Research ArticlePHYSICS

Relativistic independence bounds nonlocality

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Science Advances  12 Apr 2019:
Vol. 5, no. 4, eaav8370
DOI: 10.1126/sciadv.aav8370

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 (13). 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 (911), 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: A0/A1 on Alice’s side, B0/B1 on Bob’s side, and C0/C1 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, ΔAi2 and ΔBi2, and the covariances, C(Ai, Bj) ≝ EAiBjEAiEBj, where EAi,EBj, and EAiBj are the respective one- and two-point correlators. Charlie, on the other hand, is far from them (see Fig. 1).

Fig. 1 An illustration of RI in a tripartite scenario.

In a theory obeying generalized uncertainty relations (shown in the bottom right corner in the form of a certain positive semidefinite matrix), RI prevents Bob and Charlie from influencing Alice’s uncertainty relations, e.g., ΔA02ΔA12rjk2, through their choices j and k, i.e., rjk = r. Here, ϱijAB=C(Ai,Bj), ϱikAC=C(Ai,Ck), and ϱjkBC=C(Bj,Ck) illustrated by the arrows are the covariances of Alice-Bob, Alice-Charlie, and Bob-Charlie measurements, respectively. In the quantum mechanical formalism, a similar matrix inequality gives rise to the Schrödinger-Robertson uncertainty relations of Alice’s self-adjoint operators A^0 and A^1, as well as between the nonlocal Alice-Bob operators, A^0B^j and A^1B^j. See Materials and Methods.

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 relationΛABCdef¯¯[ΛCC(B,C)TC(A,C)TC(B,C)ΛBC(A,B)TC(A,C)C(A,B)ΛA]¯0(1)which means that ΛABC 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., ΛA, 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 Ai, Bj, and Ck, hence the natural generalization to other theories.

Provided that Bob measured Bj and Charlie measured Ck, the system as a whole is governed by a submatrix of ΛABCΛABCjkdef¯¯[ΔCk2C(Ck,Bj)C(Ck,A1)C(Ck,A0)C(Ck,Bj)ΔBj2C(Bj,A1)C(Bj,A0)C(Ck,A1)C(Bj,A1)ΔA12rjkC(Ck,A0)C(Bj,A0)rjkΔA02]¯0(2)

Here, rjk is a real number whose value guarantees that ΛABCjk¯0. Therefore, it generally depends not only on Alice’s choices but also on Bob’s j and Charlie’s k. The lower 2 × 2 submatrix in Eq. 2, which is henceforth denoted as the positive semidefinite ΛABCjk, implies that Alice’s measurements satisfy ΔA02ΔA12rjk2, as well as other uncertainty relations that depend on rjk rather than rjk2, i.e., uTΛAjku0, where 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 rjk, which, in general, we are unable to assume. In local hidden-variables theories, where A0 and A1 are classical random variables whose joint probability distribution is well defined, Eq. 2 holds for rjk = C(A0, A1), which is independent of j and k. In quantum mechanics, the Schrödinger-Robertson uncertainty relations show that rjk depends exclusively on Alice’s self-adjoint operators, in particular their commutator and anti-commutator. If Alice and Charlie share a PR-box, then rjk = (–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, ΛAjk¯0, through their j and k. Prohibiting this by requiring that Alice’s uncertainty relation as a whole, i.e., the trio ΔA0, ΔA1, and rjk 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 ΔABCjk, satisfies ΛAjkΛA, for rjkr. Swapping the roles of Alice and Bob, where Alice measures Ai, RI similarly implies ΛBikΛB, for r¯ikr¯. That is, RI means[ΔBj2C(Bj,A1)C(Bj,A0)C(Bj,A1)ΔA12rC(Bj,A0)rΔA02]¯0[ΔAi2C(Ai,B1)C(Ai,B0)C(Ai,B1)ΔB12r¯C(Ai,B0)r¯ΔB02]¯0(3)for 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 ΔA02 and ΔA12, are independent of Bob’s choices. RI, on the other hand, implies that ΛA 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|ϱ00ϱ10ϱ01ϱ11|j=0,1(1ϱ0j2)(1ϱ1j2)|ϱ00ϱ01ϱ10ϱ11|i=0,1(1ϱi02)(1ϱi12)(4)where ϱijC(Ai,Bj)/(ΔAiΔBj) is the Pearson correlation coefficient between Ai and Bj.

