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 *ℓ*_{1} 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.

## I. INTRODUCTION

For finite sets $X,Y,A$ and $B$, a *quantum correlation* is an element of the set

where $HA$ and $HB$ range over all finite-dimensional Hilbert spaces and a POVM (positive operator-valued measure) on a Hilbert space $H$ is a collection of positive semidefinite operators on $H$ that sum to identity. For each $X,Y,A,B$, the set $Cq(X,Y,A,B)$ is convex, as can be seen by taking direct sums, but there are $X,Y,A,B$ such that it is not closed.^{1} We write *C*_{q} for the union of $Cq(X,Y,A,B)$ over all finite $X,Y,A$ and $B$.

A *strategy* is a tuple $S=(\psi ,A,B)$ such that $\psi \u2208HA\u2297HB$ is a state (i.e., a unit vector), $A=Aax$ is a collection of POVM on $HA$, and $B=Bby$ is a collection of POVM on $HB$. (The finite-dimensional Hilbert spaces $HA$ and $HB$ as well as the index sets $X,Y,A,$ and $B$ are generally left implicit in the notation.) Given a strategy $S$, we say that $S$ *induces* the correlation $(Cx,y,a,b=\psi Aax\u2297Bby\psi )xyab$.

The study of the set of quantum correlations *C*_{q} 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 $\psi $ that is a tensor product $\psi =\psi A\u2297\psi B\u2208HA\u2297HB$, is of importance in the foundations of quantum mechanics. The fact that *C* ⊊ *C*_{q}, as first shown by Bell^{2} 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 *C*_{q} introduced in Ref. 9 and called the *synchronous set* $Cqs$. It is defined as the union of all $Cqs(X,A)$, where $Cqs(X,A)$ is the subset of $Cq(X,X,A,A)$ that contains all those correlations *C* that satisfy *C*_{x,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 $G$ is specified by a distribution *ν* on $X\xd7Y$ and a function $D:X\xd7Y\xd7A\xd7B\u21920,1$. A nonlocal game gives rise to a linear function on $Cq(X,Y,A,B)$ through the quantity

Given a game $G$, one is interested in its *quantum value*$\omega q(G)$, which is defined as the supremum over all *C* ∈ *C*_{q} of $\omega q(G;C)$. A game $G$ such that $X=Y$, $A=B$, *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 $\omega q(G;C)=1$ can only be obtained by a $C\u2208Cqs$. Synchronous games arise naturally in applications; see, e.g., the classes of graph homomorphism games^{10} 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 $Cqs$ retains most of the interesting geometric aspects of *C*_{q}, 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 $S\lambda =(\psi \lambda ,A\lambda ,B\lambda )\lambda \u2208\Lambda $ and a measure *μ* on Λ such that for each *λ*, $\psi \lambda =\psi me(d\lambda )$ with

where $d\lambda =dim(H\lambda )$ and $ui:1\u2264i\u2264d\lambda $ is an orthonormal family in $H\lambda $, and each measurement $Aa\lambda ,x$ and $Bb\lambda ,y$ consists entirely of projections, and moreover, for all *x*, *y*, *a*, *b*, we have

When $\psi me(d\lambda )$ takes the form in (1), we can express

where Tr(·) is the usual matrix trace and *X*^{T} denotes the transpose with respect to the basis $ui$. 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* ∈ *C*_{q} 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 $S=(\psi ,A,B)$ that are *almost* synchronous, where the default to synchronicity is measured by the quantity

where *ν* is some distribution on $X$. This averaged *ℓ*_{1} distance is motivated by applications to nonlocal games, which we describe below. Informally, our main result is that any strategy $S$ that induces a correlation *C* is well-approximated by a convex combination of strategies $S\lambda $ each using a maximally entangled state, where the approximation is controlled by *δ*_{sync}(*C*; *ν*) for any *ν* (*ν* also enters in the measure of approximation between $S$ and the $S\lambda $) and, crucially for applications, does not depend on the dimension of $\psi $ or the size of the sets $X$ and $A$. In particular, each $S\lambda $ gives rise to a synchronous correlation *C*^{λ} such that *∫*_{λ}*C*^{λ} ≈ *C* in a suitable *ℓ*_{1} sense. Moreover, and crucially for the applications that we describe next, specific structural properties of the $S\lambda $, such as algebraic relations between some of the measurement operators, can be transferred to the strategy $S$. A simplified version of our theorem specialized to the case of a single measurement can be stated as follows:

