Exploring Bell inequalities for the device-independent
certification of multipartite entanglement depth
Techniques developed for device-independent characterizations allow one to certify certain physical properties of quantum systems without assuming any knowledge of their internal workings. Such a certification, however, often relies on the employment of device-independent witnesses catered for the particular property of interest. In this work, we consider a one-parameter family of multipartite, two-setting, two-outcome Bell inequalities and demonstrate the extent to which they are suited for the device-independent certification of genuine many-body entanglement (and hence the entanglement depth) present in certain well-known multipartite quantum states, including the generalized Greenberger-Horne-Zeilinger (GHZ) states with unbalanced weights, the higher-dimensional generalizations of balanced GHZ states, and the states. As a byproduct of our investigations, we have found that, in contrast with well-established results, provided trivial qubit measurements are allowed, full-correlation Bell inequalities can also be used to demonstrate the nonlocality of weakly-entangled unbalanced-weight GHZ states. Besides, we also demonstrate how two-setting, two-outcome Bell inequalities can be constructed—based on the so-called GHZ paradox—to witness the entanglement depth of various graph states, including the ring graph states, the fully-connected graph states, and some linear graph states etc.
Entanglement Horodecki et al. (2009) is a feature of quantum theory that has no analogue in classical theory. Various tasks that were thought to be impossible in classical theory are now made possible by utilizing this precious resource. For example, secure secret keys between two remote parties can be established with the help of pairs of entangled states Ekert (1991). Even an unknown quantum state can be teleported Bennett et al. (1993) from one place to another intact if one has access to maximally entangled states. These examples involve only entanglement between two parties but the concept of entanglement goes beyond this. With more complicated entanglement structures Lu et al. (2018), such as that endowed by the two-dimensional graph states Briegel and Raussendorf (2001), one can perform universal quantum computation Raussendorf et al. (2003) through a one-way quantum computer Raussendorf and Briegel (2001). Entanglement can thus be seen as useful resources for a variety of different tasks.
To utilize the advantages of entangled states, one must first prepare the desired entangled states relevant to the protocols. However, imperfections always arise during the preparation stage and hence one may end up having something very different from the desired target state. How can one be sure that the prepared state is indeed useful for the tasks in mind? Even if one only wishes to certify that the state is entangled, an analogous question remains, namely, how can one perform such a certification when the measurement devices themselves may be subjected to imperfections Rosset et al. (2012)?
Tools developed for device-independent quantum information (DIQI) Brunner et al. (2014); Scarani (2012) are tailored-made to address such problems. More precisely, the statistics derived from locally measuring the prepared state allow one to reach nontrivial conclusions about the system with minimal assumptions — specifically, no assumption about the internal workings of any of the devices involved is needed. For instance, whenever the measurement statistics observed in a Bell experiment show a violation of Bell inequalities Bell (1964), one can immediately conclude that the shared state is entangled Bancal et al. (2011). Moreover, the amount of entanglement present can be lower bounded from the violation of Bell inequalities Moroder et al. (2013); Chen et al. (2016); Cavalcanti and Skrzypczyk (2016); Chen et al. (2018). In some cases, it is even possible to certify nontrivial properties of the measuring apparatus Mayers and Yao (2004); Chen et al. (2016); Cavalcanti and Skrzypczyk (2016); Renou et al. (2018); Bancal et al. (2018).
Going beyond the bipartite scenario, the structure of entanglement can be much more complicated Lu et al. (2018). For example, the entanglement may not involve all subsystems but only a subset of them. In other words, an -partite state that is entangled may only contain -body entanglement, with . To capture this, the notion of entanglement depth Sørensen and Mølmer (2001), which is closely related to that of -producibility Gühne et al. (2005), has been introduced. More recently, a device-independent certification of the entanglement depth was theoretically shown to be feasible Liang et al. (2015) (see also Refs. Nagata et al. (2002); Yu et al. (2003); Aloy et al. (2018)) for certain multipartite quantum states, such as the family of Greenberger-Horne-Zeilinger (GHZ) qubit states.
