Maximal qubit violation of n-locality inequalities in a star-shaped quantum network

Bell's theorem was a cornerstone for our understanding of quantum theory and the establishment of Bell non-locality played a crucial role in the development of quantum information. Recently, its extension to complex networks has been attracting growing attention, but a deep characterization of quantum behavior is still missing for this novel context. In this work we analyze quantum correlations arising in the bilocality scenario, that is a tripartite quantum network where the correlations between the parties are mediated by two independent sources of states. First, we prove that non-bilocal correlations witnessed through a Bell-state measurement in the central node of the network form a subset of those obtainable by means of a local projective measurement. This leads us to derive the maximal violation of the bilocality inequality that can be achieved by arbitrary two-qubit quantum states and arbitrary local projective measurements. We then analyze in details the relation between the violation of the bilocality inequality and the CHSH inequality. Finally, we show how our method can be extended to the n-locality scenario consisting of n two-qubit quantum states distributed among n + 1 nodes of a star-shaped network.


Introduction
Since its establishment in the early decades of the last century, quantum theory has been elevated to the status of the 'most precisely tested and most successful theory in the history of science' [1]. And yet, many of its consequences have puzzled-and still do-most of the physicists confronted with it. At the heart of many of the counter-intuitive features of quantum mechanics is quantum entanglement [2], nowadays a crucial resource in quantum information and computation [3] but that also plays a central role in the foundations of the theory. For instance, as shown by the celebrated Bellʼs theorem [4], quantum correlations between distant parts of an entangled system can violate Bell inequalities, thus precluding its explanation by any local hidden variable (LHV) model, the phenomenon known as quantum non-locality.
Given its fundamental importance and practical applications in the most varied tasks of quantum information [5], not surprisingly many generalizations of Bellʼs theorem have been pursued over the years. Bellʼs original scenario involves two distant parties that upon receiving their shares of a joint physical system can measure one out of possible dichotomic observables. Natural generalizations of this simple scenario include more measurements per party [6] and sequential measurements [7], more measurement outcomes [8], more parties [9,10] and also stronger notions of quantum non-locality [11][12][13][14]. All these different generalizations share the common feature that the correlations between the distant parties are assumed to be mediated by a single common source of states (see, for instance, figure 1(a)). However, as it is often in quantum networks [15], the correlations between the distant nodes is not given by a single source but by many independent sources which distribute entanglement in a non-trivial way across the whole network and generate strong correlations among its nodes (figures 1(b)-(d)). Surprisingly, in spite of its clear relevance, such networked scenario is far less explored.
The simplest networked scenario is provided by entanglement swapping [16], where two distant parties, Alice and Charlie, share entangled states with a central node Bob (see figure 1(b)). Upon measuring in an entangled basis and conditioning on his outcomes, Bob can generate entanglement and non-local correlations among the two other distant parties even though they had no direct interactions. To contrast classical and quantum correlation in this scenario, it is natural to consider classical models consisting of two independent hidden variables (figures 1(b)), the so-called bilocality assumption [17,18]. The bilocality scenario and generalizations to networks with an increasing number n of independent sources of states (figures 1(d)), the so called n-locality scenario [19][20][21][22][23][24][25][26], allow for the emergence of a new kind of non-local correlations. For instance, correlations that appear classical according to usual LHV models can display non-classicality if the independence of the sources is taken into account, a result experimentally demonstrated in [27,28]. However, previous works on the topic have mostly focused on developing new tools for the derivation of inequalities characterizing such scenarios and much less attention has been given to understand what are the quantum correlations that can be achieved in such networks.
That is precisely the aim of the present work. We consider in details the bilocality scenario and the bilocality inequality derived in [17,18] and characterize the non-bilocal behavior of general qubit quantum states when the parties perform different kinds of projective measurements. First of all we show that the correlations arising in an entanglement swapping scenario, i.e. when Bob performs a Bell-state measurement (BSM), form a strict subclass of those correlations which can be achieved by performing local projective measurements in all stations. Focusing on this wider class of correlations, we derive a theorem characterizing the maximal violation of the bilocality inequality [17,18] that can be achieved from general two-qubit quantum states shared among the parties. This leads us to obtain a characterization for the violation of the bilocality inequality in relation to the violation of the CHSH inequality [29]. Finally we show how our maximization method can be extended to the star network case [19], a n-partite generalization of the bilocality scenario, deriving thus the maximum violation of the n-locality inequality that can be extracted from this network.

