A appendices

# Elemental and tight monogamy relations in nonsignalling theories

## Abstract

Physical principles constrain the way nonlocal correlations can be distributed among distant parties. These constraints are usually expressed by monogamy relations that bound the amount of Bell inequality violation observed among a set of parties by the violation observed by a different set of parties. We prove here that much stronger monogamy relations are possible for nonsignalling correlations by showing how nonlocal correlations among a set of parties limit any form of correlations, not necessarily nonlocal, shared among other parties. In particular, we provide tight bounds between the violation of a family of Bell inequalities among an arbitrary number of parties and the knowledge an external observer can gain about outcomes of any single measurement performed by the parties. Finally, we show how the obtained monogamy relations offer an improvement over the existing protocols for device-independent quantum key distribution and randomness amplification.

Introduction. It is a well established fact that entanglement and nonlocal correlations (cf. Refs. (1); (2)), i.e., correlations violating a Bell inequality (3), are fundamental resources of quantum information theory. It has been confirmed by many instances that, when distributed among spatially separated observers, they give an advantage over classical correlations at certain information-theoretic tasks, many of them being considered in the multipartite scenario. For instance, nonlocal correlations outperform their classical counterpart at communication complexity problems (4), and allow for security not achievable within classical theory (5).

Physical principles impose certain constraints on the way these resources can be distributed among separated parties; these are commonly referred to as monogamy relations. For instance, in any three–qubit pure state one party cannot share large amount of entanglement, as measured by concurrence, simultaneously with both remaining parties (6). Analogous monogamy relations, both in qualitative (7); (8); (9); (10) and quantitative (11); (12) form, were demonstrated for nonlocal correlations, with the measure of nonlocality being the violation of specific Bell inequalities. In particular, Toner and Verstraete (11) and later Toner (12) showed that if three parties , , and share, respectively, quantum and general nonsignalling correlations, then only a single pair can violate the Clauser-Horne-Shimony-Holt (CHSH) Bell inequality (13). These findings were generalized to more complex scenarios (14); (15) (see also Ref. (16)), and in particular in (14) a general construction of monogamy relations for nonsignalling correlations from any bipartite Bell inequality was proposed.

In this work, we demonstrate that nonsignalling correlations are monogamous in a much stronger sense: the amount of nonlocality observed by a set of parties may imply severe limitations on any form of correlations with other parties. That is, instead of comparing nonlocality between distinct groups of parties, we rather relate it to the knowledge that external parties can gain on outcomes of any of the measurements performed by the parties (see Fig. 1). To be more illustrative, consider again parties , , and performing a Bell experiment with observables and outcomes. We construct tight bounds between the violation of certain Bell inequalities (9) among any pair of parties, say and , and classical correlations that the third party can establish with outcomes of any measurement performed by or . This means that the amount of any correlations — classical or nonlocal — that could share with or is bounded by the strength of the Bell inequality violation between and . Our monogamies are further generalized to the scenario with an arbitrary number of parties [ scenario] with nonlocality measured by the recent generalization of the Bell inequalities (9) presented in Ref. (10). The obtained monogamy relations are logically independent from, and in fact stronger than, the existing relations involving only nonlocal correlations, as a bound on nonlocal correlations does not necessarily imply any nontrivial constraint on the amount of classical correlations.

Our new monogamy relations prove useful in device-independent protocols (17); (18); (19); (20); (21). First, we show that they impose tight bounds on the guessing probability, the commonly used measure of randomness, that are significantly better than the existing ones (9); (10). We then argue that this translates into superior performance in protocols for device-independent quantum key distribution (DIQKD) (29) using measurements with more than two outputs. Finally, we show that they allow for a generalization of the results of (19) on randomness amplification to any number of parties and outcomes, demonstrating, in particular, that arbitrary amount of arbitrarily good randomness can be amplified in a bipartite setup.

