## Abstract

If nature allowed nonlocal correlations other than those predicted by quantum mechanics, would that contradict some physical principle? Various approaches have been put forward in the past two decades in an attempt to single out quantum nonlocality. However, none of them can explain the set of quantum correlations arising in the simplest scenarios. Here, it is shown that generalized uncertainty relations, as well as a specific notion of locality, give rise to both familiar and new characterizations of quantum correlations. In particular, we identify a condition, relativistic independence, which states that uncertainty relations are local in the sense that they cannot be influenced by other experimenters’ choices of measuring instruments. We prove that theories with nonlocal correlations stronger than the quantum ones do not satisfy this notion of locality, and therefore, they either violate the underlying generalized uncertainty relations or allow experimenters to nonlocally tamper with the uncertainty relations of their peers.

## INTRODUCTION

Quantum mechanics stands out in enabling strong, nonlocal correlations between remote parties. On the one hand, these quantum correlations cannot, in any way, be explained by models of classical physics. On the other hand, quantum theory remains rather elusive about their physical origin (*1*–*3*). If nature allowed nonlocal correlations other than those predicted by quantum mechanics, would that break any known physical principle? This question becomes all more important when the predictions of quantum mechanics are experimentally verified time and again.

Initially, it was speculated that those correlations excluded by quantum mechanics violate relativistic causality—the principle that dictates that experiments can be influenced only by events in their past light cone and can influence events only in their future light cone. However, it was shown that other theories may exist, whose correlations, while not realizable in quantum mechanics, are nevertheless nonsignaling and are hence consistent with relativistic causality (*1*).

Over the past 20 years, many efforts have been invested in a line of research aimed at quantitatively deriving the strength of quantum correlations from basic principles. For example, it was shown that violations of the Bell-CHSH (Clauser-Horne-Shimony-Holt) inequality (*4*) beyond the quantum limit, known as Tsirelson’s bound, are inconsistent with the uncertainty principle (*5*). Popescu-Rohrlich–boxes (PR-boxes), the hypothetical models achieving the maximal violation of the Bell-CHSH inequality (*1*), would allow distributed computation to be performed with only one bit of communication (*6*), which looks unlikely but does not violate any known physical law. Similarly, in stronger-than-quantum nonlocal theories, some computations exceed reasonable performance limits (*7*), and there is no sensible measure of mutual information between pairs of systems (*8*). Last, it was shown that superquantum nonlocality does not permit classical physics to emerge in the limit of infinitely many microscopic systems (*9*–*11*), and also violates the exclusiveness of local measurement outcomes in multipartite settings (*12*). However, none of these and other principles that have been proposed (*2*) can explain the set of one- and two-point correlators that fully characterize the quantum probability distributions witnessed in the simplest bipartite two-outcome scenario.

A consequence of relativistic causality within the framework of probabilistic theories is known as the no-signaling condition—the local probability distributions of one experimenter (marginal probabilities) are independent of another experimenter’s choices (*1*). While the no-signaling condition is insufficient to single out quantum correlations, it is shown here that an analogous requirement applicable in conjunction with generalized uncertainty relations is satisfied exclusively by quantum mechanical correlations.

## RESULTS

In what follows, we first assume (in the next section) that generalized uncertainty relations are valid within the theory in question. These uncertainty relations broaden the meaning of uncertainty beyond the realm of quantum mechanics and give rise to the Schrödinger-Robertson uncertainty relation when applied to the latter. Then, in the “Independence” section, we additionally assume a certain form of independence—we name relativistic independence (RI)—meaning here that local uncertainty relations cannot be affected at a distance. The above assumptions accord well with experimental observations but generalize the underlying theoretical model beyond the quantum formalism.