Scenario
In the following we will mostly consider the bilocality scenario, which classical description in terms of a directed acyclic graph is shown in figure 1(b). It consists of three spatially separated parties (Alice, Bob and Charlie) whose correlations are mediated by two independent sources of states. In the quantum case, Bob shares two pairs of entangled particles, one with Alice and another with Charlie. Upon receiving their particles Alice, Bob and Charlie perform measurements labeled by the random variables X, Y and Z obtaining, respectively, the measurement outcomes A, B and C. The difference between Bob and the other parties is the fact that the first has in his possession two particles and thus can perform a larger set of measurements including, in particular, measurements in an entangled basis.
Any probability distribution compatible with the bilocality assumption (i.e. independence of the sources) can be decomposed as In particular, if we consider that each party measures two possible dichotomic observables (x y z a b c , , , , , 0, 1 = ), it follows that any bilocal hidden variable (BLHV) model described by equation (1) must fulfill the bilocality inequality As shown in [17,18], if we impose the same causal structure to quantum mechanics (e.g. in an entanglement swapping experiment) we can nonetheless violate the bilocality inequality (even though the data might be compatible with LHV models), thus showing the existence of a new form of quantum non-locality called quantum non-bilocality.
To that aim let us consider the entanglement swapping scenario with an overall quantum state . We notice that equation (2) is also valid if Bob performs a single measurement with four possible outcomes (instead of two dichotomic measurements). Then, we can choose the measurements operators for the different parties in the following way. Stations A and C perform single qubit measurements defined by Station B, instead, performs a complete BSM, assigning to the two bits b b 0 1 the values 00 for , 01 for , 10 for , 11 for The binary measurement B y is then defined such that it returns 1 b y -( ) , with respect to the value of y 0, 1 = . This leads to where, in the last steps, we made explicit use of the marginalization of probability p a b b c x z , , , , 0 1 ( | ) over b k y ¹ . With these state and measurements, the quantum mechanical correlations achieve a value 2 1  = > , which violates the bilocality inequality and thus proves quantum non-bilocality.

Results
3.1. Non-bilocal correlations with local projective measurements As reproduced above, in an entanglement swapping scenario Quantum Mechanics (QM) can exhibit correlations which cannot be reproduced by any BLHV model. In turn, it was recently proved [22] that an equivalent form of the bilocality inequality (equation (2)), can be violated by QM when all parties only perform a a particular case of single qubit operations (i.e. , , x z y s s s and linear combinations). Here we will prove that, given the bilocality inequality of equation (2), the non-bilocal correlations arising in an entanglement swapping scenario (exploiting the BSM protocol described in section 2) are a strict subclass of those obtainable by means of local projective measurements. This latter scenario differs from the former since station B performs the two-spin projective measurements described by which do not require any entangled operation (differently from the BSM case) and can be obtained by only performing local projective measurements. The main motivation for considering local projective measurements is the fact that they correspond to the simplest set of measurements implementable in most physical setups used to study the violation of Bell inequalities. This means that station B can be seen as composed of two independent substations B A and B C ( figure 1(c)). On the contrary, a general Local Operations and Classical Communication (LOCC) process would require substantially more experimental efforts, for instance the implementation of a feedback system which drives the settings choice in substation B C , with respect to the results of substation B A ʼs measurement. As a matter of fact, when considering linear optical setups, even complete BSMs cannot be implemented [30]. This implies that recent experiments violating the bilocality inequality [27,28] have to rely on assumptions about the measurement device and being thus device-dependent in clear opposition to the general aim of Bellʼs theorem. Finally, as pointed out in [31], when shared reference frames are added in the scenario, some particular situations may require that no internal communication is ensured between B A and B C , i.e. the presence of the local measurements scenario. From a more theoretical perspective, we stress that, so far, most of works on the topic has been focused on the case where station B performs a BSM. The possibility of emulating the non-bilocal results of a BSM (or rotated BSM) by means of an experimentally friendly subset of the LOCC operations is, in our view, a notable point.
We can now proceed with the evaluation of the main result of this section. The core of the bilocality parameter  is the computation of the expected value A B C x y z á ñ (equation (4)), that in the quantum case is given by