It is known that any four correlators, EAiBi, must satisfy Eq. 4 if they are to describe the nonlocality present in a physically realizable quantum mechanical pair of systems (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, EAi=EBj=0, independently by Tsirel’son (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 22 bound (18) on the Bell-CHSH parameter (4), ℬAB ≝ ϱ00 + ϱ10 + ϱ01 − ϱ11, 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 A^0B^j and A^iB^j, where Â1 and B^j are Alice’s and Bob’s self-adjoint operators. See Materials and Methods for the proof of this theorem and for further details.

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 Âi and B^j, the following holds|ϱ00ϱ10ϱ01ϱ11|j=0,1(1ϱ0j2)(1ϱ1j2)ηA^2|ϱ00ϱ01ϱ10ϱ11|i=0,1(1ϱi02)(1ϱi12)ηB^2(5)where ϱij(ÂiB̂jÂiB̂j)/(ΔÂiΔB̂j), and ηX^12i[X^0,X^1]/(ΔX^0ΔX^1), with X^ being either  or B^. Here, [X^0,X^1]X^0X^1X^1X^0 is the commutator of X^0 and X^1, and ΔX^2=X^2X^2 is the variance of X^. The 〈⋅〉 is the quantum mechanical expectation. Note that 12i[X^0,X^1] is self-adjoint and is therefore an observable. Moreover, |ηX^|1, where |ηX^|=1 only if the Robertson uncertainty relation of X^0 and X^1 is saturated. The proof of this theorem is given in Materials and Methods.

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 A^0B^j and A^1B^j in the special case of binary measurements. In other cases, Eq. 3 may be viewed as a generalization of the Schrödinger-Robertson uncertainty relations. As shown in Materials and Methods, inside Hilbert space, Eq. 3 describes two circles in the complex plane: one for j = 0 (red) and another for j = 1 (yellow). The circles are centered at ϱ0jϱ1j, and their respective radii are σ0jσ1j, where σij2=1ϱij2. Alice’s local uncertainty relations are confined to one or another circle depending on Bob’s choice j. Quantum mechanics satisfies RI and thus keeps Alice’s uncertainty relations independent of Bob’s choice, i.e., by allowing only those covariances ϱij 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.

Fig. 2 Geometry of bipartite RI in Hilbert space, the bounds in Eq. 5.

The η is as defined in Theorem 2, and νA^(12{A^0,A^1}A^0A^1)/(ΔA^0ΔA^1), where {X^,Y^} is the anti-commutator. Using these definitions, the Schrödinger-Robertson uncertainty relation between Alice’s observables is vA^2+ηA^21, hence the pair of bluish unit discs. Bob’s choice, j = 1 or j = 0, further confines Alice’s uncertainty, the ηA^ and vA^, to one of the circles, the yellow or the red, respectively. The extent and location of these circles are determined by the nonlocal covariances, ϱij. Quantum mechanics satisfies RI and thus keeps Alice’s uncertainty relations independent of Bob’s choices, i.e., by allowing only those covariances for which the red and yellow circles intersect. Tsirelson’s bound is an extreme configuration where these circles intersect 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 A^0=x^ and A^1=p^ are the position and momentum operators, respectively (see Materials and Methods for the complete derivation), is(22)2+(/2Δx^Δp^)21(6)

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 |AB|+|AC|4, follows as a special case of this inequality. In the same section, it is shown that the correlators in local hidden-variables theories can be similarly bounded by a variant of RI.

