y, x) for all , and for all a, b, x, y, D(a, b|x, y) = D(b, a|y, x). In this case, we write . We often abuse notation and also use ν to denote the marginal distribution of ν on .
We show elementary and generally well-known lemmata that will be useful in the proofs. The first lemma relates two different measures of state-dependent distance between measurements on .
Given a density matrix ρ on , define the canonical purification of ρ as the state
where is the spectral decomposition.
The next lemma gives conditions under which two strategies induce nearby correlations.
D. Rounding operators
We introduce two simple lemmas originally due to Connes17 (who proved them in the much more general setting of semifinite von Neumann algebras). The lemmas allow one to provide estimates on ‖f(A) − g(B)‖F when A, B are Hermitian operators on and f, g are real-valued functions. As discussed in the Introduction, these lemma were previously used in Ref. 7 to show a weaker result than we show here (which was sufficient for their purposes).
For , define by χ≥λ(x) = 1 if x ≥ λ and 0 otherwise. Extend χ≥λ to Hermitian operators on using the spectral calculus. The first lemma appears as Lemma 5.6 in Ref. 7.
The second lemma appears as Lemma 5.5 in Ref. 7.
The following result shows that an approximately consistent strategy is always close to a projective strategy. The result first appears in Ref. 18. The statement that we give here is taken from Ref. 14.
III. MAIN RESULT
The following is our main result. It states that a strategy that induces a correlation that is almost synchronous must be proportionately close, in a precise sense, to a projective maximally entangled strategy.
There are universal constants c, C > 0 such that the following holds. Let and be finite sets and ν be a distribution on . Let be a symmetric strategy and . Then, there is a measure μ on and a family of Hilbert spaces for (both depending on only) such that the following holds. For every , there is a maximally entangled state and for each x a projective measurement on such that we have the following:
- Letting ρ be the reduced density of on and ρλ be the totally mixed state on ,(19)
- provide an approximate decomposition of as a convex sum of projective maximally entangled (PME) strategies in the following sense:(20)
The key point in Theorem 3.1 is that the error estimates are independent of the dimension of and of the size of the sets and . We remark that the integral over λ can be written as a finite convex sum. This is evident from the definition of ρλ as a multiple of the projection Pλ defined in (31). Since is finite-dimensional, ρ has a discrete spectrum and Pλ takes on a finite set of values.
We note that the theorem does not imply that itself is close to a maximally entangled state. Rather, (19) implies that after tracing an ancilla, which contains the index λ, this is the case. It is not hard to see that this is unavoidable by considering a game such that there exist multiple optimal strategies for the game that are not unitarily equivalent. For example, one can consider a linear system game that tests the group generated by the Pauli matrices σX, σZ and σY; this can be obtained from three copies of the Magic Square game as in, e.g., Ref. 19 (Appendix A). This game can be won with probability 1 using any state of the form
where is an EPR pair (rank-2 maximally entangled state) and the measurement operators are block-diagonal with respect to the third system (i.e., for the first player). Crucially, the measurement operator’s dependence on the third system cannot be removed by a local unitary because the X′ and X″ components are not unitarily related. Although the strategy cannot be locally dilated to a maximally entangled strategy in the sense of Definition 2.4, it is not hard to see that it nevertheless has a decomposition of the form promised by Theorem 3.1.
Before turning to the proof, we give a pair of corollaries. The first shows that the conclusions of the theorem are maintained even without the assumption that is symmetric.
Let be a strategy. Let ν be a distribution on and . Then, the same conclusions as Theorem 3.1 hold (for different constants c, C), where ρ is chosen as the reduced density of on either (in which case the conclusions apply to ) or (in which case they apply to ).
The second corollary shows that the conclusions of the theorem imply an approximate decomposition of the correlation implied by as a convex combination of synchronous correlations.
The assumption that is projective is without loss of generality since by Lemma 2.5 any correlation can be achieved by a projective strategy. By an averaging argument, the corollary immediately implies that for any game with question distribution , there is a λ such that succeeds at least as well as in up to an additive loss poly(δ).
B. Proof of Theorem 3.1
We now prove the theorem. As in the statement of Theorem 3.1, let be a symmetric strategy. Let ρ be the reduced density of on . As a first step in the proof, we apply Proposition 2.13 to obtain a nearby symmetric projective strategy with nearly the same success probability.
For every , let
be the projection on the direct sum of all eigenspaces of ρ with the associated eigenvalue at least λ. Using Lemma 2.11, ∫λTr(Pλ)dλ = 1, so dμ(λ) = Tr(Pλ)dλ is a probability measure. Let be a Hilbert space of dimension, the rank of Pλ. We endow each with an orthonormal basis of eigenvectors of ρ that allows us to view as a subspace of for any λ′ ≤ λ, with for any λ > ‖ρ‖ and the convention .
The next lemma shows a form of approximate commutation between and .
With the preceding two lemmas in hand, we are ready to give the Proof of Theorem 3.1.
Fix a symmetric strategy for . Let be the family of projective measurements obtained in Lemma 3.4, and .
IV. APPLICATIONS TO NONLOCAL GAMES
We give two applications of Theorem 3.1. The first is to transferring “rigidity” statements obtained for PME strategies to the general case. The second is to the class of projection games.
