Entanglement Witnesses for Graph States: General Theory and Examples
Abstract
We present a general theory for the construction of witnesses that detect genuine multipartite entanglement in graph states. First, we present explicit witnesses for all graph states of up to six qubits which are better than all criteria so far. Therefore, lower fidelities are required in experiments that aim at the preparation of graph states. Building on these results, we develop analytical methods to construct two different types of entanglement witnesses for general graph states. For many classes of states, these operators exhibit white noise tolerances that converge to one when increasing the number of particles. We illustrate our approach for states such as the linear and the 2D cluster state. Finally, we study an entanglement monotone motivated by our approach for graph states.
pacs:
I Introduction
The key role of entanglement is illustrated not only by its usefulness in many quantuminformational tasks, such as measurementbased quantum computation mqc and highprecision metrology metrology , but also by its fundamental importance for excluding certain models of nature in Bell tests belltests . Nowadays, experiments have succeeded in the preparation of 14qubit systems in ion traps blatt and tenqubit systems in photonic systems pan , so the characterization of multipartite entanglement is of high interest. Especially in the case of the most interesting kind of entanglement, genuine multipartite entanglement, general treatments turn out to be difficult multient ; ghzlimit ; huberdicke ; multientrev .
In Ref. ourpaper we proposed an alternative approach to this characterization by considering a relaxed version of the problem, leading to a criterion for genuine multipartite entanglement. Besides being easily implementable as a semidefinite program, it also provides surprisingly strong analytical entanglement criteria which can then be investigated further and generalized. As a first step, this has been done for the linear cluster state in Ref. ourpaper .
In this paper, we use this approach to develop a general theory of witnesses for graph states. Graph states are a family of multiqubit states which are of eminent importance for tasks like measurementbased quantum computation mqc or quantum error correction errcorr . These states have several interesting properties, for instance they are relatively robust against decoherence and violate certain Bell inequalities maximally heindiss . Recently, several experiments succeeded in preparing graph states of several qubits with photons pan ; graphexp , and also the theory of entanglement detection for such experiments has been investigated in a number of papers entdecstab ; entdecstabpra .
The main results of our paper can be grouped into two parts: First, we provide entanglement criteria, socalled entanglement witnesses, for all graph states up to six qubits. These witnesses are optimal in the framework of Ref. ourpaper , they detect more states than the graph state witnesses known so far and thus require a lower fidelity when measured in an experiment.
Second, we extend our results to arbitrary qubit numbers by providing a general theory of how to construct witnesses for arbitrary graph states. In many cases, these witnesses improve the best known witnesses so far and have white noise tolerances that approach one for an increasing particle number. This implies that for this type of noise the state fidelity can decrease exponentially with the number of qubits, but still entanglement is present and can be detected. Moreover, this improvement comes with very low experimental costs, since it is realized by measuring one additional setting in the experiment. Furthermore, a similar improvement can be achieved for witnesses that require only two settings to be measured entdecstab , which results in improved witnesses that consist of only two experimental settings in total.
The paper is structured as follows. In Sec. II, we start by presenting the structure of entanglement in the multipartite case and introducing the notions that will be used later, such as entanglement witnesses and graph states. Then, we will briefly recall the criterion of Ref. ourpaper in Sec. III. We will show that it can be reduced to a linear program in the case of graphdiagonal states.
Having laid the foundations, we first consider a certain class of witnesses, namely the class of fully decomposable witnesses ourpaper ; lewenstein . This is done in Sec. IV. We provide entanglement witnesses for all graph states of up to six qubits in Sec. IV.1. Then, in Sec. IV.2, we present analytical construction methods. We provide examples and give an extended construction for particular states in Sec. IV.4 including further examples.
In Sec. V, we move on to another class of witnesses, the fully PPT witnesses which are easier to characterize ourpaper . Here, we do not only provide a construction method for witnesses of this class (in Sec. V.1), but we can extend it to an even larger number of graph states compared with the case of fully decomposable witnesses. We present this extension in Sec. V.3. In order to illustrate that the presented methods can be exploited further, we supply a witness for the 2D cluster state (Sec. V.5).
Finally, we discuss an entanglement monotone for genuine multiparticle entanglement coming from the approach of Ref. ourpaper and show that graph states are the maximally entangled states for this entanglement measure. In the conclusion, we disscuss our results and possible extensions for the future. In order to make this paper as readable as possible, we provide nearly all proofs in the Appendix.
Ii Setting the stage
ii.1 Multipartite entanglement
First, we discuss the structure of the set of entangled states of multipartite systems, i.e., for systems of more than two particles multientrev . For the sake of an easy illustration (cf. Fig. 1), we consider the case of three particles here. Nevertheless, the generalization to a higher number of particles is straightforward.