*There are universal constants*

*c*,

*C*> 0

*such that the following holds. Let*$H$

*be a finite-dimensional Hilbert space and*$\psi \u2208H\u2297H$

*a state. Then, there is a finite set*Λ

*, a distribution*

*μ*

*on*Λ

*, and, for each*

*λ*∈ Λ

*, a state*$\psi \lambda $

*that is maximally entangled on a subspace*$H\lambda \u2297H\lambda \u2286H\u2297H$

*such that letting*

*ρ*

*be the reduced density of*$\psi $

*on the first factor and*

*ρ*

_{λ}be

*the reduced density of*$\psi \lambda $

*on*$H\lambda \u2286H$

*,*

*Moreover, let*$A$

*be a finite set and*$Aaa\u2208A$

*be an arbitrary measurement on*$H$

*. Then, there is a projective measurement*$Aa\lambda $

*on*$H\lambda $

*such that*

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 $\psi \lambda $ 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 $H\lambda $ and the states $\psi \lambda $ depend on $\psi $ 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 *Cδ*_{sync}(*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 $C\u2208Cq\u0304$ that is also synchronous can be approximated by elements of $Cqs$; this is because any sequence of approximations to *C* taken from *C*_{q} 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 $S$, we write $\omega q(G;S)$ for $\omega q(G;C)$, where *C* is the correlation induced by $S$. Recall that the game value $\omega q(G)$ is the supremum over all strategies of $\omega q(G;S)$. The fact that the supremum is taken over *C*_{q} and not $Cqs$ 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 *C*_{x,x,a,b} = 0; indeed, such a constraint cannot be verified with absolute confidence in any statistical test.

Given a game and a strategy $S$, it is possible to obtain statistical confidence that $\omega q(G;S)\u2265\omega q(G)\u2212\epsilon $ 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 $G$ such that furthermore $\omega q(G)=1$, it follows that any strategy $S$ for $G$ such that $\omega q(G;S)\u2265\omega q(G)\u2212\epsilon $ must satisfy $\delta sync(S;\nu diag)=O(\epsilon )$, where *ν*_{diag}(*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 *δ*_{sync} will be obtained in cases when the distribution *ν* is *not* a uniform distribution, as the uniform distribution places weight $\u22481|X|$ 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 $X$ and $Y$ are disjoint; an example is the class of projection games that we consider in Sec. IV B. Examples of projection games include linear system games^{11} and games such as the low-degree test^{14} 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 $\delta sync(S;\nu )$ 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 work^{15} 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.

### B. Discussion

Given an almost synchronous strategy $S=(\psi ,A,B)$, it is not hard to show that the state and operators that underlie the strategy behave in an “approximately” cyclic manner, e.g., letting *ρ*_{A} denote the reduced density of $\psi $ on $HA$, it holds that $\Vert Aax\rho A\u2212\rho AAax\Vert 1\u22480$ 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 $X,Y,A$ or $B$. 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 $\chi \u2265\lambda (\rho )$, where *λ* is any non-negative real and $\chi \u2265\lambda $ is the indicator of the interval $[\lambda ,+\u221e)$; 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 $\chi \u2265\lambda (\rho )$. The latter is a scaled multiple of the identity and is thus directly related to a maximally entangled state.

## II. PRELIMINARIES

### A. Notation

We use $X,Y,A,B$ to denote finite sets. We use $H$ to denote a finite-dimensional Hilbert space, which we generally endow with a canonical orthonormal basis $i|i\u22081,\u2026,d$ with $d=dim(H)$. We use ‖·‖ to denote the operator norm (largest singular value) on $H$. Tr(·) is the trace on $H$ and ‖·‖_{F} is the Frobenius norm $\Vert X\Vert F=Tr(X\u2020X)1/2$ for any operator *X* on $H$, where *X*^{†} is the conjugate-transpose. A positive operator-valued measure (POVM), or *measurement* for short, on $H$ is a finite collection of positive semidefinite operators $Aaa\u2208A$ such that ∑_{a}*A*_{a} = Id. A measurement $Aa$ is *projective* if each *A*_{a} 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*)| ≤ *Cδ*^{c} 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 $X$, we write E_{x∼ν} for the expectation with respect to *x* with distribution *ν*.