To produce a genuinely multipartite entangled (GME) quantum state (one where the entanglement indeed involves all subsystems) beyond a handful of subsystems is experimentally challenging Monz et al. (2011); Lu et al. (2018). Interestingly, a device-independent certification of the presence of this strongest type of entanglement remains possible Bancal et al. (2011); Moroder et al. (2013), even for an arbitrary GME pure quantum state Zwerger et al. (2019). The generic construction provided in Ref. Zwerger et al. (2019), however, apparently requires a Bell test that has too many measurement settings, thus making it infeasible in many practical situations.
In this work, we consider Bell inequalities that require only two binary-outcome measurements per party and characterize their ability to serve as device-independent witnesses for entanglement depth for various multipartite pure states, including unbalanced-weight qubit GHZ states, higher-dimensional generalization of the GHZ states, states Dür et al. (2000), 1-dimensional cluster states Briegel and Raussendorf (2001) with periodic or closed boundary conditions, as well as a few 2-dimensional graph states. For the first two kinds of states, we make use of the 1-parameter families of witnesses proposed (but never analyzed) in Ref. Liang et al. (2015). For the graph states, we construct new Bell inequalities based on the so-called GHZ paradox Kafatos (1989); Mermin (1990a) and show that these inequalities can indeed be used to witness the GME nature of (many of) these states.
The rest of this paper is organized as follows. In Sec. II we introduce the notations used in this paper and briefly recall the definitions of producibility and entanglement depth. Then, we present our analysis of the generalized witnesses introduced in Liang et al. (2015). We also show in Sec. III.2 that those inequalities could be used to detect the genuine multipartite entanglement of generalized GHZ states. In Sec IV we present our inequalities constructed for graph states and show that the genuine multipartite entanglement of the states can be revealed. In Sec V we summarize our work and discuss some possible future research directions.
The basis for employing a device-independent witness for entanglement depth (DIWED) is a multipartite Bell experiment where a shared quantum state is locally measured to estimate the correlation present between the observed measurement outcomes. More precisely, consider spatially separated parties and with each of them allowed to perform two dichotomic (i.e., binary-outcome) measurements. We denote the measurement choice of the -th party and the corresponding measurement outcome, respectively, by and . We then use the -bit vector to describe the collection of measurement settings of all parties and label their respective measurement outcomes by . According to Born’s rule, the probability of observing outcomes given the measurement choices read as:
where is the positive-operator-valued-measure (POVM) used to describe the -th party’s -th measurement. Given the conditional probabilities, we can further define the -partite correlators as
In these notations, a (linear) Bell expression is defined by a certain linear combination of conditional probabilities with weights specified by . By maximizing this linear expression over all in (the Bell-local set Brunner et al. (2014); Scarani (2012)), one obtains a Bell inequality:
It is well-known Werner (1989) that regardless of the measurements employed and the fully separable state shared, the resulting always satisfies the above inequality. Consequently, if the observed gives rise to a violation of the Bell inequality , one can immediately conclude that the shared state is entangled — this is the observation that allows one to employ a Bell inequality as a device-independent entanglement witness Bancal et al. (2011).
ii.2 Entanglement depth, entanglement intactness, and their device-independent certification
As mentioned above, entanglement in a multipartite scenario is much more complicated. To this end, if an -partite pure state can be separated into tensor factors, i.e.,
then it is said to be -separable Horodecki et al. (2009). Producibility is a closely related concept but focuses instead on the number of parties in each subgroup: is said to be -producible Gühne et al. (2005) if the number of parties defining each tensor factor is at most -partite. For example, any -partite state is, according to the definition given, trivially 1-separable and -producible. In contrast, a fully separable -partite state is -separable as well as being -producible. These examples make it evident that one is generally interested in the smallest for which a given state is -producible but not -producible — the quantity is then known as the entanglement depth Sørensen and Mølmer (2001) of . Likewise, one is interested in the largest for which a given state is -separable but not -separable — the quantity is then known as the entanglement intactness Lu et al. (2018) of . An -partite GME state is one that has an entanglement depth of and an entanglement intactness of 1.