A threequbit state is separable with respect to some bipartition, say, , if it can be written in the form
(1) 
Here, the coefficients form a probability distribution, i.e., they are positive and sum up to one. Let us denote states of this type by . Analogously, we define the sets of states which are separable with respect to other bipartitions and denote them by and . In Fig. 1, these three sets are drawn with a dashed, green border.
A state is called biseparable, if it can be written as a convex sum of states each of which is separable with respect to some bipartition. That is, any biseparable state can be written in the form
(2) 
where, the form a probability distribution. Thus, the set of biseparable states is given by the convex hull of states that are separable with respect to some bipartition. In Fig. 1, we show this convex hull with a dashed, thick green border. Any state that is not biseparable is called genuinely multipartite entangled.
Genuine multipartite entanglement is the strongest kind of entanglement, since biseparable states can be created by entangling, say, only two of three particles and then, to create a statistical mixture, forgetting to which pairs this operation was applied. In fact, in order to detect genuine multipartite entanglement, it is not enough to apply a bipartite criterion to every possible bipartition. Instead, in order to prove that a multipartite state is entangled, one has to show that it cannot be written in the form of Eq. (2).
There is, however, no efficient way to search through all possible decomposition of this form. Thus, it was the idea of Ref. ourpaper to relax the condition of being biseparable.
More precisely, for each fixed bipartition, we consider a superset of the set of separable states which can be characterized more easily than the set of separable states itself. There are different possible choices for supersets. However, in this paper, as a superset of the states that are separable with respect to, say, partition , we select the set of states that have a positive partial transposition (PPT) with respect to partition . A state is said to be a PPT state (with respect to ), if its partial transposition
(3) 
has no negative eigenvalues. We denote a state of this type by (and analogously for the other bipartitions). We refer to a state in the convex hull of these sets of PPT states as a PPT mixture. The set of PPT mixtures is therefore the set of states that can be written in the form
(4) 
where, again, the form a probability distribution. In Fig. 1, this set is shown with a solid, thick red border.
Since every separable state is necessarily PPT PPTcriterion ; horodeckis , every biseparable state is a PPT mixture. Therefore, showing that a state is no PPT mixture implies that it is not biseparable and therefore genuinely multipartite entangled. For some prominent states affected by white noise, such as the three and the fourqubit GHZ state, the threequbit W state and the fourqubit linear cluster state, being no PPT mixture happens to be necessary and sufficient for entanglement ourpaper . This is also true for the case that the PPT states , and live on a subspace of dimension or . However, since there exist states that are PPT with respect to every bipartition and therefore of the form given in Eq. (4), but are nevertheless genuinely multipartite entangled symmstates , not every entangled state can be detected in this way.
By considering the set of PPT mixtures, we exploit that it can be characterized more easily than the set of biseparable states. Numerically, this characterization allows for the use of linear semidefinite programming (SDP) sdp — a standard problem of constrained convex optimization theory. As we will see later, for an important class of states, namely socalled graphdiagonal states, this characterization can be cast into the form of a linear program (LP) which is an even simpler program.
ii.2 Entanglement witnesses
A useful tool that is often employed in experiments to show that a state is entangled are entanglement witnesses. A witness for genuine multipartite entanglement is an observable that has a nonnegative expectation value on all biseparable states, but a negative expectation value on at least one entangled state. Therefore, measuring a negative expectation value for in an experiment proves the presence of entanglement.
Every entangled state is detected by at least one witness horodeckis . Therefore, the question whether, for a given state , there exists a witness that detects it is equivalent to the question whether is entangled.
Let us now consider a certain subclass of witnesses that is central to our approach. In the case of two particles, A and B, a decomposable witness is defined as a witness that can be written as
(5) 
where and have no negative eigenvalues, i.e., are positive semidefinite, denoted by , lewenstein . Furthermore, is the partial transposition with respect to as defined by Eq. (3).
It can be easily seen that observables of the form given by Eq. (5) are positive (or zero) on all separable states . Any separable state has a positive partial transpose and therefore
(6)  
(7) 
We can now generalize this definition to multipartite systems. A witness is called fully decomposable, if, for every strict subset of the set of all particles , is decomposable with respect to the bipartition given by and its complement . In other words, there exist positive semidefinite operators , , such that
(8) 
For example, in the case of three qubits, a fully decomposable witness can be written in three ways,
(9) 
where all operators and are positive semidefinite. Note that, e.g., the existence of the two positive operators and implies the existence of two positive operators and . One simply has to set and . Since , the two operators and defined in this way are positive. Due to , they also obey Eq. (8).
