Bell monogamy relations in arbitrary qubit networks
Characterizing trade-offs between simultaneous violations of multiple Bell inequalities in a large network of qubits is computationally demanding. We propose a graph-theoretic approach to efficiently produce Bell monogamy relations in arbitrary arrangements of qubits. All the relations obtained for bipartite Bell inequalities are tight and leverage only a single Bell monogamy relation. This feature is unique to bipartite Bell inequalities, as we show that there is no finite set of such elementary monogamy relations for multipartite inequalities. Nevertheless, many tight monogamy relations for multipartite inequalities can be obtained with our method as shown in explicit examples.
Bell inequalities are a prime example where a fundamental physics concept finds concrete practical applications. They were derived to put limits on correlations achievable within local hidden variable models but more recently their violation was linked to device-independent randomness generation, cryptography or reduction of communication complexity Brunner et al. (2014). Similarly, Bell monogamy relations, i.e. trade-offs between simultaneous violation of multiple Bell inequalities, have both fundamental and practical aspects. On the fundamental side, they were shown to exist in every no-signaling theory Scarani and Gisin (2001a, b); Barrett et al. (2005a); Masanes et al. (2006); Toner (2009); Pawłowski and Brukner (2009); Pawłowski (2010); Augusiak et al. (2014); Ramanathan and Horodecki (2014), but the principle of no-signaling alone does not single out the monogamies derived with the quantum formalism Toner and Verstraete (2006); Kurzyński et al. (2011); Ramanathan and Mironowicz (2017). On the practical side, Bell monogamy was used to obtain the optimal shrinking factor of cloning machines Pawłowski and Brukner (2009) and was shown to improve device-independent tasks such as randomness amplification and quantum key distribution Pawłowski (2010); Augusiak et al. (2014). In the latter case, it is the existence of the Bell monogamy relation that allows for secure cryptography even in the presence of signaling Pawłowski (2010). See also Ref. Barrett et al. (2005b) in this context.
Despite their importance, only a handful of Bell monogamy relations have been derived within the quantum formalism Scarani and Gisin (2001a, b); Toner and Verstraete (2006); Kurzyński et al. (2011); Ramanathan and Mironowicz (2017). A powerful approach to generate tight relations is given by the correlation complementarity Kurzyński et al. (2011). The approach involves dividing relevant observables into sets of mutually anti-commuting ones. The complexity of this task grows exponentially with the number of Bell parameters and therefore renders correlation complementarity inefficient for large networks. In fact, an efficient method to generate tight monogamy relations in arbitrary arrangements of a large number of qubits is not yet available.
Here we propose an efficient method to generate Bell monogamy relations that is applicable to arbitrary number of observers. For every collection of bipartite Bell parameters our method yields a corresponding tight monogamy relation. The approach leverages a single Bell monogamy relation (derived in Ref. Toner and Verstraete (2006)) multiple times. We therefore name this monogamy relation as elementary. We then investigate if the method generalizes to multipartite inequalities. It turns out that the situation is far more complicated already for tripartite inequalities. We construct a Bell scenario with an increasing number of observers for which the method produces a Bell monogamy relation that is not tight, even if all elementary relations for smaller number of observers are taken into account. We conclude that in tripartite scenario there is no finite set of elementary relations. Nevertheless, the method does produce many tight monogamies and hence is valuable also in the multipartite case.
Bell inequalities. We focus on a complete set of correlation Bell inequalities for observers, each choosing between two measurement settings and obtaining a dichotomic outcome Werner and Wolf (2001); Żukowski and Brukner (2002). The set is equivalent to a single general Bell inequality 111See e.g. Eq. (5) in Ref. Żukowski and Brukner (2002)., where the following upper bound on the Bell parameter was also derived:
The summation is over orthogonal local directions and which span the plane of local settings and are the quantum correlation functions of state . Therefore, if the quantum correlations admit a local hidden variable model (for measurements in the plane), i.e. they do not violate any Bell inequality. The condition is also necessary and sufficient if Horodecki et al. (1995). We shall use it as a building block for our monogamy relations.