Before turning to the results, we provide some background. Consider parties (for denoted by ), each measuring one of possible observables with outcomes (enumerated by ) on their local physical systems. The produced correlations are described by a collection of probabilities of obtaining results upon measuring . One then says that the correlations are (i) nonsignalling (NC) if any of the marginals describing a subset of parties is independent of the measurements choices made by the remaining parties and (ii) quantum (QC) if they arise by local measurements on quantum states (cf. (2)).

Elemental and tight monogamies for nonsignalling correlations. We start with the derivation of our monogamy relations in the case of nonsignalling correlations. For clarity, we begin with the simplest tripartite scenario. We will use the Bell inequality introduced by Barrett, Kent, and Pironio (BKP) (9). Denoting by the mean value of a random variable , that is, , it reads

 I2,M,dAB:=M∑α=1(⟨[Aα−Bα]⟩+⟨[Bα−Aα+1]⟩)≥d−1 (1)

with being modulo , and . For , Ineq. (1) reproduces the chained Bell inequalities (23), while for the Collins-Gisin-Linden-Massar-Popescu (CGLMP) inequalities (24). The maximal nonsignalling violation of (1) is .

The only monogamy relations for (1) have been formulated in terms of its violations between Alice and Bobs (14), which is a natural quantitative extension of the concept of -shareability (7). In the following theorem we show that the BKP Bell inequalities allow one to introduce elemental monogamies obeyed by any NC.

###### Theorem 1.

For any tripartite NC with -outcome measurements, the inequality

 I2,M,dAB+⟨[Xi−Cj]⟩+⟨[Cj−Xi]⟩≥d−1 (2)

holds for any pair and denoting or .

Interestingly, all these inequalities are tight in the sense that for any values of and saturating (2), one can find NC realizing these values. Take, for instance, a probability distribution , with being a mixture of a nonlocal model maximally violating (1) and a local deterministic one saturating it. Then, is the same distribution as the one used by or in the local model saturating (1).

The physical interpretation of our monogamies can be now concluded if we rewrite them in a bit different form. Using the fact that for any variable , (25), Ineqs. (2) transform to

 I2,M,dAB+1≥dp(Xi=Cj) (3)

for , and any pair . These relations hold if is replaced by any pair of parties and if any is added modulo to the argument of probability. The meaning of the introduced monogamy relations is now transparent. The probability that parties and obtain the same results upon measuring the th and th observables is a measure of how the outcomes of these measurements are classically correlated. Consequently, Ineqs. (2) establish trade-offs between nonlocality, as measured by (1), that can be generated between any two parties and classical correlations that the third party can share with the results of any measurement performed by any of these two parties. Furthermore, they are tight. In fact, it is known that the maximal NC violation of (1), , implies for any , meaning that at the point of maximal violation cannot share any correlations with any other party’s measurement outcomes (9). On the other hand, it is well known that at the point of no violation can be arbitrarily correlated with and . For intermediate violations, the best one can hope for is a linear interpolation between these two extreme values and this is precisely what our monogamy relations predict, see Fig. 2.

Let us now move to the general case of an arbitrary number of parties each having -outcome observables at their disposal. We will utilize the generalization of the Bell inequality (1) introduced in Ref. (10), which can be stated as

 IN,M,dA≥d−1 (4)

with . Since the form of is rather lengthy and actually not relevant for further considerations, for clarity, we omit presenting it here (see (25)). We only mention that it can be recursively determined from and that its minimal nonsignalling value is . Then, the generalization of Theorem 1 to arbitrary goes as follows.

###### Theorem 2.

For any –partite NC with -outcome measurements per site, the following inequality

 IN,M,dA+⟨[A(k)xk−A(N+1)xN+1]⟩+⟨[A(N+1)xN+1−A(k)xk]⟩≥d−1 (5)

is satisfied for any and .