For the entanglement swapping scenario we can summarize the measurements in stations A and C by
where A x and C z are general single qubit projective measurements with eigenvalues 1 and −1. When dealing with station B, it is suitable to consider its operatorial definition which is implicit in equation (7). Indeed we can consider that 1 b y -( ) is the outcome of our measurement, leading to values shown in table 1. The quantum mechanical description of the operator B y (in an entanglement swapping scenario) is thus given by which relates each value of y 0, 1 = with its correct set of outcomes. This leads to the following theorem.
Theorem 1 (Non-bilocal correlations and local projective measurements). Given the general set of local projective measurements described in equation (8), QM predictions for the bilocality parameter  which arise in an entanglement swapping scenario (where Bob performs the measurement described in equation (11)) are completely equivalent to those obtainable by performing a strict subclass of equation (8), i.e.
Proof. Let us write the Bell basis of a two qubit Hilbert space in terms of the computational basis ( 00 , 01 , 10 , 11 . From equation (11), we obtain B y y y y y y 1 2 2 1 1 00 00 01 01 10 10 11 11 00 11 01 10 10 01 11 00 This shows that the entanglement swapping scenario is equivalent to the one where station B only performs the two local projective measurements B x 1 s s = Ä , which form a strict subclass of the general set of local measurements given by equation (8). Moreover, if we consider a rotated Bell basis, then we obtain , this case still represents a strict subset of equation (8). , To further illustrate the relevance of theorem 1 we show next an example of correlations that can be achieved with local projective measurements but cannot with rotated BSMs. Let us consider the overall quantum state Table 1. Expected values for the operator B y , as implicitly defined in equation (7).

Moreover, let us define the measurements performed in stations A and C as
We can analyze the set of non-bilocal correlations in terms of the correlation parameters I and J (defined in section 2). As we show in appendix, if we consider the rotated BSM case (constrained by b b , values leads to 1.14 1 ~> , then it proves how, in this scenario, some peculiar non-bilocal correlations can only be obtained by means of local projective measurements, rather than performing a rotated BSM.
It is interesting to stress that the proof of theorem 1 also allows for convex combinations of local projective measurements. However, in principle, more general separable measurement are obtained through LOCCs.
As it turns out, this theorem has strong implications in our understanding of the non-bilocal behavior of QM. Indeed, it shows how the usual BSM protocol described in section 2, which is characteristic of the entanglement swapping scenario, is not capable of exploring the whole set of quantum non-bilocal correlations, since its expected value for the parameter  is equivalent to a subclass of those obtainable by means of local projective measurements. Moreover, it also shows that neither its extension to a rotated Bell basis is able to provide an enhancement, compared to the local projective measurement protocol. As we will show next, a better characterization of quantum correlations within the bilocality context must thus in principle take into account more general forms for Bobʼs measurements, especially when dealing with different types of quantum states.