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, namelybp(a,b|i,0)=bp(a,b|i,1)def¯¯p(a|i)ap(a,b|0,j)=ap(a,b|1,j)def¯¯p(b|j)(7)

Of course it means that one experimenter’s precision is independent of another experimenter’s choicesΔAi2=Ea2|i,jEa|i,j2=a,ba2p(a,b|i,j)(a,bap(a,b|i,j))2=aa2p(a|i)(aap(a|i))2ΔBj2=Eb2|i,jEb|i,j2=a,bb2p(a,b|i,j)(a,bbp(a,b|i,j))2=bb2p(b|j)(bbp(b|j))2(8)

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 ΛA as a whole, and therefore also rj, 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, ΔAi2, are nevertheless independent of this 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(Bj, Ck) = 0, and Alice and Charlie share a PR-box (1). The PR-boxes define EAiCk=(1)ik, EAi=0, and ECk=0. The variances are thus ΔAi2=EAi2EAi2=1 and ΔCk2=ECk2ECk2=1, and the covariances are C(A1, Ck) = (–1)k and C(A0, Ck) = 1. In this case, a permutation of Eq. 2 readsΛPRjk[ΔBj20C(A1,Bj)C(A0,Bj)011(1)kC(A1,Bj)11rjkC(A0,Bj)(1)krjk1]¯0(9)

NamelyM1ΛPRjkM1=[10ϱ1jABϱ0jAB011(1)kϱ1jAB11rjkϱ0jAB(1)krjk1]¯0(10)where M is a diagonal matrix whose (nonvanishing) terms are all ones but ΔBj. By the Schur complement condition for positive semidefiniteness, Eq. 10 is equivalent to

[1rjkrjk1]¯[ϱ1jABϱ0jAB1(1)k][ϱ1jABϱ0jAB1(1)k]T=[(ϱ1jAB)2ϱ1jABϱ0jABϱ1jABϱ0jAB(ϱ0jAB)2]+[1(1)k(1)k1](11)which renders ϱijAB=0 (positive semidefiniteness of the matrix obtained by subtracting the right-hand side from the left-hand side implies the nonnegativity of its diagonal entries from which this result follows). The inequality Eq. 11 is equivalent to − [rjk − (−1)k]2 ≥ 0 and only holds for rjk = (−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 evaluateA0A1=(A0Ck)(A1Ck)=C(A0,Ck)C(A1,Ck)=(1)0(1)k=(1)k=rjk(12)from which she could tell Charlie’s choice 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 A^i and B^j be self-adjoint operators with ±1 eigenvalues and A^i=B^j=0, whose product, A^iB^j, is similarly self-adjoint. The Schrödinger-Robertson uncertainty relations of the corresponding products, A^0B^j and A^1B^jΔA^0B^j2ΔA^1B^j2(12{A^0,A^1}C(A^0,B^j)C(A^1,B^j))2+(12i[A^0,A^1])2(13)where C(A^i,B^j)=A^iB^j, and the variance, ΔA^iB^j2=1C(A^i,B^j)2, can alternatively be written as[1A^1A^0A^0A^11]¯[C(A^1,B^j)2C(A^1,B^j)C(A^0,B^j)C(A^1,B^j)C(A^0,B^j)C(A^0,B^j)2](14)

By the Schur complement condition for positive semidefiniteness, this is equivalent to[ΔB^j2C(A^1,B^j)C(A^0,B^j)C(A^1,B^j)ΔA^12A^0A^1C(A^0,B^j)A^1A^0ΔA^02]¯0(15)because ΔB^j2=B^j2B^j2=1 and ΔA^i2=A^i2A^i2=1. This, in turn, implies

[ΔB^j2C(A^1,B^j)C(A^0,B^j)C(A^1,B^j)ΔA^12rC(A^0,B^j)rΔA^02]¯0(16)with r = 〈{A^0, A^1}〉/2. The inequalities in Eq. 3 generalize the uncertainty relation Eq. 16 to arbitrary measurements. The inequalities Eqs. 1 and 2 further extend Eq. 16 to include the remaining measurements of Alice, Bob, and Charlie.