Bipartite inequalities. Let us first consider trade-offs between simultaneous violation of a set of bipartite Bell inequalities. Each bipartite Bell parameter may involve two out of observers, each having access to a single qubit. The problem can be represented by a graph with vertices denoting the observers and edges denoting the relevant Bell parameters. An example of such a graph is given in Fig. 1. The simplest scenario of Bell monogamy is when three observers try to simultaneously violate two Bell inequalities. The statement of Bell monogamy is that the simultaneous violation is impossible and the quantitative quantum relation reads Toner and Verstraete (2006):
This monogamy relation is a straightforward application of the correlation complementarity (Appendix A).
For a general graph, one can in principle also apply correlation complementarity to find tight monogamy relations. The method requires grouping relevant observables into mutually anti-commuting sets, which is computationally demanding. Instead, we propose the following simple method to derive a tight Bell monogamy relation for every graph. Let us denote with the graph with observers represented by vertices and Bell inequalities by edges. A line graph of the initial graph is constructed by placing vertices of on every edge of , and by connecting the vertices of whenever the corresponding edges of share a vertex (Fig. 1). The properties of the line graph determine the Bell monogamy relations. Note that Bell inequalities are represented by vertices of and edges of provide information whether two Bell inequalities share a common observer. In other words, for every edge of we have monogamy relation (2), and summing them up gives a general monogamy
where the sum is over the vertices of , denotes the number of edges incident to the vertex , is the Bell parameter associated with vertex and is the total number of edges in . The factor of comes from the monogamy relation (2). We shall also refer to this method as the averaging method.
The general monogamy relation (3) turns out to be tight, i.e. the bound cannot be any smaller. This follows from the handshaking lemma stating that for any finite undirected graph . This corresponds to for all the vertices of , which can be achieved e.g. by the state , where all the spins are aligned along the axis and the measurements are all .
We emphasize that this construction is general and surprisingly simple. It applies to arbitrary graphs, i.e. an arbitrary number of observers measuring an arbitrary configuration of the bipartite Bell inequalities while in the process only monogamy relation (2) is utilized. We therefore term the monogamy relation (2) elementary.
On a side note, the elementary relation (2) has a remarkable property that all mathematically allowed values of and that saturate it are physically realizable Toner and Verstraete (2006). The general monogamy relation (3) does not share this property as simply seen by considering the triangle graph: in this case (3) gives the bound of , while each individual Bell expression can take at most the maximum Tsirelson value of . However, one may ask if the set defined by the intersection of elementary relations (2) contains values of Bell parameters that are all physically achievable. We show in Appendix B examples of configurations where all the points in the intersection are indeed realized in quantum physics.
A natural question is whether the averaging method generalizes to multipartite Bell monogamy, i.e. . In particular, we ask if there exists an elementary monogamy relation, or a finite set of elementary monogamy relations, from which tight monogamy relations could be derived in arbitrary scenario. The answer is more complex even for tripartite Bell inequalities. On one hand, there are simple monogamy relations averaging which results in tight monogamy relations. But on the other hand, there are Bell scenarios where tight monogamy relations cannot be obtained from simpler relations. We now discuss them in more detail.
Tripartite inequalities. The graph-theoretic approach from the previous section can be naturally extended to tripartite Bell inequalities. The graph is now upgraded to a hypergraph with the vertices representing observers and hyperedges connecting three observers testing violation of the Bell inequality. Fig. 2 presents examples of such hypergraphs. The two Bell monogamy relations at the bottom of Fig. 2 list all possible ways two tripartite Bell parameters may overlap, and results from the bipartite case suggest they might be of special importance. Using correlation complementarity one easily verifies that the corresponding monogamy relations hold (Appendix A):
The averaging method naturally extends. But now in the line graph, an edge connects Bell inequalities that share at least one common observer. Since the bound in both inequalities above is the same, the general monogamy relation is of the form (3) with the factor of on the right-hand side replaced by .
Note however that the physical implication of monogamy relations (4) and (5) is different than that of the bipartite relation (2). In the bipartite case, whenever one Bell inequality is violated, the other has to be satisfied. However, two tripartite Bell inequalities can be violated simultaneously. There indeed exist quantum states and measurements which give rise to any value of the Bell parameters compatible with (4) and (5) Kurzyński et al. (2011); Ramanathan and Mironowicz (2017). This suggests that there are Bell monogamy relations stronger than the two listed above.