All the properties of the three-partite monogamy relations persist for any . In particular, all inequalities (5) are tight. Moreover, they can be rewritten as

 IN,M,dA+1≥dp(A(k)xk=[A(N+1)xN+1+m]) (6)

for any , and and remain valid if the nonlocality is tested among any -element subset of parties. Analogously to the three-partite case, Ineqs. (6) tightly relate the nonlocality observed by any parties, as measured by , and correlations that party can share between measurement outcomes of any of these parties. It is worth pointing out that for it holds , and Ineqs. (5) simplify to which can be rewritten in a more familiar form as , where stand now for dichotomic observables with outcomes , while . Thus, the strength of violation of (24) imposes tight bounds on a single mean value for any and , which is also a measure of how outcomes of a measurement performed by the external party are correlated to those of for any . In particular, when (maximal nonsignalling violation), all these means are zero, while maximal correlations between a single pair of measurements, i.e., for some , make the parties unable to violate .

Bounds on randomness. Our monogamies are of particular importance for device–independent applications since they imply upper bounds on the guessing probability (GP) of the outcomes of any measurement performed by any of the parties by the additional party, here called . To be precise, assume that has full knowledge about all parties devices and their measurement choices and wishes to guess the outcomes of, say . The best can do for this purpose is to simply measure one of its observables, say the th one, and, irrespectively of the obtained result, deliver the most probable outcome of . Then, , and Ineqs. (6) imply that for any and , GP is bounded as

 maxakp(ak|xk)≡maxakp(A(k)xk=ak)≤1d(1+IN,M,dA). (7)

These bounds are tight and significantly stronger than the previously existing one,

 maxakp(ak|xk)≤1d(1+dN4(N−1)IN,M,dA) (8)

derived in Refs. (9); (10) (see Fig. 2).

Let us now discuss how the bound (7) performs in comparison to (8) in security proofs of DIQKD against no-signalling eavesdroppers. At the moment, a general security proof in this scenario is missing and the strongest proof requires the assumption that the eavesdropper is not only limited by the no-signalling principle but also lacks a long-term quantum memory (so–called bounded-storage model) (29). Assume that Alice and Bob share a two-qudit maximally entangled state and they use it to maximally violate (1) by performing the optimal measurements for this setup (see, e.g., (9)). To generate the secure key, Bob performs one more measurement that is perfectly correlated to one of Alice’s measurements. The key rate of this protocol is lower-bounded as (29), where is any upper bound on GP for nonsignalling correlations. and is the conditional Shannon entropy between Alice and Bob for the measurements used to generate the secret key. As the state is maximally entangled, this term is equal to zero. Fig. 2 compares bounds on the secret key obtained by using our bound (7) and the previous bound (8) in this protocol. We fix the key rate and compute the minimal number of measurements needed to attain this rate using these bounds as a function of the number of outputs. As shown in Fig. 2, the number of measurements when using our bound is much smaller and, in particular, decreases with the number of outputs.

Randomness amplification. Let us finally show the usefulness of our monogamy relations in randomness amplification. Assume that each party is given a sequence of bits produced by the Santha–Vazirani (SV) source (or the –source). Its working is defined as follows: it produces a sequence of bits according to

 12−ε≤p(yk|w)≤12+ε,k=1,…,n, (9)

where denotes any space-time variable that could be the cause of . Thus the bits are possibly correlated with each other retaining, however, some intrinsic randomness — we say that they are –free. The goal is now to obtain a perfectly random bit (or more generally it) from an arbitrarily long sequence of –free bits by using quantum correlations that violate the Bell inequality (24). This procedure is called randomness amplification (RA).

It is useful to recast this task in the adversarial picture (19), in which one assumes that an adversary , using the –sources, wants to simulate the quantum violation of (24) by NC, in particular the local ones. The random variable is now held by who uses it to control both the -sources and the physical devices possessed by the parties. That is, for every value of the former provides settings with probabilities obeying (45), while these devices generate the -partite probability distribution . Using (7), we can now restate and generalize Lemma 1 of (19) (see (25)).