Proof of Theorem 1

By the Schur complement condition for positive semidefiniteness, the first condition in Eq. 3 is equivalent to ΛA¯ΔBj2C(A,Bj)C(A,Bj)T. This can be normalizedM1ΛAM1=[1rr1]¯[ϱ1j2ϱ0jϱ1jϱ0jϱ1jϱ0j2]=ΔBj2M1[C(A1,Bj)C(A0,B1)][C(A1,Bj)C(A0,B1)]M1(17)where rrΔA1ΔA0 and M is a diagonal matrix whose (nonvanishing) entries are ΔA1 and ΔA0. This condition is equivalent to|rϱ0jϱ1j|(1ϱ0j2)(1ϱ1j2)(18)which follows from the nonnegative determinant of the matrix obtained by subtracting the right-hand side from the left-hand side in Eq. 17. This, together with the triangle inequality, yield|ϱ00ϱ10r+rϱ01ϱ11||rϱ00ϱ10|+|rϱ01ϱ11|j=0,1(1ϱ0j2)(1ϱ1j2)(19)

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 A^0 and A^1. Similarly, Bob’s measurements are represented by the self-adjoint operators B^j. The Schrödinger-Robertson uncertainty relations of A^0 and A^1 areΔA^02ΔA^12(12{A^0,A^1}A^0A^1)2+(12i[A^0,A^1])2(20)where ΔA^i2=A^i2A^i2 is the variance of A^i. This may alternatively be written as

ΛA^=[ΔA^12rQrQ*ΔA^02]¯0(21)where rQA^1A^0A^1A^0 with rQ* being its complex conjugate. It can be recognized that this leads to Alice’s part in the generalized uncertainty relation in Eq. 2, where rjk=(rQ+rQ*)/2 is independent of j and k.

We shall show that the RI condition, the first inequality in Eq. 3, holds in Hilbert space. This condition tells thatΛA^B^=[ΔB^j2A^1B^jA^1B^jA^0B^jA^0B^jA^1B^jA^1B^jΔA^12A^1A^0A^1A^0A^0B^jA^0B^jA^0A^1A^1A^0ΔA^02],j=0,1(22)where ΔB^j2=B^j2B^j2 is a positive semidefinite matrix. Let U* = [u1, u2, u3] be any 3 × 1 complex-valued vector, and denote |ϕ〉 as the underlying state. Note thatU*ΛA^B^U=V*V0(23)whereVu1(B^jB^j)|ϕ+u2(A^1A^1)|ϕ+u3(A^0A^0)|ϕ(24)which shows that ΛA^B^¯0, and therefore, Eq. 3 holds.

In what follows, we show that ΛA^B^¯0 implies the first bound in Eq. 5. Note thatM1ΛA^B^M1=[1ϱ1jϱ0jϱ1j1rQΔA^1ΔA^0ϱ0jrQ*ΔA^1ΔA^01]¯0,j=0,1(25)where M is a diagonal matrix whose (nonvanishing) entries are ΔB^j, ΔA^1, and ΔA^0. By the Schur complement condition for positive semidefiniteness, Eq. 25 is equivalent to[1ϱ1j2rQΔA^1ΔA^0ϱ1jϱ0jrQ*ΔA^1ΔA^0ϱ1jϱ0j1ϱ0j2]¯0,j=0,1(26)

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

(1ϱ1j2)(1ϱ0j2)({A^0,A^1}/2A^0A^1ΔA^1ΔA^0ϱ0jϱ1j)2+(12i[A^0,A^1]ΔA^1ΔA^0)2,j=0,1(27)namely(1ϱ1j2)(1ϱ0j2)ηA^2|{A^0,A^1}/2A^0A^1ΔA^1ΔA^0ϱ0jϱ1j|,j=0,1(28)where η is as defined in the theorem. This, together with the triangle inequality, implies the first bound in the theorem|ϱ00ϱ10ϱ01ϱ11|j=0,1|{A^0,A^1}/2A^0A^1ΔA^1ΔA^0ϱ0jϱ1j|j=0,1(1ϱ1j2)(1ϱ0j2)ηA^2(29)

The remaining bound similarly follows from the second RI condition in Eq. 3. It is was previously noted that for the case where A^i2=B^j2=I and 〈Ai〉 = 〈Bj〉 = 0, the inequality Eq. 27 coincides with the Schrödinger-Robertson uncertainty relations of A^0B^j and A^1B^j, the inequality Eq. 13.