### B. Strategies, correlations, and games

*strategy*$S$ is a tuple $(\psi ,A,B)$, where $\psi \u2208HA\u2297HB$ is a quantum state and $A=Aax$ $respectively,B=Bby$ is a collection of measurements on $H$ indexed by $x\u2208X$ and with outcomes $a\u2208A$ (respectively, $y\u2208Y$ and $b\u2208B$). Any strategy induces a

*correlation*, which is the collection of real numbers,

(synchronous correlations). For finite sets $X$ and $A$, a correlation $C=(Cx,y,a,b)\u2208Cq(X,X,A,A)$ is called *synchronous* if *C*_{x,x,a,b} = 0 for all $x\u2208X$ and $a,b\u2208A$ such that *a* ≠ *b*. Given a distribution *ν* on $X$, recall the definition of *δ*_{sync}(*C*; *ν*) in (4). Given a strategy $S$, we also write $\delta sync(S;\nu )$ for *δ*_{sync}(*C*; *ν*), where *C* is the correlation induced by $S$.

*symmetric*if $X=Y$, $A=B$, $\psi \u2208H\u2297H$ takes the form

*λ*

_{i}are non-negative and $ui$ orthonormal, and for every

*x*,

*a*, $Aax=(Bax)T$ with the transpose being taken with respect to the $ui$. Note that this implies that $HA=HB$ and that $\psi $ has the same reduced density on either subsystem. For a symmetric strategy, we write it as $S=(\psi ,A)$. A strategy is

*projective*if all $Aax$ and $Bby$ are projections. It is

*maximally entangled*if $\psi $ is a maximally entangled state (1) on $HA\u2297HB$. We use the acronym “PME” to denote “symmetric projective maximally entangled.”

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 $\psi \u2208H\u2297H$ of the form (7) with reduced density *ρ*, it holds that

where the transpose is taken with respect to the basis $ui$ as in (7). Now, for a PME strategy $S=(\psi ,A)$ 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 $\psi $ on either system is proportional to the identity and hence commutes with any operator, the third equality uses that all $Aax$ are projections, and the last uses that they sum to identity.

We reproduce a definition from Ref. 16.

*ɛ*,

*ν*)-dilation). Given

*ɛ*≥ 0, a distribution

*ν*on $X\xd7Y$, and two strategies $S=(\psi ,A,B)$ and $S\u0303=(\psi \u0303,A\u0303,B\u0303)$, we say that $S\u0303$ is a

*local*(

*ɛ*,

*ν*)

*-dilation*of $S$ if there exists isometries $VA:HA\u2192H\u0303A\u2297KA$ and $VB:HB\u2192H\u0303B\u2297KB$ and a state $aux\u2208KA\u2297KB$ such that

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 $X,Y,A,B$, as is the case in the context of this paper.

The following lemma implies that for any correlation *C* ∈ *C*_{q}, there is a projective (but not necessarily maximally entangled) strategy that realizes it.

*Let*$\psi $

*be a state in*$HA\u2297HB$

*. Let*$A=Aax$

*be a measurement on*$HA$

*and*$B=Bby$

*be a measurement on*$HB$

*. Then, there exists Hilbert spaces*$HAaux$

*and*$HBaux$

*, a state*$aux\u2208HAaux\u2297HBaux$

*, and two projective measurements*$A\u0302=A\u0302ax$

*and*$B\u0302=B\u0302by$

*acting on*$HA\u2297HAaux$

*and*$HB\u2297HBaux$

*, respectively, such that the following is true. If we let*$\psi \u0302=\psi \u2297aux$,

*then for all*

*x*,

*y*,

*a*,

*b*

*,*

*In addition,*$aux$

*is a product state, meaning that we can write it as*$aux=auxA\u2297auxB$

*, for*$auxA$

*in*$HAaux$

*and*$auxB$

*in*$HBaux$

*.*

A *nonlocal game* (or *game* for short) $G$ is specified by a tuple $(X,Y,A,B,\nu ,D)$ of finite *question sets* $X$ and $Y$, finite *answer sets* $A$ and $B$, a distribution *ν* on $X\xd7Y$, and a *decision predicate* $D:X\xd7Y\xd7A\xd7B\u21920,1$ that we conventionally write as *D*(*a*, *b*|*x*, *y*) for $(x,y)\u2208X\xd7Y$ and $(a,b)\u2208A\xd7B$. The game is *symmetric* if $X=Y$, $A=B$, *ν*(*x*, *y*) = *ν*(*y*, *x*) for all $x,y\u2208X\xd7Y$, and for all *a*, *b*, *x*, *y*, *D*(*a*, *b*|*x*, *y*) = *D*(*b*, *a*|*y*, *x*). In this case, we write $G=(X,A,\nu ,D)$. We often abuse notation and also use *ν* to denote the marginal distribution of *ν* on $X$.