The separability and producibility for mixed states are similarly defined. If a density matrix can be written as a convex mixture of pure states that are -separable (resp. -producible), then we say that it is -separable (resp. -producible), thus, the set of -separable (resp. -producible) states are convex. Moreover, the set of -separable (-producible) states is a subset (superset) of -separable (-producible) set for all (). With some thoughts, one realizes that the convexity of these sets translate, via Eq. (1), into the convexity of the set of that can be obtained from -separable (-producible) states (if we impose no restriction on the Hilbert space dimension). The separating hyperplane theorem then dictates that for any that does not belong to the -separable (-producible) set for some fixed (), one can construct a linear witness to separate from the corresponding set. In other words, we may use a Bell inequality, Eq. (3), or more precisely, the strength of violation of a Bell inequality to certify that a given cannot arise from any -separable (-producible) state for some fixed (), thus putting an upper bound (lower bound) on the entanglement intactness (depth) of the underlying state. Put it differently, for any Bell inequality , if one can determine -producible bound:
then an empirical observation of would certify an entanglement depth of the underlying state that is at least . Since this conclusion holds without invoking any assumption about the dimension of the underlying state , let alone the measurements employed, the inequality
serves as a DIWED by placing a lower bound of on the underlying entanglement depth. In the terminology of Ref. Curchod et al. (2015), this means that Eq. (6) is a constraint that has to be satisfied by a nonlocal quantum resource of minimal group size .
Likewise, a device-independent witness for entanglement intactness:
can be established by determining the -separable bound:
A violation of the witness given in Eq. (7) then allows one to put a device-independent upper bound of on the entanglement intactness of the underlying state.
The concepts of -separability and -producibility are evidently closely related. In fact, it is easy to see that Gühne et al. (2005) a quantum state that is -producible is necessarily -separable for while a quantum state that is -separable is necessarily -producible for . In particular, if we are to determine Eq. (5) for , it is equivalent to computing Eq. (8) for . Similarly, if we are to determine Eq. (5) for , it is equivalent to computing Eq. (8) for . Bearing this in mind, we remark that multipartite Bell inequalities whose -separable bounds have been determined can already be imported to serve as DIWEDs. A particularly worth noting example of this is the family of Mermin-Ardehali-Belinskii-Klyshko (MABK) inequalities Mermin (1990b); Ardehali (1992); Roy and Singh (1991); Belinskiĭ and Klyshko (1993); Gisin and Bechmann-Pasquinucci (1998), whose -separable bounds for an arbitrary value of have been determined, see Refs. Nagata et al. (2002); Yu et al. (2003). Turning the argument around, we see that Bell inequalities for which the -producible bounds have been determined, such as those given in Ref. Liang et al. (2015) can also be used to certify an upper bound on entanglement intactness.
ii.3 Reduction for pure states
In Ref. Liang et al. (2015), it was left as an open problem whether the entanglement depth of an arbitrary -partite pure state can be certified in a device-independent manner. Here, we shall demonstrate that the problem reduces to that of certifying device-independently the genuine -partite entanglement for all -partite GME states. To this end, let us remind that with an appropriate choice of local bases, a multipartite pure quantum state can always be cast in the form of Eq. (4) for some choices of and such that (1) is its entanglement intactness, and (2) the maximal size111Here, size refers to the number of subsystems involved in the definition of . of is the entanglement depth of .
Suppose that for an arbitrary -partite GME pure state , there exists a Bell-like inequality whose violation can be used to certify the GME nature of . Now, let be the tensor factor of which determines its entanglement depth, i.e.,
Then, a DIWED that can be used to certify the entanglement depth of is given by , when applied to the subsystems (i.e., the partial trace) of that give . If necessary, one could trivially extend this -partite DIWED to make it an -partite DIWED by lifting Pironio (2005) the corresponding Bell-inequality to involve the remaining parties.
To this end, let us remark that the recent work of Ref. Zwerger et al. (2019) has indeed provided a generic recipe for the construction of such a Bell-like inequality for an arbitrary GME pure state. Unfortunately, their construction requires one to perform a Bell test that generally involves many measurement settings. Thus, there remains the problem of finding tractable DIWEDs for an arbitrary pure GME pure state. For the family of -partite GHZ states,
the problem is solved using the family of MABK inequalities Mermin (1990b); Ardehali (1992); Roy and Singh (1991); Belinskiĭ and Klyshko (1993); Gisin and Bechmann-Pasquinucci (1998), or the witnesses given in Ref. Liang et al. (2015). For a generic multipartite pure state, however, there is no known systematic construction that involves only a few measurement settings. In what follows, we start by exploring the family of DIWEDs given in Ref. Liang et al. (2015) and its 1-parameter generalization to understand its usefulness when it comes to witnessing the GME nature of other families of states. Later, in Sec. IV, we provide a systematic construction that allows us to witness the GME nature of 1-dimensional cluster states.