### Generalized uncertainty relations

Three experimenters—Alice, Bob, and Charlie—perform an experiment, where each of them owns a measuring device. On each such device, a knob determines its mode of operation, either “0” or “1,” which allows the measurement of two physical variables: *A*_{0}/*A*_{1} on Alice’s side, *B*_{0}/*B*_{1} on Bob’s side, and *C*_{0}/*C*_{1} on Charlie’s side. Alice and Bob are close to one another, and so, they use the readings from all their devices to empirically evaluate the variances, **C**(*A*_{i}, *B*_{j}) ≝

Assume that measurements of physical variables are generally inflicted with uncertainty. This uncertainty not only affects pairs of local measurements performed by individual experimenters but also governs any number of measurements performed by groups of remote experimenters. In our tripartite setting, for example, the measurements of Alice, Bob, and Charlie are assumed to be jointly governed by the generalized uncertainty relation_{ABC} is a positive semidefinite matrix. Here, **C**(*A*, *B*), **C**(*A*, *C*), and **C**(*B*, *C*) are the empirical covariance matrices of Alice-Bob, Alice-Charlie, and Bob-Charlie measurements. The diagonal submatrices, e.g., Λ_{A}, represent the uncertainty relations governing the individual experimenters. Below and in Materials and Methods, Eq. 1 is shown to imply the quantum mechanical Schrödinger-Robertson uncertainty relations (*13*), as well as their multipartite nonquantum generalizations. Moreover, in local hidden-variables theories where all measurement outcomes preexist, Eq. 1 coincides with a covariance matrix, which is, by construction, positive semidefinite and represents the uncertainty of *A*_{i}, *B*_{j}, and *C*_{k}, hence the natural generalization to other theories.

Provided that Bob measured *B*_{j} and Charlie measured *C*_{k}, the system as a whole is governed by a submatrix of Λ_{ABC}

Here, *r*_{jk} is a real number whose value guarantees that *j* and Charlie’s *k*. The lower 2 × 2 submatrix in Eq. 2, which is henceforth denoted as the positive semidefinite *r*_{jk} rather than *u* is any two-dimensional real-valued vector.

Local hidden-variables theories, quantum mechanics, and nonquantum theories such as the hypothetical PR-boxes (*1*) obey Eq. 2. Moreover, they provide different closed forms for this *r*_{jk}, which, in general, we are unable to assume. In local hidden-variables theories, where *A*_{0} and *A*_{1} are classical random variables whose joint probability distribution is well defined, Eq. 2 holds for *r*_{jk} = **C**(*A*_{0}, *A*_{1}), which is independent of *j* and *k*. In quantum mechanics, the Schrödinger-Robertson uncertainty relations show that *r*_{jk} depends exclusively on Alice’s self-adjoint operators, in particular their commutator and anti-commutator. If Alice and Charlie share a PR-box, then *r*_{jk} = (–1)^{k}, which, in contrast to the other two theories, depends on *k*.

### Independence

In the above setting, Bob and Charlie may be able to nonlocally tamper with Alice’s uncertainty relation, *j* and *k*. Prohibiting this by requiring that Alice’s uncertainty relation as a whole, i.e., the trio *r*_{jk} would be independent of Bob’s *j* and Charlie’s *k*, leads to the set of quantum mechanical one- and two-point correlators. This condition is named RI.

By RI, the Alice-Bob system, which is governed by the lower 3 × 3 submatrix of *r*_{jk} ≝ *r*. Swapping the roles of Alice and Bob, where Alice measures *A*_{i}, RI similarly implies *i*, *j* ∈ {0, 1}. RI (Eq. 3) and the no-signaling condition are distinct and do not follow from one another. The no-signaling condition, for example, dictates that the (marginal) probability distributions of Alice’s measurements, and therefore also _{A} in its entirety must be independent of Bob’s choices, which may hold whether or not Alice’s marginal probabilities are independent of *j*. The relationship between the two conditions is discussed in more detail in Materials and Methods.

PR-boxes satisfy the no-signaling condition but violate RI (see Materials and Methods). Moreover, as stated below, RI (Eq. 3) is satisfied exclusively by the quantum mechanical bipartite one- and two-point correlators.