*success probability*of $S$ in $G$ is

### C. Consistency

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 $H$.

*Let*

*γ*,

*δ*≥ 0

*. Let*$H$

*be a Hilbert space and*$\psi \u2208H\u2297H$

*be a state. Let*$X$

*be a finite set, and for each*$x\u2208X$

*, let*$Aaxa\u2208A$

*be a projective measurement and*$Maxa\u2208A$

*be an arbitrary measurement on*$H$

*. Let*

*μ*

*be a distribution on*$X$

*. Let*

*Then,*

*δ*and that for each $x\u2208X$, $Aax$ is a projective measurement and $Maxa$ is a measurement.□

Given a density matrix *ρ* on $H$, define the *canonical purification* of *ρ* as the state

where $\rho =\u2211i\lambda iuiui$ is the spectral decomposition.

*Let*$\psi \u2208HA\u2297HB$

*and*$Aa$

*and*$Ba$

*be measurements on*$HA$

*and*$HB$,

*respectively. Let*

*ρ*

_{A}

*and*

*ρ*

_{B}

*be the reduced density of*$\psi $

*on*$HA$

*and*$HB$,

*respectively. Let*$\psi A\u2208HA\u2297HA$

*and*$\psi B\u2208HB\u2297HB$

*be the canonical purifications of*

*ρ*

_{A}

*and*

*ρ*

_{B}

*, respectively. Then,*

*ρ*

_{A}=

*KK*

^{†}and

*ρ*

_{B}=

*K*

^{†}

*K*, this concludes the proof.□

The next lemma gives conditions under which two strategies induce nearby correlations.

*Let*$S=(\psi ,A,B)$

*be a strategy, let*$A\u0302=A\u0302ax$

*be a family of POVM on*$HA$

*, and let*$S\u0302=(\psi ,A\u0302,B)$

*. Let*

*ρ*

_{A}

*be the reduced density of*$\psi $

*on*$HA$

*and*$\psi A\u2208HA\u2297HA$

*be the canonical purification of it. Let*$SA=(\psi A,A)$

*. Let*

*ν*

*be a distribution on*$X\xd7Y$

*and*$\delta =\delta sync(SA;\nu A)$,

*where*

*ν*

_{A}

*is the marginal of*

*ν*

*on*$X$

*. Let*

*Let*

*C*

*be the correlation induced by*$S$

*and*$C\u0302$

*by*$S\u0302$

*. Then,*

*ρ*

_{A}=

*ρ*

_{B}. Then, $\psi A=\psi $. As a first step in the proof, we show that

*x*,

*a*, the second uses that $\u2211bBby=Id$ for all

*y*, and the third uses that $\u2211aAax=Id$ for all

*x*. Next, we observe that

*ρ*

_{A}=

*ρ*

_{B}. Plugging back into (12), this shows (11). For the second step, we show

*γ*using $\u2211bBby=\u2211aAax=1$ and the definition of

*γ*. An analogous calculation gives

*η*−

*δ*)

^{2}≤

*γ*so

### D. Rounding operators

We introduce two simple lemmas originally due to Connes^{17} (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 $H$ 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 $\lambda \u2208R$, define $\chi \u2265\lambda :R\u2192R$ by *χ*_{≥λ}(*x*) = 1 if *x* ≥ *λ* and 0 otherwise. Extend *χ*_{≥λ} to Hermitian operators on $H$ using the spectral calculus. The first lemma appears as Lemma 5.6 in Ref. 7.

*Let*

*ρ*

*be a positive semidefinite operator on a finite-dimensional Hilbert space. Then,*

*where the integral is taken with respect to the Lebesgue measure on*$R+$

*.*

The second lemma appears as Lemma 5.5 in Ref. 7.

*Let*

*ρ*,

*σ*

*be positive semidefinite operators on a finite-dimensional Hilbert space. Then,*

### E. Orthonormalization

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.