Iii A one-parameter family of DIWEDs
In Ref. Liang et al. (2015), the following family of Bell expressions was proposed as the first examples of a DIWED:
where is an -bit vector of twos. While the analysis in Ref. Liang et al. (2015) concerned predominantly the case of , we show in Appendix A.1 that has a local bound of 1 for all . When , Eq. (11) leads to the well-known Clauser-Horne-Shimony-Holt (CHSH) Bell inequality Clauser et al. (1969):
Let denote the maximal quantum value of . It was shown in Ref. Liang et al. (2015) that for , the -producible bound of , cf. Eq. (5), is precisely and thus depends only on , but not on . Numerically, for , we have found that the maximal quantum value of can always be achieved by considering the -partite GHZ state and the following ansatz Werner and Wolf (2001); Liang et al. (2015) of measurement observables:
where and . The resulting quantum value, computed via , then reads as:
where the analytic form of the optimal parameter and optimal quantum value for can be found in Appendix B. Note that for these values of , appear to increase monotonically with as well as , thus showing a gap between the and producible bounds for . In Ref. Liang et al. (2015), the presence of such a gap was numerically verified for but only for the case of .
iii.1 Effectiveness for some families of qubit states
Under the assumption that there always exist the aforementioned gaps between successive -producible bounds and noting that the maximal quantum violation of a full correlation Bell inequality—such as the one associated with Eq. (11)—is always achievable Werner and Wolf (2001) using a GHZ state, we deduce that the GME nature of a GHZ state can always be certified using the DIWED
by setting to be .
In Ref. Liang et al. (2015), the DIWEDs of Eq. (15) with were also considered in conjunction with the states , and the 1-dimensional cluster states (i.e., graph states corresponding to a ring graph or a linear chain ). Unfortunately, the numerical results presented therein suggested that for (), the witness can, at best, be used to certify that ( and ) is entangled.222Note that in Table II of Ref. Liang et al. (2015), the certifiable entanglement depth of was mistaken to be 3, even though the result presented in Table III in the corresponding Supplemental Material clearly indicate that this should be 2. In other words, the witness of Eq. (15) with essentially failed to detect any of the more-than-two-body entanglement present in all these multipartite qubit states.
Would tuning the parameter help in providing a better lower bound on the entanglement depth of any of these states? In the case of , this indeed turned out to be of some use. By numerically maximizing Liang and Doherty (2007) the quantum value of for over the POVMs used to define qubit measurements, we have found that for (some instances of) , the optimized quantum value of for with indeed exceeds the corresponding 2-producible bound , thus correctly certifying the GME nature of while improving—in comparison with the DIWED—the entanglement depth certifiable for , and . The details of these findings are shown in Fig. 1.
Unfortunately, tuning the parameter does not seem to help at all for the device-independent certification of the entanglement depth of the 1-dimensional cluster states. In Sec. IV, we return to these states and present other DIWEDs that are naturally suited for them.
Now, let us focus, instead, on the following family of unbalanced-weight GHZ states:
where defined in Eq. (10) corresponds to the special case of .
For , it is known Gisin (1991) that all such states violate the CHSH Bell inequality. However, for , it is also known Żukowski et al. (2002) that for odd and , cannot violate any full correlation Bell inequality with rank-1 qubit projective measurement. It thus seems unlikely to certify the entanglement depth of all these GME states using the DIWED of Eq. (11). In fact, our numerical results, see Fig. 2, suggest that the interval of for which we can correctly certify the GME nature of shrinks as increases. Again, tuning the value of the parameter does not seem to help.