###### Theorem 3.

Let be a nonsignalling probability distribution for any . Then for any and :

 ∑ak,w|p(ak,w|x)−˜p(ak)p(w|x)|≤(d−1)2+1dQM(x)IN,M,dA, (10)

where for any , describes correlations between outcomes obtained by party and the random variable for the measurements choice , and is taken in the probability distribution observed by the parties. Finally, , where with minimum taken over those measurement settings that appear in .

It then follows that if correlations violate maximally the Bell inequality (24), then the dits observed by the parties are perfectly random and uncorrelated from (19).

Let us now show that one can amplify partially random input bits to almost perfectly random dits by using QC that produce arbitrarily high violation of . To generate one of the measurement settings, each party uses its SV source times. Hence for any , (cf. Ref. (19)). Then, there is a state and measurement settings (9); (10) such that for large ,

 IN,M,dA≈λ(d)/M≤λ(d)/2r−1, (11)

where is a function of . After plugging everything into (10), one checks that its r.h.s. tends to zero for iff . As a result, QC violating (11) can be used to amplify randomness of any -source provided . In particular, for , the above reproduces the value found in (19), and, because is a strictly decreasing function of , the larger , the lower the critical epsilon for this method to work. Notice, however, that is independent of , so almost perfectly random dits are obtained from partially random bits. This means that using the setup from Ref. (19) we can in fact achieve both amplification and expansion of randomness simultaneously.

Recently, with the same Bell inequality but for , the critical epsilon was shifted from to (20). We will now show that by using a slightly different approach the critical epsilon can be almost doubled. To this end, we exploit the fact that only measurement settings out of all possible appear in . However, to generate them a common source has to be used. Assuming then that this is the case, (instead of ) uses of the SV source are enough to generate all measurement settings in . Thus, , which together with (11) imply that the right-hand side of (10) vanishes for iff , and in particular .

Conclusions. We have presented a novel class of monogamy relations, obeyed by any nonsignalling physical theory. They tightly relate the amount of nonlocality, as quantified by the violation of Bell inequalities (9); (10), that parties have generated in an experiment to the classical correlations an external party can share with outcomes of any measurement performed by the parties. Such trade–offs find natural applications in device-independent protocols and here we have discussed how they apply in quantum key distribution (cf. also Ref. (26)) and generation and amplification of randomness. We have finally showed that bipartite quantum correlations allow one to amplify –free its for any .

Our results provoke further questions. First, it is natural to ask if analogous monogamies hold for quantum correlations, and, in fact, such elemental monogamies can be derived in the simplest (3,2,2) scenario (see (25)). From a more fundamental perspective, it is of interest to understand what is the (minimal) set of of monogamy relations generating the same set of multipartite correlations as the no-signalling principle.

Acknowledgments. Discussions with Gonzalo De La Torre are gratefully acknowledged. This work is supported by NCN grant 2013/08/M/ST2/00626, FNP TEAM, EU project SIQS, ERC grants QITBOX, QOLAPS and QUAGATUA, the Spanish project Chist-Era DIQIP. This publication was made possible through the support of a grant from the John Templeton Foundation. R. A. also acknowledges the Spanish MINECO for the support through the Juan de la Cierva program.

## Appendix A appendices

Here we present detailed proofs of Theorems 1, 2, and 3 of the main text. Also, in the simplest scenario we provide elemental monogamies for quantum correlations.

## Appendix B Appendix A: Monogamy relations

### b.1 Monogamy relations for nonsignalling correlations

Let us start with a simple fact. Recall for this purpose that is the standard mean value of a random variable , that is, and stands for modulo .

###### Fact 1.

It holds that for any random variable ,

 (a) ⟨[Ω]⟩+⟨[−Ω−1]⟩=d−1, (12) (b) ⟨[Ω]⟩+⟨[−Ω]⟩=d[1−p([Ω]=0)]. (13)
###### Proof.