*Let*0 ≤

*δ*≤ 1

*. Let*$\psi $

*be a state on*$H\u2297H$

*whose reduced densities on either subsystem are identical. Let*

*k*

*be an integer and*

*Q*

_{1}, …,

*Q*

_{k}

*be positive semidefinite operators on*$H$

*such that*∑

_{i}

*Q*

_{i}= Id

*. Let*

*Then, there exists orthogonal projections*

*P*

_{1}, …,

*P*

_{k}

*on*$H$

*such that*∑

_{i}

*P*

_{i}= Id

*and*

## 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* $X$ *and* $A$ *be finite sets and* *ν* be *a distribution on* $X$*. Let* $S=(\psi ,A)$ *be a symmetric strategy and* $\delta =\delta sync(S;\nu )$*. Then, there is a measure* *μ* *on* $R+$ *and a family of Hilbert spaces* $H\lambda \u2286H$ *for* $\lambda \u2208R+$ *(both depending on* $\psi $ *only) such that the following holds. For every* $\lambda \u2208R+$, *there is a maximally entangled state* $\psi \lambda \u2208H\lambda \u2297H\lambda $ *and for each* *x* *a projective measurement* $Aa\lambda ,x$ *on* $H\lambda $ *such that we have the following:*

*Letting**ρ**be the reduced density of*$\psi $*on*$H$*and**ρ*_{λ}*be the totally mixed state on*$H\lambda \u2286H$*,*(19)$\rho =\u222b\lambda \rho \lambda d\lambda .$- $S\lambda =(\psi \lambda ,A\lambda )$
*provide an approximate decomposition of*$S$*as a convex sum of projective maximally entangled (PME) strategies in the following sense:*(20)$Ex\u223c\nu \u2211a\u222b\lambda TrAax\u2212Aa\lambda ,x2\rho \lambda d\mu (\lambda )\u2264C\delta c.$

The key point in Theorem 3.1 is that the error estimates are independent of the dimension of $H$ and of the size of the sets $X$ and $A$. We remark that the integral over *λ* can be written as a finite convex sum. This is evident from the definition of *ρ*_{λ} as a multiple of the projection *P*_{λ} defined in (31). Since $H$ is finite-dimensional, *ρ* has a discrete spectrum and *P*_{λ} takes on a finite set of values.

We note that the theorem does not imply that $\psi $ 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 *σ*_{X}, *σ*_{Z} and *σ*_{Y}; 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 $\varphi +$ is an EPR pair (rank-2 maximally entangled state) and the measurement operators are block-diagonal with respect to the third system (i.e., $X=XA1A2\u2032\u229700A3+XA1A2\u2033\u229711A3$ 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.

### A. Corollaries

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 $S$ is symmetric.

*Let* $S=(\psi ,A,B)$ *be a strategy. Let* *ν* *be a distribution on* $X$ *and* $\delta =\delta sync(S;\nu )$*. Then, the same conclusions as Theorem 3.1 hold (for different constants* *c*, *C**), where* *ρ* *is chosen as the reduced density of* $\psi $ *on either* $HA$ *(in which case the conclusions apply to* $Aax$*) or* $HB$ *(in which case they apply to* $Bby$*).*

The second corollary shows that the conclusions of the theorem imply an approximate decomposition of the correlation implied by $S$ as a convex combination of synchronous correlations.

*Let*$S=(\psi ,A,B)$

*be a projective strategy. Let*

*ν*

*be a distribution on*$X$

*,*$\delta =\delta sync(S;\nu )$

*, and*$S\lambda $

*and*

*μ*

*be the family of strategies and the measure obtained from Theorem 3.1, respectively. Let*

*C*

*(respectively,*

*C*

^{λ}

*) be the correlation induced by*$S$

*(respectively,*$S\lambda $

*). Let*$\nu \u0303$

*be any distribution on*$X\xd7X$

*with marginal*

*ν*

*. Then,*

The assumption that $S$ is projective is without loss of generality since by Lemma 2.5 any correlation $C\u2208Cq(X,A)$ can be achieved by a projective strategy. By an averaging argument, the corollary immediately implies that for any game $G$ with question distribution $\nu \u0303$, there is a *λ* such that $S\lambda $ succeeds at least as well as $S$ in $G$ up to an additive loss poly(*δ*).