A concrete such Bell monogamy relation is presented at the top left corner of Fig. 2. It is a very condensed graph where four observers aim at testing four tripartite Bell inequalities. The red graph is the line graph of the original hypergraph. Since each and there are edges of in total, the averaging method predicts , whereas the tight bound is Kurzyński et al. (2011):
Since the monogamy relation (6) involves four Bell parameters and it is bounded by , it shares with relation (2) its physical implication. Namely, if one Bell inequality is violated, another must be satisfied. It is therefore interesting to augment the set of monogamy relations with inequality (6) and verify which tight monogamy relations follow from the averaging method. In fact, since (4) is a special case of (6), it is sufficient to replace one with the other. Likewise, (5) is a special case of the following monogamy relation, presented at the top right corner of Fig. 2,
This leads us to ask whether a finite set of elementary monogamy relations exists, i.e. such a set that the averaging method produces tight monogamy relation for arbitrary hypergraph. Note that when adding (6) and (7) to the set of elementary relations, the line graph method must be suitably updated. Since the relations (6) and (7) involve more than two Bell parameters, an edge in the line graph may connect more than two vertices. Therefore the line graph needs to be upgraded to a hypergraph. In contrast to the bipartite case, there may be more than one line hypergraph for each original hypergraph. We shall take into account all possible line hypergraphs in the averaging method.
We may attempt to construct the set of elementary relations by a brute force algorithm searching over all hypergraphs with vertices and hyperedges, each covering vertices (Appendix C). In principle, if this algorithm were to be run for infinitely large , it returns a set of elementary monogamy relations . We shall now argue that such a set must in fact be infinite, in stark contrast to the case of bipartite Bell inequalities.
The infinite set. The idea is to construct a set of hypergraphs with increasing number of vertices, for which it will be shown that their corresponding monogamy relations obtained using the averaging method are not tight even if the set is composed of all elementary monogamy relations with smaller number of observers. Part of the set we study is depicted in Fig. 3. We consider cyclic hypergraphs that involve an odd number of Bell inequalities, , which are tested by observers.
Assume that the algorithm above returned a finite set and let be the highest number of observers involved in any elementary monogamy relation in . We choose to begin our analysis with the graph that has the number of vertices higher than . In this way we rule out the case that is (a subgraph of a hypergraph) already present in . Therefore the only way of obtaining a bound on the monogamy relation corresponding to is to combine graphs or subgraphs of monogamy relations in that simultaneously are the subgraphs of . Hence we study the subgraphs of . The only non-trivial ones are connected graphs involving consecutive Bell parameters. The case of is covered by the monogamy relation (5), for which the bound is achieved, e.g., if the first three particles are in the Greenberger-Horne-Zeilinger (GHZ) state Kafatos (1989). For any higher , the corresponding monogamy relation has to have the bound of at least as this is the number obtained if the triples of particles tested in every second Bell parameter are in the GHZ state. Since our method is averaging these monogamy relations, it follows that the Bell monogamy relation corresponding to has the bound of at least . A concrete example how the bound of is obtained is presented in Fig. 4.
We now show that the bound is not tight, i.e. there is no quantum state and measurements achieving it (recall that is odd). We point out properties that such a hypothetical state would have to satisfy and show that they are contradictory. Let us label the Bell parameters in by index . A way to obtain the bound is presented in Fig. 4 and involves summing up pairs of consecutive Bell parameters , for , with . Therefore, saturation of the bound of implies saturation of every constituent monogamy, i.e. for all ’s in question. Recall that the bound is proved by partitioning observables that enter the upper bound (1) into 4 groups, each of mutually anti-commuting observables (Appendix A). According to correlation complementarity the constituent monogamy is saturated if each group of anti-commuting observables saturates the bound of . In particular, we have , where is defined as
with e.g. denotes Pauli- operator acting on the qubit . Since there is an odd number of Bell parameters, each must be exactly . It is perhaps worth a comment that one cannot proceed any further using correlation complementarity alone. We shall now utilize the relation between correlations and marginal expectation values introduced previously in the context of non-local hidden variable theories Leggett (2003); Gröblacher et al. (2007); Branciard et al. (2008). We construct an observable that on one hand necessarily has high expectation value but on the other hand it must have small average as it anti-commutes with observables that enter and , leading to a final contradiction.