Both equations follow from the very definition of . To prove (a) we notice that , and hence

 ⟨[−Ω−1]⟩ = d−1∑i=1ip([Ω]=d−i−1) (14) = d−2∑i=0(d−i−1)p([Ω]=i) = (d−1)d−2∑i=0p([Ω]=i)−d−2∑i=0iP([Ω]=i) = (d−1)d−1∑i=0p([Ω]=i)−⟨[Ω]⟩ = (d−1)−⟨[Ω]⟩,

where the second equality is a consequence of changing of the summation index, the fourth one stems from the definition of and rearranging terms, and the last equality follows from normalization.

To prove (b), we write

 ⟨[Ω]⟩+⟨[−Ω]⟩ = d−1∑i=1i[p([Ω]=i)+p([−Ω]=i)] (15) = d−1∑i=1i[p([Ω]=i)+p([Ω]=d−i)] = d−1∑i=1ip([Ω]=i)+d−1∑i=1(d−i)p([Ω]=i) = dd−1∑i=1p([Ω]=i) = d[1−p([Ω]=0)],

where the second equality is a consequence of the fact that , while the third equality follows from shifting of the summation index in the second sum. ∎

Let us now move to the proofs of the monogamy relations. In the tripartite case we make use of the Barrett, Kent, and Pironio (BKP) (9) inequality

 I2,M,dAB=M∑α=1(⟨[Aα−Bα]⟩+⟨[Bα−Aα+1]⟩)≥d−1, (16)

where the convention that is assumed.

###### Theorem 1.

For any three-partite nonsignalling correlations with measurements and outcomes per site and any pair , the following inequality

 I2,M,dAB+⟨[Xi−Cj]⟩+⟨[Cj−Xi]⟩≥d−1 (17)

is satisfied with denoting either or .

###### Proof.

Let us start with the case of and then notice that for a random variable it holds that (see Fact 1). Consequently,

 M∑β=1β≠i(⟨[Cj−Aβ−1]⟩+⟨[Aβ−Cj]⟩)−(M−1)(d−1) (18)

is equal to zero. The fact that for any and it holds that allows us to rewrite (18) in the following way

 i−1∑β=1(⟨[Cj−Aβ−1]⟩+⟨[Aβ+1−Cj]⟩) +M∑β=i+1(⟨[Aβ−Cj−1]⟩+⟨[Cj−Aβ]⟩)−(M−1)(d−1).

Then, by adding to both sides of the above and rearranging some terms in the resulting expression, one obtains

 ⟨[Ai−Cj]⟩+⟨[Cj−Ai]⟩ =i−1∑β=1(⟨[Cj−Aβ−1]⟩+⟨[Aβ+1−Cj]⟩) +M−1∑β=i(⟨[Aβ+1−Cj−1]⟩+⟨[Cj−Aβ]⟩) +⟨[A1−Cj]⟩+⟨[Cj−AM]⟩−(M−1)(d−1). (20)

In an analogous way, we may decompose :

 I2,M,dAB = i−1∑α=1(⟨[Aα−Bα]⟩+⟨[Bα−Aα+1]⟩) (21) +M−1∑α=i(⟨[Aα−Bα]⟩+⟨[Bα−Aα+1]⟩) +⟨[AM−BM]⟩+⟨[BM−A1−1]⟩.

In the last step of these manipulations, we add line by line Eqs. (B.1) and (21) in order to finally obtain

 I2,M,dAB+⟨[Ai−Cj]⟩+⟨[Cj−Ai]⟩ = i−1∑α=1(⟨[Cj−Aα−1]⟩+⟨[Aα−Bα]⟩+⟨[Bα−Aα+1]⟩+⟨[Aα+1−Cj]⟩) (22) +M−1∑α=i(⟨[Cj−Aα]⟩+⟨[Aα−Bα]⟩+⟨[Bα−Aα+1]⟩+⟨[Aα+1−Cj−1]⟩) +⟨[Cj−AM]⟩+⟨[AM−BM]⟩+⟨[BM−A1−1]⟩+⟨[A1−Cj]⟩ −(M−1)(d−1).