Nonlocality and Heisenberg uncertainty

An interesting corollary of Theorem 2 is that there is a bound, a generalization of Tsirelson’s 22 bound, for different values of η and ηB^. In particular,||221max{ηA^2,ηB^2}(30)

A geometrical view of this bound is given in Fig. 2. Application of Eq. 30 to A^0=x^ and A^1=p^, the position and momentum operators, yields||221(/2Δx^Δp^)2(31)which follows from the definition of ηA^ and the identity [x^,p^]=i. This elucidates the interplay between the extent of nonlocality and the Heisenberg uncertainty principle. The greater the uncertainty Δx^Δp^, the stronger the nonlocality may get, where Tsirelson’s 22 bound corresponds to the limit Δx^Δp^.

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(22)2+|r|21(32)where, as before, rrΔA1ΔA0. In quantum mechanics where r = rQ in Eq. 21, this relation assumes an explicit form(22)2+|A^0A^1A^0A^1ΔA^0ΔA^1|21(33)

Proof. Assume that ϱij = (−1)ijϱ, a configuration underlying the maximal Bell-CHSH parameter, i.e., ℬ = 4ϱ. RI in Eq. 3 implies Eq. 18, which, in this case, yields[r(1)jϱ2]2(1ϱ2)2(34)

That is|r|2+(22)22(1)jrϱ21(35)where we have used the identity ϱ = ℬ /4. Averaging Eq. 35 for 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 asMQ[M1ΛAM1R0R0T02×202×2M1ΛAM1R1R1T]¯0(36)where M is a diagonal matrix whose (nonvanishing) entries are ΔA1 and ΔA0, and RjT=[ϱ0j,ϱ1j]. RI may further restrict the underlying correlators when the off-diagonal blocks do not vanish. Locality is implied, for example, byML[M1ΛAM1R0R0TR0R1TR1R0TM1ΛAM1R1R1T]¯0(37)

In particularuMLuT=420(38)where u = [1, 1, 1, −1] and ϱ00+ϱ10+ϱ01ϱ11 is the Bell-CHSH parameter.

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 AiBj , i, j = 0, 1, where Ai and Bj are Alice’s and Bob’s measurement outcomes. Therefore, the joint probabilities of A0 and A1, and of B0 and B1, exist, and the correlators satisfy the Bell-CHSH inequality. As mentioned in the main text, here, the parameter r = C(A0 , A1), and ΔA02ΔA12r2. However, this form of the uncertainty relation cannot be saturated but for the trivial case of deterministic A0 and A1.

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 Si and Tj, where the indices i and j represent sets of choices of measuring devices in each region. As in the bipartite case, we may write ΛS(i) and ΛT(j) for the uncertainty relations underlying the sets of measurements i in S and j in T. The covariances between Si and Tj 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[ΛT(0)R0TR0ΛS]¯0,[ΛT(1)R1TR1ΛS]¯0(39)

But also in the converse direction, uncertainty relations in T are independent of choices in S[ΛTR¯0TR¯0ΛS(0)]¯0,[ΛTR¯1TR¯1ΛS(1)]¯0(40)