Let us consider . We introduce observable , with normalized vector parallel to . It has expectation value , because . Similarly, we find observable with expectation value following from . For the two observables and have no overlapping qubits and can be measured simultaneously. Their product satisfies the lower bound (Appendix D):
At the same time one verifies that observable together with observables entering and form a pairwise anti-commuting set. Therefore, by the correlation complementarity we have
Inequalities (9) and (10) contradict . Summing up, there is no state and measurements which achieve the bound of derived from the averaging method applied on a sequence of cyclic hypergraphs . Accordingly, the set of elementary tripartite monogamy relations must contain an infinite number of monogamy relations — at least those that correspond to all ’s with odd .
Optimistic coda. We showed a simple method to produce Bell monogamy relations for an arbitrary arrangement of observers and Bell inequalities. For bipartite inequalities, the monogamy relations obtained are tight and leverage a single elementary monogamy relation derived in Ref. Toner and Verstraete (2006). Using a combination of correlation complementarity and inequalities for marginal expectation values, we showed that in the multipartite case, however, there is no finite set of elementary monogamy relations from which tight relations corresponding to arbitrary complicated graphs could be obtained. Nevertheless, the method is still useful as it does produce non-trivial tight Bell monogamy relations. For example, a tight monogamy relation for a completely connected graph is obtained by averaging (6) only, combining elementary monogamy relations of different types can give tight relations, e.g. top of Fig. 5, as well as tight relations for higher number of observers, e.g. bottom of Fig. 5 and Appendix E.
Acknowledgements.Acknowledgments. This work is supported by the Singapore Ministry of Education Academic Research Fund Tier 2 project MOE2015-T2-2-034 and NCN Grant No. 2014/14/M/ST2/00818. M.C.T acknowledges support from the NSF-funded Physics Frontier Center at the JQI and the QuICS Lanczos Graduate Fellowship. R.R. acknowledges support from the research project “Causality in quantum theory: foundations and applications” of the Fondation Wiener-Anspach and from the Interuniversity Attraction Poles 5 program of the Belgian Science Policy Office under the grant IAP P7-35 photonics@be.
Appendix A Correlation complementarity and Bell monogamy relations
Here we give examples how correlation complementarity yields Bell monogamy relations. In particular, we detail groups of anti-commuting observables that lead to monogamy relations (2) and (5), the latter used in the proof that the set of elementary tripartite relations is infinite.
Given a set of mutually anti-commuting observables , correlation complementarity states that , where are the expectation values in the state Kurzyński et al. (2011); Tóth and Gühne (2005); Wehner and Winter (2008, 2010). In the first example, let us show how to use correlation complementarity to prove Eq. (2):
The relevant observables that enter into the upper bound (1) are (for ) and (for ).
These eight observables can be partitioned into two sets, namely and , each containing only mutually anti-commuting observables.
Eq. (2) follows by direct application of correlation complementarity to these two sets.
Similarly, is upper bounded by the sum of squared expectation values of observables. They can be arranged into groups of mutually anti-commuting observables, e.g. the columns of the following table:
Correlation complementarity gives a bound of 1 for each group, and hence a bound of 4 for .
Appendix B Configurations with quantum trade-off completely characterized by Steinmetz solids
We shall give two examples of configurations where the intersection of the elementary monogamy relations precisely captures the trade-off relation within quantum theory. The first example is the star configuration with arbitrary number of observers, see Fig. 6, and the second example involves four observers in the chain configuration, see Fig. 7. The resulting quantum sets of allowed values of Bell parameters are intersections of cylinders having the same radii, the sets known as the Steinmetz solids. We note that Toner and Verstraete already realized that the Steinmetz solid corresponding to the triangle configuration is not the quantum set, it contains points which cannot be realized in quantum physics Toner and Verstraete (2006).
b.1 Star configuration
Consider observers arranged in a star network with a central Alice and remaining observers, Fig. 6. We show that any set of values that obeys for is achievable by a suitable shared quantum state and measurements.
Let us take a Cartesian coordinate system in dimensions with each axis giving the value of . Clearly, any point in the intersection of the cylinders can have only one coordinate, say , larger than the local bound of . At this point, the remaining Bell expressions can attain at most the value . The shared quantum state achieving these values is given as
with and , with a parameter . The relevant correlations are in the place and give
For the range , we see that violates the local bound of , and every relation of the form is saturated.