Now, let us make the connection between the notions of PPT mixtures and fully decomposable witnesses by citing the following lemma of Ref. ourpaper , which is based on (lewenstein, , Theorem 3). For the sake of completeness, we present it again here.
Lemma 1.
is a PPT mixture if and only if every fully decomposable witness is nonnegative on .
Proof.
“If”: Let us show that if a state is no PPT mixture, there is a fully decomposable witness that detects it. We note that the set of PPT mixtures is convex and compact. Therefore, for any state outside of it, there exists a witness that detects it and is positive on the set of PPT mixtures. Moreover, an operator which is positive on all states that are PPT with respect to a fixed (but arbitrary) bipartition is decomposable with respect to this fixed (but arbitrary) bipartition lewenstein . Thus, for any .
“Only if”: A reasoning as in Eq. (6) shows that fully decomposable witnesses are nonnegative on any state that is PPT with respect to some bipartition. Therefore, these witnesses are also nonnegative on all PPT mixtures.
∎
In the language of constrained optimization theory, the search for a fully decomposable witness with negative expectation value on is the dual problem to the search for a decomposition into PPT states as in Eq. (4).
In the following, we will often use the subclass of the set of fully decomposable witnesses that we obtain when we require that . We call them fully PPT witnesses and they are defined by
(10) 
Fully PPT witnesses are easier to characterize analytically than fully decomposable witnesses. This is due to the fact that, in order to show that an operator is a fully decomposable witness, one has to find a positive operator for every and prove the positivity of the corresponding obtained from Eq. 8. For fully PPT witnesses, it suffices to show that the partial transpose of with respect to any possible bipartition is positive.
Before we state the criterion for genuine multipartite entanglement of Ref. ourpaper , we finish our introduction to the terms used in this paper by recalling some facts about the widelyused class of graph states.
ii.3 Graph states
Graph states are defined by mathematical graphs in the following way heindiss . Given a graph that is defined by a set of vertices which correspond to qubits and a set of edges that connect some of these vertices (cf. the examples in Table 1). We denote the number of vertices by .
Then, one can define a set of operators
(11) 
where is the neighborhood of qubit , i.e., the set of all qubits that are connected to qubit by an edge. Furthermore, and are the Pauli operators and , respectively, that act on qubit .
No. 1 — Bell state  No. 2 —  No. 3 — 




No. 4 —  No. 5 —  No. 6 — 



No. 7 —  No. 8 —  No. 9 — 



No. 10  No. 11 —  No. 12 — 



No. 13 —  No. 14 —  No. 15 



No. 16  No. 17  No. 18 — 



No. 19  

The operators commute and generate a set of socalled stabilizer operators which consists of elements, i.e.,
(12) 
This means that every operator can be written as a product of some generators , in which every generators appears once or not at all. Note that due to , also, e.g., the product of with itself is included in the definition of Eq. (12). In particular, the identity operator is contained in .
To every graph we can then associate a graph state that is uniquely defined by
(13) 
Thus, is the unique state that is an eigenstate to eigenvalue of all generators . Moreover, every graph also defines a socalled graph state basis, whose elements are denoted by and which are defined by
(14) 
Consequently, . Moreover, projectors on these vectors can be written as
(15) 
In the following, we will refer to states that are diagonal in a graph state basis as graphdiagonal states.
Note that two graph states that belong to two different mathematical graphs can still be physically equivalent, i.e., equivalent under local unitary transformations (LUequivalent) and permutations of qubits hein ; vandennest . For example, this is the case for the star graph No. 9 of Table 1 and the fully connected graph, in which each of the six vertices is connected with every other vertex. Both graphs describe a state which is LUequivalent to the GHZ state of six qubits.
It has been shown that, when taking into account states of up to six qubits, there are 19 LUequivalence classes of connected graph states hein . Note that the equivalence classes of up to eight qubits have been characterized in Ref. cabellographstates . Table 1 shows one representative state of each LUequivalence class. Any graph state of six or less qubits can therefore be mapped by local unitaries and permutations onto the state associated to some graph in Table 1. The local unitaries that one has to apply for this mapping are given in Ref. hein . In this way, one can also transform a witness for any state in a particular LUequivalence class into a witness of any other state in the same class.
In order to improve readability, we will drop the subscript in the following and write . Nevertheless, it is important to keep in mind that all partial transpositions are to be understood w.r.t the computational basis. This ensures that the only Pauli matrix that is changed under transposition is , whose transpose is .
As before, we will refer to , where is a subset of the set of all qubits and its complement, as a bipartition of our system.