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 A0 or A1, and Bob and Charlie in T measure (Bl, Ck) or (Bl,Ck), RI in Eq. 39 holds forΛT(0)[ΔCk2C(Ck,Bl)C(Ck,Bl)ΔBl2],ΛT(1)[ΔCk2C(Ck,Bl)C(Ck,Bl)ΔBl2],ΛS[ΔA12rrΔA02](41)whereR0T=[C(A1,Ck)C(A0,Ck)C(A1,Bl)C(A0,Bl)],R1T=[C(A1,Ck)C(A0,Ck)C(A1,Bl)C(A0,Bl)](42)

Theorem 4

The RI condition (Eq. 39) with the matrices in Eqs. 41 and 42 implies|ζ01(l,k)ζ01(l,k)|(1ζ11(l,k))(1ζ00(l,k))+(1ζ11(l,k))(1ζ00(l,k))(43)whereζij(l,k)[ϱikACϱjkACϱlkBCϱilABϱjkACϱlkBCϱjlABϱikAC+ϱilABϱjlAB]/(1(ϱlkBC)2)(44)and ϱijXYC(Xi,Yj)/(ΔXiΔYj). Note that letting ϱAC = ϱBC = 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 yieldsΛABC[ΔCk2C(Ck,Bl)C(Ck,A1)C(Ck,A0)C(Ck,Bl)ΔBl2C(Bl,A1)C(Bl,A0)C(Ck,A1)C(Bl,A1)ΔA12rC(Ck,A0)C(Bl,A0)rΔA02]¯0(45)and similarly for k′ and l′. This is equivalent toM1ΛABCM1=[1ϱlkBCϱ1kACϱ0kACϱlkBC1ϱ1lABϱ0lABϱ1kACϱ1lAB1rϱ0kACϱ0lABr1]¯0(46)where rr/(ΔA1ΔA0) and M is a diagonal matrix whose (nonvanishing) entries are ΔCk, ΔBl, ΔA1, and ΔA0. By the Schur complement condition for positive semidefiniteness, Eq. 46 is equivalent to[1rr1]¯[ϱ1kACϱ0kACϱ1lABϱ0lAB]T[1ϱlkBCϱlkBC1]1[ϱ1kACϱ0kACϱ1lABϱ0lAB](47)which holds if and only if the determinant of the matrix obtained by subtracting the right-hand side from the left-hand side in Eq. 47 is nonnegative. Carrying out this calculation for 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(Ck, Bj) = 0, then by RIAB2+AC28(48)and therefore also |AB|+|AC|4, where both Bell-CHSH parameters, AB and AC , are for the same pair, A0, A1.

Proof. Substituting ϱjkBC=0 in Eq. 47 implies2(1±r)=uT[1rr1]uuT[ϱ1kACϱ1jABϱ0kACϱ0jAB][ϱ1kACϱ1jABϱ0kACϱ0jAB]Tu=[ϱ0jAB±ϱ1jAB]2+[ϱ0kAC±ϱ1kAC]2(49)for uT = [1, ±1]. Therefore4[ϱ00AB±ϱ10AB]2+[ϱ00AC±ϱ10AC]2+[ϱ01AB±ϱ11AB]2+[ϱ01AC±ϱ11AC]212AB2+12AC2(50)from which the theorem follows.

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, C(Mik,Mjl)=0, where Mik stands in for the kth physical variable measured by the ith experimenter. In this case, the generalized uncertainty relations underlying Alice measurements A0, A1 and the n other measurements M1i1,,Mnin are described byEmbedded Image(51) where ϱi,ksC(Ai,Msk)/(ΔAiΔMsk). This matrix is obtained as an extension of Eq. 2 following a normalization similar to the one in previous sections. In this case, Alice’s uncertainty relations are governed by the parameter ri1,,in, which may depend on the choices of all of the other experimenters.