Non-bilocality maximization criterion
We will now explore the maximization of the bilocality inequality considering that Bob performs the projective measurements described by equation (8). It is convenient to consider station B as a unique station composed of the two substations B A and B C , which perform single qubit measurements on one of the qubits belonging to the entangled state shared, respectively, with station A or C (see figure 1(c)).
Let A perform a general single qubit measurement and similarly for B A , B C and C. We can define these measurements as , , The coefficients t nm can be used to define a real matrix T  that lead to the following result: Lemma 1 (Bilocality Parameter with Local Projective Measurements). Given the set of general local projective measurements described in equation (17) and defined the general quantum state AB BC   Ä accordingly to equation (18), the bilocality parameter  is given by · and a two qubit quantum state ñ described by equation (18). We can write where we made use of the properties of the Pauli matrices i s . Given the set of local measurements described in equation (17), and the definitions of I and J (showed in equation (3)), the proof comes from a direct application of equation (20) to the quantum mechanical expectation value: Next we proceed with the maximization of the parameter  over all possible measurement choices, that is, the maximum violation of bilocality we can achieve with a given set of quantum states. To that aim, we introduce the following Lemma.  is also an eigenvalue of 2  , and vice versa.
We can then apply the operator M from the left, obtaining which shows that Mv  is an eigenvector of 2  with eigenvalue λ. The opposite statement can be analogously proved. , We can now enunciate the main result of this section.
Theorem 2 (Bilocality Parameter Maximization). Given the set of measurements described in equation (17), the maximum bilocality parameter that can be extracted from a quantum state AB can be written as where t A 1 and t A 2 (t C 1 and t C 2 ) are the two greater (and positive) eigenvalues of the matrix T T Proof. We will prove theorem 2, following a scheme similar to the one used by Horodecki [32] for the CHSH inequality. Let us introduce the two pairs of mutually orthogonal vectors and let us apply equation (25) to (19) where the maximization is done over the variables n n b b n n b b , , , , , l l and 3 l its eigenvalues and let us write n A  and n A ¢  in an eigenvector basis. If we define k T n 0 || , our problem can be written in terms of Lagrange multipliers related to the maximization of a function f, given the constraints g i f n n k n k n g n n n g n n n g n n n n , , 1, 1, , , 32 where we considered that finding the values that maximize f x | ( )| is equivalent to find these values for f x | ( )|. Let us now introduce the scaled vectors k n where we made use of the lemma 2. The maximization over the last two variables leads to an analogous Lagrange multipliers problem with similar solutions, thus proving the theorem. , This theorem generalizes the results of [33] (which dealt with some particular classes of quantum states in the entanglement swapping scenario) to the more generic case of any quantum state in the local measurements scenario (which, in a bilocality context, includes the correlations obtained through entanglement swapping). It represents an extension of the Horodecki criterion [32] to the bilocality scenario, taking into account a general class of local measurements which can be performed in station B. Our result thus shows that, as far as we are concerned with the optimal violations of the bilocality inequality provided by given quantum states, local projective measurements or a BSM (in the right basis) are fully equivalent.