In contrast with the results of Ref. Żukowski et al. (2002), we have nonetheless found that by allowing trivial (degenerate qubit) measurements, for (at least) , it is possible to demonstrate the nonlocality of all entangled . As an example, let us consider . Note that each nontrivial qubit observable can be parametrized by a unit vector where , are, respectively, the polar angle and the azimuthal angle of . To exhibit the nonlocality of the aforementioned 3-partite state, it suffices for the three parties to set their first measurement observable to be the one associated, respectively, with the unit vectors of (0.7734 rad., 0.6767 rad.), (0.7457 rad., 0.1533 rad.) and (0.2295 rad., 0.8300 rad.) while the second observable, respectively, as , , and .
iii.2 Effectiveness for higher-dimensional GHZ states
Next, we discuss the effectiveness of using 11 to certify the ED of the -partite -dimensional GHZ states . Importantly, each of these GME states can actually be seen as the direct sum of -qubit GHZ state residing in disjoint qubit subspaces (and a product state if is odd):
where is the -qubit GHZ state acting on the local qubit subspace spanned by , if is odd but vanishes otherwise.
In view of this, it is not surprising that the certifiable ED apparently depends on the parity of the local Hilbert space dimension.
In particular, the very same maximal quantum value of Eq. (11), i.e., given in Eq. (14), is attainable using whenever the local Hilbert space dimension is even. Explicitly, this can be achieved using the following choice of block-diagonal qudit observables:
where , are defined in Eq. (13), the -th qubit observable in each of these direct sums acts on the qubit subspace spanned by , and the full correlator is computed as . Thus, the DIWED of Eq. (11) can correctly certify the ED of such states, as with the case for .
On the other hand, when the local Hilbert space dimension is odd, the ED certifiable using the DIWED of Eq. (11)—based on our numerical results—is not tight. However, these bounds on ED do become tighter and approach the actual ED as the Hilbert space dimension increases. Specifically, with the following choice of block-diagonal qudit observables:
we recover the best quantum value that we have found for these states.
The effectiveness of the DIWED of Eq. (11) in certifying the entanglement depth of for odd can then be decided by verifying if the following inequality holds true for the given local Hilbert space dimension and for some :
To this end, note that if the above equation holds for some value of , , and , then it must also hold for the same combination of , , and . In Fig. 3, we plot for each , the value of and the smallest value of found by optimizing the choice of (with the optimal value denoted by ). As with the case of , we have found that for with , tuning the parameter can sometimes lead to a tighter lower bound on the ED of these latter states.
Iv DIWED for graph states
iv.1 Graph states and stabilizers
We now return to the problem of certifying, in a device-independent manner, the GME nature of graph states. Specifically, we shall demonstrate how one can construct a non trivial Bell-inequality suited for certifying the ED of certain qubit graph states. Before that, let us first recall from Ref. Hein et al. (2004a) the definition of a qubit graph state. For a given graph, with vertices and edges, consider the following tensor product of qubit operators
where is the set of neighbors of vertex , , , and represent, respectively, the Pauli matrix , , and acting on the -th, -th, and -th qubit. For simplicity, we have omitted in the above equation the identity operator acting on the remaining (non-neighboring) qubits. The of Eq. (21) is known as a stabilizer of the corresponding graph state , which is defined as the simultaneous eigenstate of all stabilizers .
We should further introduce the stabilizer set formed by all possible combinations of the product of :
where is a subset of vertices of the graph G. Notice that since commutes with for all , the order of the products does not matter. The corresponding is evidently also the eigenstate of all .
iv.2 Nonlocality of graph states
The reason of introducing is that we can use its elements to construct a Bell operator Braunstein et al. (1992), which can be translated into a Bell inequality naturally suited for . The idea was first proposed in Gühne et al. (2005), where they summed over all elements of to obtain the Bell operator:
To get a Bell expression in the form of Eq. (3), one associates—to each term —the Pauli matrices , respectively, with measurement setting . Each term is then mapped to a correlator with . Importantly, depending on the graph G, the corresponding correlator of may be a marginal correlator, i.e. one involves only nontrivial measurements on a strict subset of subsystems and hence . Graphs of nodes arranged in a ring (straight line)—giving rise to the -partite ring (linear) graph states ()—are some explicit examples of this kind, see Figure 4. By determining the local bound of the corresponding Bell expression, one then obtains a Bell inequality whose maximal quantum value of is guaranteed to be achievable by the underlying graph state Gühne et al. (2005).