**Theorem 1.** The conditions (Eq. 3) imply*A*_{i} and *B*_{j}.

It is known that any four correlators, *3*). In addition, all the sets of these correlators permitted by Eq. 4 are possible within quantum mechanics. This result was proven when assuming quantum mechanics and vanishing one-point correlators, *14*), Landau (*15*), and Masanes (*16*). More recently, Eq. 4 has been derived for the case of binary measurements from the first level of the Navascues-Pironio-Acin (NPA) hierarchy (*17*). We show without assuming any of these that this bound (in the form of Landau) originates from RI (Eq. 3). Moreover, it is now clear that Eq. 4 must hold not only for binary but also for other, both discrete and continuous, variables. Consequently, Tsirelson’s *18*) on the Bell-CHSH parameter (*4*), ℬ_{AB} ≝ ϱ_{00} + ϱ_{10} + ϱ_{01} − ϱ_{11}, applies to any type of measurement. For example, Alice’s and Bob’s measurements may be the position and momentum of some wave function. Quantum theory satisfies the RI condition (Eq. 3) and is therefore subject to Eq. 4. Furthermore, in the case of binary ±1 measurements whose one-point correlators vanish, the first Alice-Bob uncertainty relation in Eq. 3 is given in quantum mechanics by the Schrödinger-Robertson uncertainty relations of _{1} and

Surprisingly, within the quantum formalism Eq. 4 is a special case of another bound with two extra terms.

**Theorem 2.** In quantum theory, where the Alice and Bob measurements are represented by the self-adjoint operators Â_{i} and

### Local uncertainty relations and nonlocal correlations

The geometry of bipartite RI in Hilbert space is illustrated in Fig. 2. The left picture of Fig. 2 is the geometry underlying the first bound in Eq. 5. This bound arises from the two uncertainty relations (Eq. 3), which, from within quantum mechanics, coincide with the Schrödinger-Robertson uncertainty relations of *j* = 0 (red) and another for *j* = 1 (yellow). The circles are centered at ϱ_{0j}ϱ_{1j}, and their respective radii are σ_{0j}σ_{1j}, where *j*. Quantum mechanics satisfies RI and thus keeps Alice’s uncertainty relations independent of Bob’s choice, i.e., by allowing only those covariances ϱ_{ij} for which the red and yellow circles intersect. Tsirelson’s bound (the right picture of Fig. 2), for example, is attained when the region of intersection collapses to a single point at the origin.

RI implies that the extent of nonlocality is governed by local uncertainty relations. The interplay between nonlocality as quantified by the Bell-CHSH parameter, ℬ, and Heisenberg uncertainty, where

It is known that a complete characterization of the set of quantum correlations must follow from inherently multipartite principles (*19*). As shown in Materials and Methods, RI applies to any number of parties with any number of measuring devices. This allows us, for example, to derive a generalization of Eq. 4 for the Alice-Bob, Alice-Charlie, and Bob-Charlie one- and two-point correlators in a tripartite scenario. The property known as monogamy of correlations, the

## DISCUSSION

Within a class of theories obeying generalized uncertainty relations, RI was shown to reproduce the complete quantum mechanical characterization of the bipartite correlations in two-outcome scenarios, and potentially in much more general cases as straightforward corollaries of our approach. To fully characterize the set of quantum correlations would generally require analyzing the uncertainty relation Eq. 1 in an elaborate multipartite setting, accounting for all the parties’ cross-correlations and assuming RI (this point, as well as some other technical issues, is discussed in detail in Materials and Methods). All these imply that stronger-than-quantum nonlocal theories may either be incompatible with the uncertainty relations analyzed above or allow experimenters to nonlocally tamper with the uncertainty relations of other experimenters.

## MATERIALS AND METHODS

### No-signaling condition and RI

A consequence of relativistic causality in probabilistic theories is the no-signaling condition (*1*). Consider the Bell-CHSH setting where *a* and *b* are the outcomes of Alice’s and Bob’s measurements. The joint probability of these outcomes when Alice measured using device *i* and Bob measured using device *j* is denoted as *p*(*a*, *b* | *i*, *j*). The no-signaling condition states that one experimenter’s marginal probabilities are independent of another experimenter’s choices, namely

Of course it means that one experimenter’s precision is independent of another experimenter’s choices

The no-signaling condition thus implies that the variances of one experimenter in the Alice-Bob uncertainty relations (Eq. 3) are independent of the other experimenter’s choices.

RI implies that one experimenter’s uncertainty relation is altogether independent of the other experimenter’s choices, i.e., that Λ_{A} as a whole, and therefore also *r*_{j}, are independent of *j*. This does not necessarily imply the no-signaling condition, as there may exist, for example, marginal distributions *p*(*a* | *i*, *j*) that depend on Bob’s *j* whose variances, *j*. This shows that RI does not at all require us to assume the no-signaling condition.