A. Application to rigidity
As mentioned in the Introduction, Theorem 3.1 allows one to transfer rigidity statements shown for PME strategies to general strategies. We do not have a general all-purpose statement demonstrating this. Instead, we give two simple corollaries that are meant to describe sample applications. The first corollary considers a situation important in complexity theory, where one aims to show that a large family of measurements that constitute a successful strategy in a certain game must in some sense be consistent with a single larger measurement that “explains” it; see Subsection IV A 1. The second corollary considers a typical midpoint in a proof of rigidity, where one uses the game condition to derive certain algebraic relations on the measurements that constitute a successful strategy, which are then shown to impose a further structure; see Subsection IV A 2.
1. Application to showing classical soundness
Our first application arises in complexity theory when one is trying to show that quantum strategies in a certain nonlocal game obey a certain “global” structure. We first state the corollary and then describe a typical application of it.
Let be a symmetric game. Suppose given the following:
finite sets and ,
a joint distribution p on ,
for every , a function , the collection of subsets of , such that for any fixed (x, y), the sets gxy(a) for are pairwise disjoint, and
a convex monotone non-decreasing function ,
and suppose that given these data, the following statement holds:
- For every ω ∈ [0, 1] and symmetric PME strategy that succeeds with probability ω in , there is a family of measurements on , indexed by and with outcomes , such thatwhere .(38)
Then, the same statement extends to arbitrary symmetric projective strategies , with the right-hand side in (38) replaced by κ(ω − poly(δ)) − poly(δ), where .
Note that using Lemma 2.8, the guarantee (38) can equivalently be expressed in terms of a state-dependent distance between and , with px the conditional distribution p(x, ·)/p(x). The condition that the strategy should be symmetric projective is very mild, as projectivity can always be obtained by applying Naimark dilation (Lemma 2.5) and symmetry is generally obtained as a consequence of symmetry in the game.
The loss in quality of approximation guaranteed by the corollary depends polynomially on . In many cases, this quantity can be bounded directly from a high success probability in the game. This is the case if, for example, the distribution ν is such that ν(x, x) ≥ cν(x) for some c > 0 and all x, where recalling that by slight abuse of notation, we use ν(·) to denote the marginal on either player. In this case, any strategy such that has and so no further assumption is necessary.
The assumption made in the proposition is typical of a rigidity result and is specifically meant to illustrate the potential applicability of our result to a setting such as that of the low-individual degree test of Ref. 14, which forces successful strategies in a certain game to necessarily have a specific “global” structure. For purposes of illustration, we state an over-simplified version of the main result from Ref. 14 result as follows:
To apply Corollary 4.1 to the setting of Theorem 4.2, let the game in Corollary 4.1 be the “degree-d low individual degree game” from Theorem 4.2. Let be a singleton and be the set of m-variate polynomials over of individual degree at most d. Let p be uniform over . For every and , let gxy(a) be the collection of polynomials that evaluate to a at x. Then, (39) gives (38) with κ(ω) = poly(m) · (poly(ɛ) + poly(d/q)), where ɛ = 1 − ω. To conclude, we note that for the specific game , the condition ν(x, x) ≥ cν(x) for some c > 0 mentioned earlier holds, which allows us to bound δsync by O(ɛ).22 In conclusion, Corollary 4.1 shows that to prove Theorem 4.2, provided one is willing to accept a small loss in the approximation quality, it is sufficient to prove it for PME strategies. As observed in the Introduction, this allows for a significant simplification in the technical steps of the proof.
We give the proof of the corollary.
For each y, b, define
and note that and
where the last inequality is by (41).□
2. Application to showing algebraic relations
Our second application concerns rigidity statements that go through algebraic relations, as is exemplified by the rigidity proofs for games such as the CHSH game, the magic square game, as well as more general classes of games; see, e.g., Ref. 20 for an exposition of this approach.
Let be a symmetric nonlocal game. Suppose that , and for any symmetric projective strategy in , for , is a two-outcome measurement that can be represented as an observable . Suppose that the following statement holds for some concave monotone non-decreasing function .
Since the aim of the corollary is to give a “toy” application of our results, we sketch the proof but omit the details.
B. Extension to projection games
Theorem 3.1 applies to almost consistent symmetric strategies. In this section, we give an example of how the results of the theorem can be applied to a family of games such that success in the game naturally implies a bound on consistency. This partially extends the main result in Ref. 21, with the caveat that our result applies only to projection games, and not the more general “weak projection games” considered in Ref. 21; it is not hard to see that this is necessary to obtain a “robust” result of the kind we obtain here.
A game is a projection game if for each , there is such that D(a, b|x, y) = 0 if b ≠ fxy(a).
I thank Laura Mančinska, William Slofstra, and Henry Yuen for comments and Vern Paulsen for pointing out typos in an earlier version. This work was supported by NSF CAREER Grant No. CCF-1553477, AFOSR YIP Award No. FA9550-16-1-0495, MURI Grant No. FA9550-18-1-0161, and the IQIM, an NSF Physics Frontiers Center (NSF Grant No. PHY-1125565) with support of the Gordon and Betty Moore Foundation (Grant No. GBMF-12500028).
Conflict of Interest
The author has no conflicts of interest to disclose.
Data sharing is not applicable to this article as no new data were created or analyzed in this study.
In fact, for this game, c is only inverse polynomial in m, which still suffices, given the form of κ.