*ν*, $\nu \u0303$, $S\lambda $, and

*μ*as in the statement of the corollary. Conjugating $Bby$ by a unitary if necessary, we assume without loss of generality that the reduced densities of $\psi $ on either subsystem are identical. For every

*λ*, define a symmetric strategy $S\u0303\lambda =(\psi \lambda ,A)$ and let $C\u0303\lambda $ be the associated correlation. We first show that

*δ*. This bounds (25). Together with a similar bound for the term in (26), this shows (23). A similar proof shows (24).

*λ*to the strategy $S=(\psi \lambda ,A\lambda ,B)$ here and $A\u0302$ in Lemma 2.10 is

*A*here. Since $\psi \lambda $ is maximally entangled and

*A*

^{λ}supported on its support,

*δ*in Lemma 2.10 equals 0. Applying the lemma followed by Jensen’s inequality gives

### B. Proof of Theorem 3.1

We now prove the theorem. As in the statement of Theorem 3.1, let $S=(\psi ,A)$ be a symmetric strategy. Let *ρ* be the reduced density of $\psi $ on $H$. 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.

*There is a projective symmetric strategy*$S\u2032=(\psi ,B)$

*such that letting*$\delta \u2032=\delta sync(S\u2032,\nu )$,

*then*

*δ*′ =

*O*(

*δ*

^{1/8})

*and*

*x*, let $\delta x=1\u2212\u2211a\psi Aax\u2297(Aax)T\psi $. By definition of

*δ*, it holds that

*x*,

*x*, $Aaxa$ and $Bax$ are measurements. This shows

*δ*′ =

*O*(

*δ*

^{1/8}), as claimed.□

For every $\lambda \u2208R+$, let

be the projection on the direct sum of all eigenspaces of *ρ* with the associated eigenvalue at least *λ*. Using Lemma 2.11, *∫*_{λ}Tr(*P*_{λ})*dλ* = 1, so *dμ*(*λ*) = Tr(*P*_{λ})*dλ* is a probability measure. Let $H\lambda $ be a Hilbert space of dimension, the rank of *P*_{λ}. We endow each $H\lambda $ with an orthonormal basis of eigenvectors of *ρ* that allows us to view $H\lambda $ as a subspace of $H\lambda \u2032$ for any *λ*′ ≤ *λ*, with $H\lambda =0$ for any *λ* > ‖*ρ*‖ and the convention $H0=H$.

The next lemma shows a form of approximate commutation between $Bax$ and $P\lambda $.

*The following holds:*

*where*

*δ*′ =

*O*(

*δ*

^{1/8})

*is as in Lemma 3.4.*

*m*. Define a family of unitaries $Ubx$ indexed by $x\u2208X$ and $b\u2208A$ by

*x*is taken with respect to the (marginal of) the game distribution

*ν*, the expectation over

*b*is uniform over $Zm$, the first equality uses the equality E

_{b}

*e*

^{2iπ(a−a′)/m}=

*δ*

_{a,a′}(the Kronecker

*δ*) for all $a,a\u2032\u2208Zm$ and the fact that for every

*x*, $Baxa$ is projective, and the second uses identity (8).

*P*

_{λ},

*x*and

*b*, the third line is the Cauchy–Schwarz inequality, and for the last inequality, the first term is bounded using (34) and the second term is bounded using $\Vert \rho 1/2\Vert F2=1$. The claim follows since from definition (33), we get by expanding the left-hand side that for each

*x*and

*λ*,

With the preceding two lemmas in hand, we are ready to give the Proof of Theorem 3.1.

Fix a symmetric strategy $S=(\psi ,A)$ for $G$. Let $Bax$ be the family of projective measurements obtained in Lemma 3.4, $S\u2032=(\psi ,B)$ and $\delta \u2032=\delta sync(S\u2032;\nu )$.

*P*

_{λ}is a projection for each

*λ*, and that $Bax$ is a projective measurement for all

*x*and for the last line, we used (32).