### PR-boxes violate RI

Consider a tripartite setting where Bob and Charlie are uncorrelated, **C**(*B*_{j}, *C*_{k}) = 0, and Alice and Charlie share a PR-box (*1*). The PR-boxes define **C**(*A*_{1}, *C*_{k}) = (–1)^{k} and **C**(*A*_{0}, *C*_{k}) = 1. In this case, a permutation of Eq. 2 reads

Namely*M* is a diagonal matrix whose (nonvanishing) terms are all ones but

*r*_{jk} − (−1)^{k}]^{2} ≥ 0 and only holds for *r*_{jk} = (−1)^{k}. Such a theory therefore violates RI.

However, the PR-box example teaches us something profound. In this model, complementarity (i.e., the inability to measure both local variables in the same experiment) must be assumed in both Alice’s and Charlie’s ends; otherwise, Alice, for example, may evaluate*k*. Lack of complementarity immediately leads to signaling in the case of PR-boxes, but as we have seen, the weaker assumption of uncertainty leads to a problem with RI.

### Schrödinger-Robertson uncertainty relations and the generalized uncertainty relations Eqs. 1 to 3

Let

By the Schur complement condition for positive semidefiniteness, this is equivalent to

*r* = 〈{

### Proof of Theorem 1

By the Schur complement condition for positive semidefiniteness, the first condition in Eq. 3 is equivalent to *M* is a diagonal matrix whose (nonvanishing) entries are

The second inequality in Eq. 4 is similarly obtained by swapping the roles of Alice and Bob, i.e., from the second RI condition in Eq. 3.

### Proof of Theorem 2

In the Hilbert space formulation of quantum mechanics, Alice’s measurements are represented by the self-adjoint operators

*j* and *k*.

We shall show that the RI condition, the first inequality in Eq. 3, holds in Hilbert space. This condition tells that*U** = [*u*_{1}, *u*_{2}, *u*_{3}] be any 3 × 1 complex-valued vector, and denote |ϕ〉 as the underlying state. Note that

In what follows, we show that *M* is a diagonal matrix whose (nonvanishing) entries are

This, in turn, is equivalent to the requirement that the determinant of this matrix is nonnegative, i.e., that

_{Â} is as defined in the theorem. This, together with the triangle inequality, implies the first bound in the theorem

The remaining bound similarly follows from the second RI condition in Eq. 3. It is was previously noted that for the case where *A*_{i}〉 = 〈*B*_{j}〉 = 0, the inequality Eq. 27 coincides with the Schrödinger-Robertson uncertainty relations of

### Nonlocality and Heisenberg uncertainty

An interesting corollary of Theorem 2 is that there is a bound, a generalization of Tsirelson’s _{Â} and

A geometrical view of this bound is given in Fig. 2. Application of Eq. 30 to

More generally, RI implies a close relationship between nonlocality as quantified by the Bell-CHSH parameter and the uncertainty parameter *r* in Eq. 3. This is summarized in the next theorem.

### Theorem 3

By RI*r* = *r*_{Q} in Eq. 21, this relation assumes an explicit form

*Proof*. Assume that ϱ_{ij} = (−1)^{ij}ϱ, a configuration underlying the maximal Bell-CHSH parameter, i.e., ℬ = 4ϱ. RI in Eq. 3 implies Eq. 18, which, in this case, yields

That is*j* = 0 and *j* = 1 implies the theorem.

### Locality from RI

The preceding sections forged a theory-free notion of nonlocality in the form of correlators that satisfy RI. Can locality (as appearing in classical statistical theories), which is normally defined by means of Bell inequalities, be similarly characterized? We will show that locality is, in some sense, a variant of RI.

The first RI condition in Eq. 3 may alternatively be written as*M* is a diagonal matrix whose (nonvanishing) entries are

In particular*u* = [1, 1, 1, −1] and

The nonvanishing off-diagonal matrices in Eq. 37 essentially render the underlying uncertainty relations of both experimenters ineffective. To see how, note that the matrix in Eq. 37 (but not that in Eq. 36) is the covariance of the four products *A*_{i}*B*_{j} , *i*, *j* = 0, 1, where *A*_{i} and *B*_{j} are Alice’s and Bob’s measurement outcomes. Therefore, the joint probabilities of *A*_{0} and *A*_{1}, and of *B*_{0} and *B*_{1}, exist, and the correlators satisfy the Bell-CHSH inequality. As mentioned in the main text, here, the parameter *r* = **C**(*A*_{0} , *A*_{1}), and *A*_{0} and *A*_{1}.