What we have arrived at is basically the sum of Bell expressions but ‘distributed’ among three parties in such a way that Bob and Charlie measure only a single observable. It was shown in (14) that the minimal value such an expression can achieve over nonsignalling correlations is precisely its classical bound . As a result, , finishing the proof for the case .

If in Ineq. (17), then it suffices to rewrite the Bell expression from (16) as

 I2,M,dAB=M∑α=1(⟨[Bα−Aα+1]⟩+⟨[Aα+1−Bα+1]⟩), (23)

add to it the zero expression (18) with replaced by , and repeat the above manipulations. This completes the proof. ∎

Now let us move to the general scenario. The inequality of interest is now the one from Ref. (10), namely:

 IN,M,dA=1MM∑αN−1=1IN−1,M,dA(1)…A(N−1)(αN−1)∘A(N)αN−1 (24) ≥d−1.

where . The notation means insertion of to the average with the opposite sign to the one of with any , while is the same Bell expression as in (24), but for parties, and with observables of the last party relabeled as with .

###### Theorem 2.

For any -partite nonsignalling correlations with -outcome measurements per site, the following inequality

 IN,M,dA+⟨[A(k)xk−A(N+1)xN+1]⟩+⟨[A(N+1)xN+1−A(k)xk]⟩≥d−1 (25)

is satisfied for any and .

###### Proof.

The recursive formula in Ineq. (24), which for convenience we restate here

 IN,M,dA=1MM∑αN−1=1IN−1,M,dA(1)…A(N−1)(αN−1)∘A(N)αN−1, (26)

allows us to demonstrate the theorem inductively. The case of has already been proved as Theorem 1, so we consider . Exploiting Eq. (26), one can express as

 I3,M,dA(1)A(2)A(3)=1MM∑α2=1I2,M,dA(1)A(2)(α2)∘A(3)α2. (27)

It is clear that for every

 I2,M,dA(1)A(2)(α2) = M∑α1=1(⟨[A(1)α1−A(2)α1+α2−1]⟩ (28) +⟨[A(2)α1+α2−1−A(1)α1+1]⟩)≥d−1

is a Bell inequality equivalent to (16), in which the observables of the second party have been relabelled according to . It must then fulfil the monogamy relations (17) (with ) independently of the value of . In order to see it in a more explicit way, let us consider the case , and in Eq. (22) just rename , , and , and also for the first party, while for the second one. Then, for those observables for which we use the rule to get with some and , and later replace the latter by another variable (this is just with outcomes shifted by a constant). With the aid of formula (23) the same reasoning can be repeated for .

Now, we prove that each term in Eq. (27) fulfills (25) for , that is that the inequalities

 I2,M,dA(1)A(2)(α2)∘A(3)α2+⟨[A(k)xk−A(4)x4]⟩+⟨[A(4)x4−A(k)xk]⟩≥d−1 (29)

hold for any , any pair , and any .

First assume . Let us write explicitly as

 I2,M,dA(1)A(2)(α2)∘A(3)α2=M∑α1=1(⟨[A(1)α1−A(2)α1+α2−1+A(3)α2]⟩ +⟨[A(2)α1+α2−1−A(1)α1+1−A(3)α2]⟩).

For any fixed , the last party measures solely a single observable, and therefore we treat as a single variable, or, in other words, for any , is a -outcome observable [recall that in Eq. (B.1) all variables are modulo ]. Effectively, (29) is a three-partite inequality of the form (25) (with ) that has just been proven.

In the case we insert the third party into the alternative expression (23) and further apply the same reasoning as above.