Theorem 6

RI impliess=1n|s|2n(1+r+1r)22n(52)where sϱ0,iss+ϱ1,iss+ϱ0,jssϱ1,jss is the Bell-CHSH parameter of Alice and the sth 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 ri1,,in=rj1,,jn=r. By the Schur complement condition for positive semidefiniteness, Eq. 51 is equivalent to[1rr1]¯s=1n[ϱ0,issϱ1,iss][ϱ0,issϱ1,iss](53)and similarly

[1rr1]¯s=1n[ϱ0,jssϱ1,jss][ϱ0,jssϱ1,jss](54)

Both Eqs. 53 and 54 imply2(1±r)s=1n(ϱ0,iss±ϱ1,iss)2,2(1±r)s=1n(ϱ0,jss±ϱ1,jss)2(55)which are obtained similarly to Eq. 49. By norm equivalence2n(1±r)(s=1n|ϱ0,iss±ϱ1,iss|)2,2n(1±r)(s=1n|ϱ0,jss±ϱ1,jss|)2(56)

Last, invoking the triangle inequalitys=1n|s|s=1n|ϱ0,iss+ϱ1,iss|+|ϱ0,jssϱ1,jss|2n(1+r)+2n(1r)22n(57)

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

Alice’s uncertainty relations are represented by the 2 × 2 lower submatrix ΛA in the generalized uncertainty relation Eq. 3. This shows that Eq. 3 is more stringent than any uncertainty relation derived exclusively from ΛA¯0. Consider, for example, a generalized uncertainty relation of the form[ΛDCCTΛA]¯0(58)where 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 ΛA¯CTΛD1C, which, unless C vanishes, is tighter than ΛA¯0.

As shown in the preceding sections, from within quantum mechanics, the inequality ΛA¯0, which follows from the lower 2 × 2 submatrix in Eqs. 2 and 3, is equivalent to the Schrödinger-Robertson uncertainty relations underlying Alice’s observables A^0 and A^1. That quantum mechanics obey generalized uncertainty relations like Eq. 3, and more generally Eq. 58, implies that any uncertainty relation derived from ΛA¯0 makes only a small part of the story. There are many more restrictions arising from our approach, all of which are tighter than the Schrödinger-Robertson uncertainty relation that are obeyed by Alice’s observables. One such uncertainty relation is given below.

Let D=A^im, where A^i is one of Alice’s observables, i = 0, 1, and m is an integer, m > 1. From within quantum mechanics, the generalized uncertainty Eq. 58 is now given by[ΔA^im2C(A^im,A^1)C(A^im,A^0)C(A^1,A^im)ΔA^12C(A^1,A^0)C(A^0,A^im)C(A^0,A^1)ΔA^02]¯0(59)where C(A^i,A^j)A^iA^jA^iA^j. The quantities ΔA^im2 and C(A^im,A^1) in Eq. 59 involve higher statistical moments of the underlying observables. The inequality Eq. 59 is equivalent toΛA=[ΔA^12C(A^1,A^0)C(A^0,A^1)ΔA^02]¯ΔA^im2[C(A^1,A^im)C(A^0,A^im)][C(A^im,A^1)C(A^im,A^0)](60)by the Schur complement condition for positive semidefiniteness. Let vT[1,±1]/2 and note thatvTΛAv=12ΔA^12+12ΔA^02±[12{A^1,A^0}A^1A^0]12ΔA^im2|C(A^1,A^im)±C(A^0,A^im)|2(61)

ThereforeΔA^12+ΔA^022|12{A^1,A^0}A^1A^0|+1ΔA^im2|C(A^1,A^im)±C(A^0,A^im)|2(62)

This uncertainty relation is to be contrasted withΔA^12+ΔA^022|12{A^1,A^0}A^1A^0|(63)which follows from ΛA¯0 using similar arguments. Note also that much like the Maccone-Pati uncertainty relations (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 rj 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 rj in Alice’s ΛAj.