*x*and

*λ*. To justify this application, we evaluate

*dμ*(

*λ*) and $A\u0303a\lambda ,x$ and (8), the second uses that $Bax$ is projective, and the last line is by (32). For each

*x*and

*λ*, let $Aa\lambda ,x$ be the projective measurement that is associated with $A\u0303a\lambda ,x$ by Proposition 2.13. Using Jensen’s inequality and (36), the proposition gives the guarantee that

*λ*, the strategy $S\lambda =(\psi \lambda ,A\lambda )$ is a PME strategy by definition, and (20) follows by combining (28), (35), and (37).□

## 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* $G=(X,A,\nu ,D)$ *be a symmetric game. Suppose given the following:*

*finite sets*$Y$*and*$B$*,**a joint distribution**p**on*$X\xd7Y$*,**for every*$(x,y)\u2208X\xd7Y$,*a function*$gxy:A\u21922B$*, the collection of subsets of*$B$*, such that for any fixed*(*x*,*y*),*the sets**g*_{xy}(*a*)*for*$a\u2208A$*are pairwise disjoint, and**a convex monotone non-decreasing function*$\kappa :[0,1]\u2192R+$*,*

*and suppose that given these data, the following statement holds:*

*For every**ω*∈ [0, 1]*and symmetric PME strategy*$S=(\psi ,A)$*that succeeds with probability**ω**in*$G$,*there is a family of measurements*$Mby$*on*$H$*, indexed by*$y\u2208Y$*and with outcomes*$b\u2208B$*, such that*(38)$E(x,y)\u223cp\u2211a\psi Aax\u2297M[gxy(a)]y\psi \u2265\kappa (\omega ),$*where*$M[gxy(a)]y=\u2211b\u2208gxy(a)Mby$*.*

*Then, the same statement extends to arbitrary symmetric projective strategies* $S\u2032=(\psi ,A)$*, with the right-hand side in* *(38)* *replaced by* *κ*(*ω* − poly(*δ*)) − poly(*δ*), *where* $\delta =\delta sync(S\u2032;\nu )$*.*

Note that using Lemma 2.8, the guarantee (38) can equivalently be expressed in terms of a state-dependent distance between $Aax$ and $Max=Ey\u223cpxM[gxy(a)]y$, with *p*_{x} the conditional distribution *p*(*x*, ·)/*p*(*x*). The condition that the strategy $S\u2032$ 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 $\delta sync(S\u2032;\nu )$. 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 $\omega q(G;S)\u22651\u2212\epsilon $ has $\delta sync(S;\nu A)\u2264\epsilon /c$ 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:

*Suppose that a symmetric strategy*$S=(\psi ,A)$

*succeeds in the “degree-*

*d*

*low individual degree game”*$Gld$

*, which has*$X=Fqm$

*and*$A=Fq$

*, with probability at least*1 −

*ɛ*

*. Then, there exists a projective measurement*$G=Gg$

*whose outcomes*

*g*

*are*

*m*

*-variate polynomials over*$Fq$

*of individual degree at most*

*d*

*such that*

To apply Corollary 4.1 to the setting of Theorem 4.2, let the game $G$ in Corollary 4.1 be the “degree-*d* low individual degree game” $Gld$ from Theorem 4.2. Let $Y=y$ be a singleton and $A$ be the set of *m*-variate polynomials over $Fq$ of individual degree at most *d*. Let *p* be uniform over $Fqm\xd7Y$. For every $x\u2208Fqm$ and $a\u2208Fq$, let *g*_{xy}(*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 $Gld$, the condition *ν*(*x*, *x*) ≥ *cν*(*x*) for some *c* > 0 mentioned earlier holds, which allows us to bound *δ*_{sync} 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.

*ω*denote its success probability. For each $\lambda \u2208R+$, let $S\lambda =(\psi \lambda ,A\lambda )$ be the PME strategy promised by Theorem 3.1 and

*ω*

_{λ}be its probability of success in $G$. Let $Mb\lambda ,y$ be the family of measurements promised by the assumption of Corollary 4.1 such that

*dμ*(

*λ*) and using that

*κ*is assumed to be convex monotone, it follows that

*ω*′ =

*ω*− poly(

*δ*) by Corollary 3.3.

*x*,

*y*and

*a*. Averaging over

*λ*,

*λ*,

*x*, and

*a*,

For each *y*, *b*, define

and note that $Mby\u22650$ 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* $G=(X,A,\nu ,D)$ *be a symmetric nonlocal game. Suppose that* $0,1\u2286X$, *and for any symmetric projective strategy* $S=(\psi ,A)$ *in* $G$*, for* $x\u22080,1$*,* $A0x,A1x$ *is a two-outcome measurement that can be represented as an observable* $Ax=A0x\u2212A1x$*. Suppose that the following statement holds for some concave monotone non-decreasing function* $\kappa :[0,1]\u2192R+$*.*