### RI in general multipartite settings

Suppose that some experimenters are located at spacetime region *S* and some others at spacetime region *T*. Each experimenter has an arbitrary number of measuring devices. We shall denote the vectors of measurements in *S* and *T* by *S*_{i} and *T*_{j}, where the indices *i* and *j* represent sets of choices of measuring devices in each region. As in the bipartite case, we may write Λ_{S}(*i*) and Λ_{T}(*j*) for the uncertainty relations underlying the sets of measurements *i* in *S* and *j* in *T*. The covariances between *S*_{i} and *T*_{j} may similarly be expressed by a matrix *R*.

RI dictates that uncertainty relations in *S* are independent of choices in *T*. Therefore, *S* is independent of whether *j* = 0 or *j* = 1 in *T*. This is expressed mathematically by

But also in the converse direction, uncertainty relations in *T* are independent of choices in *S*

Below, we use these to derive a bound on the quantum mechanical, Alice-Bob, Alice-Charlie, and Bob-Charlie, one- and two-point correlators. The relation thus obtained generalizes Eq. 4 in this tripartite setting.

We note that Eqs. 39 and 40 do not represent the most general approach for characterizing nonlocal correlations. Nevertheless, they facilitate analyses and particularly the derivation of the theorems that follow. A complete characterization of the set of quantum correlations would require analyzing Eq. 1 in a general multipartite setting. In such a case, the cross-correlations between the *S* and *T* subsets would have to be accounted for. To some degree, this is practiced in the derivation of Theorem 4, where it is assumed that Bob and Charlie are correlated. Disconnecting them by making their correlations zero leads to the well-known monogamy relation in Theorem 5.

In the tripartite case, where Alice in *S* measures either *A*_{0} or *A*_{1}, and Bob and Charlie in *T* measure (*B*_{l}, *C*_{k}) or

### Theorem 4

The RI condition (Eq. 39) with the matrices in Eqs. 41 and 42 implies^{AC} = ϱ^{BC} = 0 in Eq. 43 recovers the bound on the Alice-Bob correlators, the first inequality in Eq. 4.

*Proof*. Substituting Eq. 41 into Eq. 39 yields*k*′ and *l*′. This is equivalent to*M* is a diagonal matrix whose (nonvanishing) entries are *k*,*l* and then for *k*′,*l*′ and invoking the triangle inequality yield Eq. 43.

The next theorem shows that the bound Eq. 43 implies monogamy of correlations. This means that breaking of monogamy necessarily violates RI.

### Theorem 5

If Charlie and Bob are uncorrelated, **C**(*C*_{k}, *B*_{j}) = 0, then by RI*A*_{0}, *A*_{1}.

*Proof*. Substituting *u*^{T} = [1, ±1]. Therefore

### Monogamy of correlations in general multipartite settings

The above result is a special case of the more general scenario where any number of experimenters is correlated with Alice but uncorrelated among themselves. Suppose that there are *n* experimenters whose measurements are uncorrelated, *k*th physical variable measured by the *i*th experimenter. In this case, the generalized uncertainty relations underlying Alice measurements *A*_{0}, *A*_{1} and the *n* other measurements

### Theorem 6

*RI implies**s*th experimenter. Tsirelson’s bound and the monogamy property of correlations follow from this inequality as special cases for *n* = 1 and *n* = 2, respectively.

*Proof*. If RI holds, then

Both Eqs. 53 and 54 imply

Last, invoking the triangle inequality

### Tighter than Schrödinger-Robertson uncertainty relations following from Eq. 3

Alice’s uncertainty relations are represented by the 2 × 2 lower submatrix Λ_{A} in the generalized uncertainty relation Eq. 3. This shows that Eq. 3 is more stringent than any uncertainty relation derived exclusively from *D* is an invertible *n* × *n* matrix and *C* is *n* × 2 cross-covariance matrix. By the Schur complement condition for positive semidefiniteness, this inequality is equivalent to *C* vanishes, is tighter than

As shown in the preceding sections, from within quantum mechanics, the inequality

Let *i* = 0, 1, and *m* is an integer, *m* > 1. From within quantum mechanics, the generalized uncertainty Eq. 58 is now given by

Therefore

This uncertainty relation is to be contrasted with*20*), these additive inequalities do not become trivial in the case where the state coincides with an eigenvector of one of the observables.