Note that, for any graph state , the operator is a witness entdecstab . We will refer to as the projector witness of bourennane .
Iii The Entanglement Criterion
In this section, we recall the criterion for genuine multipartite entanglement originally introduced in Ref. ourpaper . We then specialize it to graphdiagonal states in Lemma 2 which is our main result in this section.
Lemma 1 naturally leads to an entanglement criterion which asks whether a given state is detected by a fully decomposable witness or not. Given a multipartite state , we consider the optimization problem
(16)  
In this minimization, the free parameters are given by and an operator for every strict subset of the set of all qubits. In practice, it is only necessary to ensure the existence of positive operators and for partitions, since the two partitions and are equivalent, as argued before in the threequbit case [cf. Eq. (9)].
A negative minimum in Eq. (16) indicates that is detected by the fully decomposable witness for which the minimum is obtained. Thus, it is entangled and, in particular, no PPT mixture.
As mentioned before, this minimization can be performed numerically by an SDP. Variations of the program in Eq. (16) have been discussed in Ref. ourpaper . There, it was applied to some important states as the W and the GHZ state for three and four qubits, and, for four qubits, the linear cluster state, the singlet and the Dicke state of two excitations. Its white noise tolerance turned out to be higher than in previous criteria. Moreover, the case in which no fully tomography, but only a restricted set of observables has been measured, was considered. In the next subsection, we will show that in the case of graphdiagonal states, the program reduces to an LP.
iii.1 Graphdiagonal states
If we are only interested in graphdiagonal states, the corresponding search for an optimal fully decomposable entanglement witness can w.l.o.g. be restricted to graphdiagonal witnesses, for which also the operators and are graphdiagonal. This is summarized in the following lemma.
Lemma 2.
For any graph diagonal state , the search for an optimal fully decomposable entanglement witness given by Eq. (16), can w.l.o.g. be restricted to a graphdiagonal form, i.e., to a linear program given by
(17)  
(18) 
The proof is given in Sec. A.1 of the Appendix.
This lemma has the following important implications: First, the optimization problems simplifies to a linear program, which are in general easier to solve than general semidefinite programs. Second, it provides a great simplification in order to derive analytic witnesses, because we know that there is an optimal witness which is diagonal in the graph state basis. Also, checking positivity of any operator simplifies to verifying nonnegativity within the graph state basis. Instead of testing positivity of a whole matrix, it is enough to consider products of generators and sums thereof [cf. Eq. (15)]. Third, let us point out that this lemma also implies that, if a state is a PPT mixture, each PPT state in its decomposition can be assumed to be graphdiagonal as well. Finally, note that a similar statement as Lemma 2 holds for PPT witnesses as well.
Iv Fully decomposable witnesses
In this section, we present a general theory for fully decomposable witnesses of graph states. First, in Sec. IV.1, we provide fully decomposable witnesses for all LUequivalence classes of graph states up to six qubits. These witnesses are obtained by the criterion of Eq. (16). The graph states are given in Table 1, while the witnesses’ white noise tolerances are given in Table 2. The witnesses can be found in Appendix B.
Moreover, we introduce an analytical construction method for fully decomposable witnesses of general graph states in Sec. IV.2. This construction method is a generalization of the linear cluster state witnesses in Ref. ourpaper and is, as one of this section’s main results, formulated in Lemma 3.
We provide specific examples in Sec. IV.3. Finally, we show how to construct witnesses that detect even more states using the witnesses of Lemma 3. This result is given as Lemma 5 in Sec. IV.4. Again, we give examples in Sec. IV.5.
iv.1 Graph states up to 6 qubits
We now apply the criterion of Eq. (16) to certain graph states. To this end, we implemented it as a semidefinite program using the parser YALMIP yalmip in combination with the solver modules SeDuMi sedumi or SDPT3 sdpt3 in MATLAB. The program we wrote is called PPTMixer and can be found online matlabcentral .
As mentioned before, there are 19 LUequivalence classes of connected graph states up to six qubits. We apply our criterion to one state of each class (cf. Table 1), obtaining the witnesses given in Appendix B. By applying the rules in Ref. hein , it is possible to transform these into witnesses for any graph state of up to six qubits.
Let us have a closer look at the witnesses of Appendix B. A widelyused indicator for how robust a witness is to noise in an experiment is the socalled white noise tolerance. It is defined in the following way: For a given state and a given witness , the white noise tolerance is the maximal amount of white noise, such that the state is still detected by the witness . Note that the criterion of Eq. (16) provides witnesses with the highest possible white noise tolerance among all fully decomposable witnesses. This can be seen by noting that both and reach their minimum for the same normalized witness , since is independent of . Thus, the witness that one obtains for the state is also a witness for . In Table 2, we give the witness tolerances of these witnesses.