In order to show (25) for , we use the fact that the Bell inequality (24) for is invariant under the exchange of the first and the third party (10), meaning that we can, analogously to Eq. (27), write it down as

 I3,M,dA(1)A(2)A(3)=1MM∑α2=1I2,M,dA(3)A(2)(α2)∘A(1)α2. (31)

Now, it is enough to repeat the above reasoning to complete the proof of the monogamy relations (25) for .

Having it proven for , let us now assume that the theorem is true for parties (any -partite nonsignalling probability distribution). In order to complete the proof we again refer to the recursive formula (26). By grouping together the last two parties, each term in the sum in Eq. (26) is effectively an –partite Bell expression for which we have just assumed (25) to hold for any and . Performing the summation over and dividing further by we obtain (25) for any and . The case can be reached by using the fact that is invariant under exchange of the last and the th party (10). ∎

### b.2 Elemental monogamies for quantum correlations

Let us now discuss the case of quantum correlations in which case similar monogamy relations are also expected to hold. Their derivation, however, is much more cumbersome and we only consider the simplest scenario and derive quantum analogs of the nonsignalling monogamies (17). To this end, we use a one-parameter modification of the CHSH Bell inequality (13) with the latter being a particular case of (16) with . Here, for convenience, we write it down in its “standard” form:

 ˜IαAB:=α(⟨A1B1⟩+⟨A1B2⟩)+⟨A2B1⟩−⟨A2B2⟩≤2α (32)

with . Here, and are local quantum observables with eigenvalues and for some state and local observables . Actually, one proves the following more general theorem, generalizing the result of Ref. (11) for the Bell inequality (32).

###### Theorem 3.

Any three-partite quantum correlations with two dichotomic measurements per site must satisfy the following inequalities

 α2max{(˜IαAB)2,(˜IαAC)2}+min{(˜IαAB)2,(˜IαAC)2} ≤4α2(1+α2) (33)

and

 (˜IαAB)2+4⟨AiCj⟩2≤4(1+α2) (34)

for any and .

###### Proof.

The proof is nothing more but a slight modification of the considerations of Ref. (11) (see also Ref. (27)). Nevertheless, we attach it here for completeness.

We start by noting that the monogamy regions, that is, the two-dimensional sets of allowed (realizable) within quantum theory pairs for Ineq. (3) and with fixed and for Ineq. (34), must be convex. Therefore, as it is shown in Ref. (11) (see also Ref. (28)), every point of their boundaries can be realized with a real three-qubit pure state and real local one-qubit measurements. Recall that the latter assumes the form

 X=x⋅σ (35)

with being a unit vector and denoting a vector consisting of the standard Pauli matrices and .

Then, it follows from a series of papers (27); (11); (22) that for a given two-qubit state , the maximal value of over local, real, and traceless observables [i.e., those of the form (35)] measured by Alice and Bob , amounts to

 maxAi,Bj(˜IαAB)=2√α2λ1+λ2. (36)

Here, denote the eigenvalues of put in a decreasing order, i.e., , and is the following Å½reducedÅ½ correlation matrix

 TAB=(⟨σx⊗σx⟩AB⟨σx⊗σz⟩AB⟨σz⊗σx⟩AB⟨σz⊗σz⟩AB). (37)

We added the subscript in (37) to indicate that the mean values are taken in the state . In particular, one can similarly compute the maximal value of a single average in the state over local observables and of the form (35) to be

 maxA,B⟨AB⟩=λ1. (38)

Equipped with these facts, we can now turn to the proof of the inequalities (3) and (34). We start from the first one and note that it suffices to demonstrate it in the case of , in which it becomes

 α2(˜IαAB)2+(˜IαAC)2≤4α2. (39)

The opposite case will follow immediately by exchanging .

Let then be a pure real three-qubit state. By and we denote its subsystems arising by tracing out the third and the second party, respectively, and by and the corresponding correlation matrices [cf. Eq. (37)]. Finally, let and be eigenvalues of