*For every*

*ω*∈ [0, 1]

*and symmetric PME strategy*$S=(\psi ,A)$

*that succeeds with probability*

*ω*

*in*$G$,

*it holds that*

*Then, the same statement extends to arbitrary symmetric projective strategies*$S\u2032=(\psi ,A)$

*, with the right-hand side in*

*(44)*

*replaced by*

*κ*(

*ω*+ poly(

*δ*)) + poly(

*δ*),

*where*

*with*

*q*

*being the uniform distribution on*$0,1\u2286X$

*and*

*ν*

*being the marginal of the game distribution on*$X$

*.*

Since the aim of the corollary is to give a “toy” application of our results, we sketch the proof but omit the details.

*ω*denote its success probability. For each $\lambda \u2208R+$, let $S\lambda =(\psi \lambda ,A\lambda )$ be the PME strategy promised by Theorem 3.1 and

*ω*

_{λ}be its probability of success in $G$. Furthermore, let $S\u0303\lambda =(\psi \lambda ,A)$. First, we claim that by an argument similar to the derivation of (22) in the proof of Corollary 3.3, it holds that

*δ*in (45) involves measuring almost synchronicity under

*q*. We may then achieve the desired conclusion as follows. First, we note that

*κ*(

*ω*− poly(

*δ*)) using the assumption made in the corollary, Jensen’s inequality, and the fact that by Corollary 3.3, it holds that

*∫*

_{λ}

*ω*

_{λ}

*dλ*≥

*ω*− poly(

*δ*). This concludes the proof.□

### 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 $G=(X,Y,A,B,\nu ,D)$ is a *projection game* if for each $(x,y)\u2208X\xd7Y$, there is $fxy:A\u2192B$ such that *D*(*a*, *b*|*x*, *y*) = 0 if *b* ≠ *f*_{xy}(*a*).

*There are universal constants*

*c*,

*C*> 0

*such that the following holds. Let*$G=(X,Y,A,B,\nu ,D)$

*be a projection game and*$S=(\psi ,A,B)$

*be a strategy for*$G$

*that succeeds with probability*1 −

*ɛ*

*, for some*0 ≤

*ɛ*≤ 1

*. Then, there is a measure*

*μ*

*on*$R+$

*and a family of Hilbert spaces*$H\lambda \u2286HA$

*for*$\lambda \u2208R+$

*(both depending on*$\psi $

*only) such that the following holds. For every*$\lambda \u2208R+$,

*there is a PME strategy*$S\lambda =(\psi \lambda ,A\lambda ,B)$

*for*$G$

*such that*$\psi \lambda $

*is a maximally entangled state on*$H\lambda \u2297H\lambda $,

*and moreover, if*

*ω*

_{λ}

*is the success probability of*$S\lambda $

*in*$G$

*, then*

*x*,

*y*, $Aax$ and $Bby$ are projective measurements. For each $x\u2208X$ and $a\u2208A$, let

*ν*

_{x}is the conditional distribution of

*ν*on $Y$, conditioned on

*x*. The assumption that $G$ is a projection game implies that for every

*x*, $\u2211aBax\u2264Id$. Let $\psi A$ and $\psi B$ be the canonical purifications of the reduced density of $\psi $ on $HA$ and $HB$, respectively. Using Lemma 2.9,

*μ*, $H\lambda \u2286HA$, and

*A*

^{λ}be as promised by the theorem. Since $dim(HB)\u2265d$, for each

*λ*, we can find a purification $\psi \lambda AB$ of

*ρ*

_{λ}=

*P*

_{λ}/Tr(

*P*

_{λ}) on $HA\u2297HB$; note that $\psi \lambda AB$ is maximally entangled.

*x*, $\u2211aBax\u2264Id$ to bound the second by 1. Using that $Aa\lambda ,x\rho \lambda =\rho \lambda Aa\lambda ,x$, since

*ρ*

_{λ}is totally mixed and $Aa\lambda ,x$ is supported on it, we have that

## ACKNOWLEDGMENTS

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).

## AUTHOR DECLARATIONS

### Conflict of Interest

The author has no conflicts of interest to disclose.

## DATA AVAILABILITY

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

## REFERENCES

_{1}, II

_{∞}, III

_{λ}, λ ≠ 1

In fact, for this game, *c* is only inverse polynomial in *m*, which still suffices, given the form of *κ*.