iv.3 Two-setting DIWED for graph states based on GHZ paradox
Evidently, from an experimental points of view, it would be desirable to make use of a Bell inequality with fewer measurement settings (or fewer expectation values to be measured). To this end, we follow the proposal of Ref. Scarani et al. (2005) to construct Bell inequalities by choosing only a subset of the elements in , specifically those that allow us to demonstrate the so-called GHZ paradox Kafatos (1989); Mermin (1990a). Let us stress that as with the work of Ref. Gühne et al. (2005), Ref. Scarani et al. (2005) only concerns the demonstration of nonlocality for graph states, but here we wish to go a step further by using such a Bell inequality as a DIWED to witness the ED of the corresponding .
Consider, without loss of generality, a situation where distinct correlators take on their maximal value,
For simplicity, in Eq. (24) and in the following arguments, we write all the correlators as -partite correlators, but this is only for the convenience of presentation, rather than a necessary requirement of the arguments. In particular, some of these correlators could well be marginal correlators that involve less than parties. A GHZ paradox—also known as a proof of nonlocality without inequality—arises if Eq. (24) implies also for Bell-local correlations but for certain quantum correlation.
To appreciate when such a “paradox" may occur, let us consider extremal333For the case of nonextremal satisfying Eq. (24), one first decomposes into all extremal of satisfying Eq. (24) and repeats the following arguments. points of , i.e., those where the outcome of the -th measurement of the -th party is deterministic, taking either for all , and all . Then, it follows from Eq. (2) that each equation in Eq. (24), such as,
imposes nontrivial constraints on the value of . In particular, if is chosen such that
then by virtue of Eq. (24) and Eq. (26), we must also have . In other words, if is chosen as the string of inputs where each only appears an odd number of times in Eq. (24), then these conditions of perfect correlations guarantee also .
To complete the argument of the paradox, one must find quantum correlation satisfying Eq. (24) and giving . While the constraints of Eq. (24) are easily enforced by choosing the observables associated with each correlator to form a stabilizer of the graph state of interest, the latter constraint of perfect anticorrelation (and the requirement of having only two measurements per party) can only be achieved with a careful selection of the stabilizers corresponding to Eq. (24). For any such selection, a nontrivial Bell inequality violated quantum-mechanically up to the algebraic maximum of can then be constructed as:
A proof of this is given in Appendix A.2. In what follows, we provide a construction of such Bell inequalities for various graph states where each of the first correlators is identified with an element of while the last term is identified with the product of all these chosen ’s.
iv.3.1 Ring graphs
As a first example, let us consider . From its stabilizer set , we choose the three stabilizers , , and their product to define our Bell operator, i.e.,
Writing these operators explicitly while associating with the 1st measurement and with the 2nd measurement gives:
Taking the sign of each term into account, we obtain
which is exactly a representative of the MABK Bell expression Mermin (1990b); Ardehali (1992); Roy and Singh (1991); Belinskiĭ and Klyshko (1993); Gisin and Bechmann-Pasquinucci (1998), whose quantum 2-producible bound Curchod et al. (2015) is known Nagata et al. (2002) to be . Note that when this bound is saturated, it may still be possible to certify genuine tripartite entanglement in a device-independent manner if marginal distributions are taken into account Bhattacharya et al. (2017).
The resulting Bell inequality thus has the following properties:
thus making a DIWED that can be used to certify the GME nature of .
More generally, we propose to consider the following Bell operator, each involving stabilizers from and their product:
In Appendix C, we show that each of these Bell operators only involves two different qubit measurements per party444Note that, if any of the parties performs only one measurement during the Bell experiment, as we show in Appendix F, such a Bell expression can never serve as a DIWED for the corresponding graph state as the 2-producible bound always coincides with the Tsirelson bound.. For the case of odd , these are always and . The same observation applies for party 2, 3, , in the case of even , but for party 1, , and , these become, respectively, , and .
Throughout, we shall adopt the following convention in mapping a stabilizer to the corresponding correlator: if in the Bell operator (and hence the list of stabilizers) considered, only and are involved, we associate these operators, respectively, as the 1st and 2nd measurement of the -th party; if instead, only and are involved, we associate them, respectively, as its 1st and 2nd measurement; and if only and are involved, we associate them, respectively, as its 1st and 2nd measurement.