Moreover, there is a freedom to control the parameters for such that not all of them achieve the maximum possible value of in this situation. To reduce the value of any individual , we replace in (13) the state at position by the noisy state . The new state then gives
By controlling the noise levels we see that the strategy allows to achieve every possible point within the Steinmetz solid. The intersection of the cylinders is therefore precisely the shape of the set of -party quantum correlations projected onto the space of two-party (CHSH) Bell parameters.
b.2 Chain configuration
Here we provide another example where the Steinmetz solid completely characterizes quantum trade-offs between violations of Bell inequalities. The configuration is illustrated in Fig. 7 and involves three bipartite Bell parameters and , and two elementary monogamy relations:
The intersection of the two cylinders above can be split into one of the following parameter regions:
We shall analyze them one by one. Note that cases (P2) and (P4) are obtained by symmetric interchange of qubits and , while case (P5) is the region within the local set which is evidently realizable within quantum theory.
Case (P1) To realize the region defined by (P1), we consider the correlations in the plane of the state
with and . The values of Bell parameters are:
Considering without loss of generality the region where , we choose the following real coefficients
for angles given as
which are well-defined for any values and . For , Eq. (B.2) gives .
Case (P2) The region (P2) is realized by the state
with , and . The corresponding Bell parameters calculated in the plane are:
Case (P3) The region (P3) is realized by a state similar to that defined in Sec. B.1, about the star configuration. We compute the values of the Bell parameters from the correlations in the plane for the state
with and , and parameter . This gives
To obtain smaller values of , we add noise to the qubits at positions and as in Sec. B.1.
Appendix C Counting all elementary monogamy relations
This section details a simple brute force algorithm to list all elementary monogamy relations. We shall focus on , in which case the first nontrivial graph has and , giving rise to monogamy relation (4). The set of elementary monogamy relations is denoted as and we now add to it the first member given by (4). The algorithm then enters a loop as follows:
construct all the hypergraphs with vertices and hyperedges
let index loop over all these hypergraphs, and let denote the hypergraph corresponding to
for each find the bound on the monogamy relation , corresponding to , using averaging of the elementary relations in
check if the bound is tight
if it is tight, move on to the next
if it is not tight, add to and remove from all monogamy relations that correspond to the subgraphs of and have the same bound as
loop over [its maximum number is ], then loop over
Appendix D Correlations versus local expectation values
Consider dichotomic observables , measured on different sets of particles. If , then
By assumption and can be measured simultaneously. Let us denote by and the measurement outcomes obtained in a single experimental run. They satisfy:
Averaging over many runs of the experiment gives
Since the average of modulus is at least the modulus of average, we have
If signs of and are the same, then . The left inequality in Eq. (27) then implies
By our assumption, this time both sides are non-positive. Multiplying the inequality with itself results in Eq. (24). In (24) we use the operator notation to stress that and are measured on different particles. ∎
Appendix E Monogamy relations for higher number of observers
Here we show how to combine -partite Bell monogamy relations, obtained from correlation complementarity, to get tight -partite relations.
We start from the monogamy relation for three-party inequalities in Eq. (6), and use it to derive monogamy between four-party inequalities. Recall Fig. 2, which presents configuration corresponding to Eq. (6). We construct the network for four-party inequalities by adding a new party, labeled ‘0’, who takes part in the four-party Bell experiment with each of the sets in , as well as with a copy . We derive the following tight monogamy relation from (6):
To this end, we first recall that Eq. (6) was proven by grouping the relevant observables into the following anti-commuting sets
where the last line in the table indicates the four observables obtained from the previous lines by an interchange at each site , i.e., , etc, making a total of eight anti-commuting observables in each of the four sets. In order to prove Eq. (30) we form the requisite eight sets of anti-commuting observables by adjoining to each of the above sets and to the corresponding sets for the qubits.
In addition to the above four sets, we obtain four more sets of anti-commuting observables from each of the above sets by interchanging , i.e., , etc.
It is readily verified that the observables within each column in Tab. 32 anti-commute. Moreover, the construction extends so that starting from any network where one has derived a monogamy relation from correlation complementarity for -party inequalities, one can obtain a tight monogamy relation for ()-party inequalities in a “tree” network, with a central qubit and with the two “leaves” corresponding to the original -party network. Furthermore, the derived inequalities are tight and completely characterize the quantum trade-off relation (see Appendix B), which is ensured by considering the correlations in the plane of the following state Kurzyński et al. (2011); Ramanathan and Mironowicz (2017):
where and normalized, denotes the hyperedge and denotes the all- state on the (remaining) qubits in the network.
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