### The measurability of *r*_{j} in a bipartite setting

In what follows, we examine RI from a different perspective. As mentioned in the main text, this condition may be viewed as the requirement that one experimenter’s uncertainty relations are independent of another experimenters’ choices. We claim that if it were not so, relativistic causality would have been necessarily violated. Our argument is based on the measurability of *r*_{j} in Alice’s

**Lemma 1.** There exists an *r*_{jk} that is independent of *j* and *k* such that Eq. 2 holds with **C**(*C*_{k}, *B*_{j}) = 0 if and only if the four intervals [*d*_{jk}(−), *d*_{jk}(+)], *j*, *k* ∈ {0,1}, with the *d*_{jk}(−) and *d*_{jk}(+) given below, all intersect.

*Proof*. The inequality Eq. 2 may be written as*M* is a diagonal matrix whose nonvanishing entries are

Namely, Eq. 66 holds if and only if*j*, *k ϵ* {0, 1}. It thus follows that *r* and *j*, *k* such that

Conversely, if there is such

Lemma 1 shows that in the absence of Charlie, *r*_{j} in a bipartite Alice-Bob setting satisfies

Let *D*_{j} be the range of admissible *r*_{j} in Eq. 70. Unless *D*_{0} ∩ *D*_{1} ≠ ∅, RI cannot be satisfied. We shall show that whenever the two intervals *D*_{0} and *D*_{1} do not intersect, in which case RI fails, signaling takes place. Define

It can be recognized that this ϵ is the smallest of the four possible numbers

Assume now that the intervals *D*_{0} and *D*_{1} do not intersect and thus ϵ > 0. Here is a procedure that Alice may, in principle, follow for detecting a signal from Bob using her local measurements. Let τ be a set of local parameters describing Alice’s nontrivial system (for practical reasons, τ can be discretized). The precision is represented for any physical variable *A* by the variance *A* in many trials of an experiment while reproducing time and again the same set τ.

For any real parameter θ ϵ [–π, π], Alice is able to evaluate

Her uncertainty relation Eq. 1 dictates that this quantity is bounded from below

Suppose that Alice and Bob agree in advance to repeat the underlying experiment *N* times, for a sufficiently large *N*. Alice may choose a new set τ and a device with which to measure in the beginning of each trail. All this time, Bob uses only one of his devices, say the *j*th one. Using the measurement outcomes from all these trails, Alice may approximate *g*(θ, τ) for each τ in the domain of these parameters. According to Eq. 75, Alice may then evaluate *g*(θ, τ) . In practice, her estimate is accurate up to an error term, δ_{j}, of the order *N*

Alice may therefore be able to evaluate a number whose magnitude is as large as ϵ and whose sign tells whether Bob measured first using *j* = 0 and then using *j* = 1 or the opposite. Of course, if independence holds, in which case ϵ = 0, Alice will not detect any signal from Bob via her local uncertainty relations.

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## REFERENCES AND NOTES

**Acknowledgments:**We are grateful to N. Gisin and A. Elitzur for helpful discussions and comments that improved the overall presentation of the idea. We especially wish to express our gratitude to Y. Aharonov for many insightful discussions. In addition, we wish to thank anonymous reviewers for very helpful comments and suggestions.

**Funding:**A.C. acknowledges support from Israel Science Foundation grant 1723/16. E.C. acknowledges support from the Engineering Faculty in Bar Ilan University.

**Author contributions:**Both authors developed the concepts and worked out the mathematical proofs.

**Competing interests:**The authors declare that they have no competing interests.

**Data and materials availability:**All data needed to evaluate the conclusions in the paper are present in the paper. Additional data related to this paper may be requested from the authors.

- Copyright © 2019 The Authors, some rights reserved; exclusive licensee American Association for the Advancement of Science. No claim to original U.S. Government Works. Distributed under a Creative Commons Attribution NonCommercial License 4.0 (CC BY-NC).