Thus, for odd , the resulting Bell expression reads as:
where is a triparite correlator involving only the first three parties555Here and below, we use the symbol to indicate the trivial measurement setting. and is a short hand to denote the additional terms that need to be included to make the Bell expression invariant under arbitrary cyclic permutation of parties (cf. Ref. Grandjean et al. (2012)).
In a similar manner, we note from that it only involves (see Appendix C for details). We thus arrive at the following Bell expression for :
For , involves only and for . The corresponding Bell expression is (see Appendix C for details):
Using the numerical technique of Ref. Moroder et al. (2013), upper bound on the -producible bounds of for all can be computed. The results are summarized in Table 1. In particular, it is worth noting that the (upper bound of the) -producible bound always appear to be , thus showing that the constructed Bell inequality can indeed serve as a DIWED for .
iv.3.2 Fully-connected graphs
Apart from , the symmetry presented in fully-connected, i.e., complete graphs also allows us to derive a simple yet nontrivial family of Bell inequalities for the corresponding graph states. For concreteness, let us denote the state of an -partite fully connected graph by . Although is known Hein et al. (2004b) to be LU equivalent to —which we already know how to certify its entanglement depth using the DIWED of Sec. III—the simplicity of this family of Bell expressions warrants a separate discussion. To this end, consider the Bell operator
which is easily verified to involve both and measurements for all parties. We thus have the following Bell expression:
where we have used to indicate the additional terms that need to be included to ensure that the first terms are invariant under arbitrary cyclic permutation of parties. Note that when , is again a special case of the MABK inequalities Mermin (1990b); Belinskiĭ and Klyshko (1993).
A remark is now in order. In our previous construction of Bell inequalities based on the GHZ paradox, the value of is determined by all other terms appearing in the Bell expression. However, this is not the case for . In fact, the GHZ paradox can already be exhibited using , , and alone. The remaining terms are included to ensure that all parties perform two alternative measurements. Despite this difference, the local bound and quantum bound are easily shown, respectively, to be and (see Appendix A.2 for a proof of the local bound).
iv.3.3 Other types of graphs
Beyond the two families of graph states presented above, appropriate DIWED can also be constructed to certify (at least for up to 6) the entanglement depth of , i.e., the graph state associated with a linear chain (see Figure 4). Since is LU equivalent to , its entanglement depth can already be certified using the DIWED discussed in Sec. III. Likewise, we omit the discussion of , since it is LU equivalent to if we allow also the permutation of parties. On the other hand, our construction leads to the following Bell expressions, respectively, for and 6:
In Appendix D, we provide the Bell operators and the explicit from of the individual stabilizers leading to these Bell expressions. The -producible bounds for these Bell expressions are summarized in Table 1. One should note that, in general, the subset of chosen to construct a DIWED via our procedure is not unique. In particular, we provide in Appendix D an alternative (inequivalent) Bell expression that is also suitable for the certification of the entanglement depth of . In this regard, note also that some of these choices may lead to a Bell expression where one (or more) of the parties only has one fixed measurement setting. In Appendix F, we show that such Bell expressions are generally not useful for a device-independent certification of the entanglement depth of the corresponding graph state, as its Tsirelson bound—assuming some mild condition holds—can also be saturated using a biseparable state.
|4.0602||+ 2||+ 2||*6||-|
|+3||+ 3||+ 3||+ 3||*7|
For completeness, we have also considered the graph states associated with the three remaining 4-node (connected) graphs, see Figure 5 (i), (ii) and the star graph, as well as a 5-node graph [Figure 5 (iii)], and a 6-node graph [Figure 5 (iv)]. The graph state associated with the 4-node star graph state is known Hein et al. (2004b) to be LU equivalent to , whose entanglement depth is certifiable using the DIWED of Sec. III, we thus omit it from the following discussion. Likewise, the graph state associated with G1, and G2, are both known Hein et al. (2004b) to be LU equivalent to (and hence to if we also allow the permutation of parties), a DIWED constructed for can thus also be used to witness the entanglement depth of and .
The Bell expressions that can be used as DIWEDs for the remaining two states are, respectively,