Lemma 1. There exists an rjk that is independent of j and k such that Eq. 2 holds with C(Ck, Bj) = 0 if and only if the four intervals [djk(−), djk(+)], j, k ∈ {0,1}, with the djk(−) and djk(+) given below, all intersect.djk(±)ϱ0jABϱ1jAB+ϱ0kACϱ1kAC±[1(ϱ0jAB)2(ϱ0kAC)2][1(ϱ1jAB)2(ϱ1kAC)2](64)

Proof. The inequality Eq. 2 may be written asM1ΛABCjkM1=[1ϱjkBCϱ1kACϱ0kACϱjkBC1ϱ1jABϱ0jABϱ1kACϱ1jAB1rjkϱ0kACϱ0jABrjk1]¯0(65)where rjkrjk/(ΔA0ΔA1) and M is a diagonal matrix whose nonvanishing entries are ΔCk, ΔBj, ΔA1, and ΔA0. As ϱ0jBC=C(Ck,Bj)/(ΔBjΔCk)=0, the Schur complement condition for positive semidefiniteness implies that Eq. 65 is equivalent to[1rjkrjk1][ϱ1kACϱ1jABϱ0kACϱ0jAB][ϱ1kACϱ1jABϱ0kACϱ0jAB]T¯0(66)which holds if and only if the diagonal entries obey, 1(ϱ1jAB)2(ϱ1kAC)20,i=0,1, and the determinant of this matrix satisfies[1(ϱ0jAB)2(ϱ0kAC)2][1(ϱ1jAB)2(ϱ1kAC)2](rjkϱ0jABϱ1jABϱ0kACϱ1kAC)20(67)

Namely, Eq. 66 holds if and only if|rjkϱ0jABϱ1jABϱ0kACϱ1kAC|[1(ϱ0jAB)2(ϱ0kAC)2][1(ϱ1jAB)2(ϱ1kAC)2](68)for j, k ϵ {0, 1}. It thus follows that rjkϵ|djk(),djk(+)|. If these intervals all intersect, then there is r and rr/ΔA0ΔA1, which are independent of j, k such that rjk=r. In particularmaxj,kdjk()rminj,kdjk(+)(69)

Conversely, if there is such rjk=r, then the underlying intervals necessarily intersect.

Lemma 1 shows that in the absence of Charlie, ϱ1kAC=ϱjkBC=0, the parameter rj in a bipartite Alice-Bob setting satisfiesϱ0jϱ1j(1ϱ0j2)(1ϱ1j2)rjϱ0jϱ1j+(1ϱ0j2)(1ϱ1j2)(70)where ϱ1j=C(Ai,Bj)/(ΔAiΔBj) and rjrj/(ΔA0ΔA1).

Let Dj be the range of admissible rj in Eq. 70. Unless D0D1 ≠ ∅, RI cannot be satisfied. We shall show that whenever the two intervals D0 and D1 do not intersect, in which case RI fails, signaling takes place. DefineϵminwjDj|w0w1|(71)

It can be recognized that this ϵ is the smallest of the four possible numbersϵ=|ϱ00ϱ10ϱ01ϱ11±(1ϱ002)(1ϱ102)±(1ϱ012)(1ϱ112)|(72)

Assume now that the intervals D0 and D1 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 ΔA2(τ). This ΔA2(τ) can be evaluated empirically by measuring 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 evaluateg(θ,τ)cos(θ)2ΔA0(τ)ΔA1(τ)+sin(θ)2ΔA1(τ)ΔA0(τ)(73)

Her uncertainty relation Eq. 1 dictates that this quantity is bounded from belowminτg(θ,τ)max{0,rjsin(2θ)}(74)which follows from [cosθ,sinθ]ΛAj[cosθ,sinθ]T0. That Alice may reach rj means that for some θ, a subset of parameters τ* saturating Eq. 74 existsminτg(θ,τ)=(θ,τ*)=rjsin(2θ)(75)which also implies that ΛAj is a singular matrix and therefore ΛA02(τ*)ΛA12(τ*)=rj2.

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 jth 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 rj, an estimate of rj, using the approximated minimum of g(θ, τ) . In practice, her estimate is accurate up to an error term, δj, of the order 𝒪(1/N), i.e., rj=rj+δj. It now follows that for sufficiently large N|r0r1|=|r0r1+δ0δ1||ϵ+𝒪(1/N)|(76)

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.
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