The relation between the non-bilocality and non-locality of sources
We will now characterize quantum non-bilocal behavior with respect to the usual non-locality of the states shared between A, B and B, C. Let us start from equation (24) and separately consider Bell non-locality of the states AB  and BC  . We can quantify it by evaluating the greatest CHSH inequality violation that can be obtained with these states. Let us define the CHSH inequality as If we apply the criterion by Horodecki et al [32], we obtain where we defined t t t , , 1 and t C 2 accordingly to equation (24). From a direct comparison of (24) and (37) we can write Proposition 1.
This result shows that if the two sources cannot violate the CHSH inequality then they will also not violate the bilocality inequality. Thus, in this sense, if our interest is to check the non-classical behavior of sources of states, it is just enough to check for CHSH violations (at least if Bob performs a BSM or local projective measurements). Notwithstanding, we highlight that this does not mean that the bilocality inequality is useless, since there are probability distributions that violate the bilocality inequality but nonetheless are local according to a LHV model and thus cannot violate any usual Bell inequality [18,27]. This fact can be explained considering that the bilocality inequality is derived under the assumption of independence of the two sources of states, which introduces a supplementary constraint. A probability distribution which violates the bilocality inequality, then, could nonetheless admit a decomposition in term of LHV models, since they do not rely on this supplementary assumption and can then include a wider class of probability distributions.
Next we consider the reverse case: is it possible to have quantum states that can violate the CHSH inequality but cannot violate the bilocality inequality? That turns out to be the case. To illustrate this phenomenon, we start considering two Werner states in the form v v . In this case, indeed, in order to have a non-local behavior between A and B (B and C) we must have v in order to witness non-bilocality. This example shows that on one hand it might be impossible to violate the bilocality inequality although one of AB  or BC  is Bell non-local (for instance v 1 A = and v 0 C = ). It also shows that, when one witnesses non-locality for only one of the two states, it can be possible, at the same time, to have non-bilocality by considering the entire network (for instance v Another possibility is the one described by the following Proposition Proposition 2. Given a tripartite scenario and such that 1, Proof. We will prove this point with an example. Let us take This shows how it is possible to have non-local quantum states which nonetheless cannot violate the bilocality inequality (with local measurements). However, we highlight that, since the bilocal set is a subset of the local one, certainly there are bilocality inequalities other than (2) that are violated by these non-local states.
All these statements provide a well-defined picture of the relation between the CHSH inequality and the bilocality inequality in respect to the quantum states AB BC   Ä . We indeed derived all the possible cases of quantum non-local correlations which may be seen between couples of nodes, or in the whole network (according to the CHSH and bilocality inequalities). This characterization is shown in figure 2, in terms of a Venn diagram.
We finally notice that if A and B share a maximally entangled state while B and C share a generic quantum state, then it is easier to obtain a bilocality violation in the tripartite network rather than a CHSH violation between the nodes B C and C. Indeed it is possible to derive where we made use of the following Lemma Lemma 3. Given the parameters t t t , , 1 and t C 2 defined in equation (24), it holds t t t t 0 , , , 1 46 Proof. This proof will be divided in two main points.
As discussed in [34], if we apply a local unitary U U U 1 2 = Ä to the initial quantum state ñ, the matrix T  will transform accordingly to where U 1 and U 2 are orthogonal 3×3 real matrices which represent, respectively, the spin rotations properties of U 1 and U 2 . According to the Singular Decomposition Theorem, it is always possible to choose U 1 and U 2 such that U T U T 1 2 ˜i s diagonal, thus demonstrating point 1.
It is important to stress that we can always rotate our Hilbert space in a way that U U    † so we can take  ¢ without loss of generality.
(2) If T  is diagonal, then the eigenvalues of T T Given the definitions of t A 1 and t A 2 (t C 1 and t C 2 ) described in equation (24), the lemma is proved. ,

Extension to the star network scenario
We now generalize the results of theorem 2, to the case of a n-partite star network. This network is the natural extension of the bilocality scenario, and it is composed of n sources sharing a quantum state between one of the n stations A i and a central node B (see figure 1(d)). The bilocality scenario corresponds to the particular case where n=2. The classical description of correlations in this scenario is characterized by the probability decomposition As shown in [19], assuming binary inputs and outputs in all the stations, the following n-locality inequality holds  We will now derive a theorem showing the maximal value of parameter star  that can be obtained by local projective measurements on the central node and given arbitrary bipartite states shared between the central node and the n parties.
Theorem 3 (Optimal violation of the n-locality inequality). Given single qubit projective measurements and defined the generic quantum state ...
accordingly to equation (18), the maximal value of star  is given by Proof. In our single qubit measurements scheme the operator B can be written as As pointed out in [19], this allows us to write Introducing the pairs of mutually orthogonal vectors a a n a a n 2 cos & 2 sin , 55 The evaluation of the correlation parameters I J , for the rotated BSM case will then lead to where k k n n , , , Nevertheless, when assuming equation (A7), one obtains J 1 5 sin sin On the contrary, we can consider the local projective measurement case by relaxing the orthogonality assumption of vectors b b , . This means that we can define the four generic independent vectors , if we choose, at the same time, the values of , , , A C A C q q f f described in equation (A7), then we are able to obtain non-bilocal correlations which could not be reached by means of a rotated BSM.
We stress that this fact does not prove that some scenarios exist where the maximal amount of non-bilocality can be obtained only with local projective measurements rather than by performing a rotated BSM. It proves, instead, that some peculiar set of non-bilocal correlations (defined by the corresponding values of the correlation parameters I and J), can only be addressed by means of local projective measurements.