Now, let us present some of these witnesses as examples. Note that the SDP yields witnesses whose trace is normalized to one. In order to make the structure of the witnesses more evident, we renormalized them for each state , such that .
For the GHZ states of three to six qubits (cf. states No. 2, No. 3, No. 5 and No. 9 in Table 1), we obtain the wellknown projector witnesses . Since it is known that is biseparable for ghzlimit , these witnesses have the maximal possible white noise tolerance.
The linear cluster state of four qubits , labelled as state No. 4, is detected by the witness
(19) 
where we defined , being the generators of the stabilizer group of , for the sake of a compact notation. Note that, alternatively, one can write in the graph state basis. We will gain a deeper understanding of the structure of this witness in the next section.
Strikingly, the similar state No. 6, which we call state, is detected by a similar witness which has, however, some additional terms. The witness is given by
(20) 
For the symmetrized version of this state, state No. 11 (or state), we obtain a witness with even more terms, namely
(21) 
The special structure of these witnesses motivates an analytical inverstigation. In fact, we will gain more insight on the witness and in Section IV.4.
state  white noise tolerance 

No. 1, Bell state  
No. 2,  
No. 3,  
No. 4,  
No. 5,  
No. 6,  
No. 7,  
No. 8,  
No. 9,  
No. 10  
No. 11,  
No. 12,  
No. 13,  
No. 14,  
No. 15  
No. 16  
No. 17  
No. 18,  
No. 19 
iv.2 Analytical construction methods
In this section, we present an analytical method to construct fully decomposable witnesses for arbitrary graph states. This construction method results in witnesses which are a generalization of the linear cluster state witnesses in Eq. (19). First, we recapitulate these witnesses in Sec. IV.2.1, before we then generalize it to arbitrary graph states in Lemma 3 of Sec. IV.2.2.
iv.2.1 Linear cluster state
We have pointed out that the witness of Eq. (19) is a witness for the fourqubit linear cluster state. For the sevenqubit linear cluster state shown in Fig. 2 a), there exists a similar witness
(22) 
is a fully decomposable witness. However, since was not obtained from our SDP, but via Lemma 3, there are — most likely — fully decomposable witness for with a higher white noise tolerance. This is in contrast to which was obtained by the semidefinite program and therefore has the maximal white noise tolerance among the fully decomposable witnesses.
has a very particular structure. The qubits whose generators appear in the witness are indicated with red circles in Fig. 2 a). Let us denote the set of these qubits by . One can see that each two qubits in have at least two other qubits between them. Moreover, the terms in Eq. (IV.2.1) all contain two or more minus signs. It turns out that witnesses of this kind can be constructed for general graph states.
iv.2.2 Arbitrary graph states
The construction in Eq. (IV.2.1) can be generalized in the following way:
Lemma 3.
Given a connected graph state . Let be a subset of the set of all qubits such that any two qubits in are neither neighbors of each other nor have a neighbor in common. We define . Let be the sum over all vectors of length with elements that contain at least two elements which equal , i.e., . In this case,
(23) 
is a fully decomposable witness for .
For the detailed proof, we refer to Sec. A.2 of the Appendix. Its main idea is to construct a suitable positive operator for every subset , such that is positive semidefinite.
Furthermore, the proof takes advantage of the fact that, besides , all terms in Eq. (23) are invariant under any partial transposition , since the identity is diagonal in any basis and there are no two generators in the product that are neighbors of each other. However, products of nonneighboring generators are only tensor products of the Pauli matrices , and the identity all of which are invariant under transposition. Moreover, the proof is simplified by being diagonal in the graph state basis.
Note that in many cases, the choice of subset is not unique. For the sevenqubit linear cluster state, instead of the choice which results in the witness of Eq. (IV.2.1), the choices or would also be valid. However, these sets would lead to witnesses that have a lower white noise tolerance.
As for the linear cluster witnesses of Ref. ourpaper , it turns out that for many graph states, the white noise tolerances of witnesses constructed according to Lemma 3 converge to one for an increasing particle number. More precisely, this is the case for graph states that can be defined for an arbitrary number of qubits such that, when increasing the number of qubits, also the number of qubits in grows. This includes the 2D cluster state for qubits and the ring cluster state. It does not include GHZ states, since for any number of qubits, no set (of more than one qubit) that contains only qubits with nonoverlapping neighborhoods can be defined on the GHZ state. Let us formulate this observation as a corollary:
Corollary 4.
Let be a graph state of qubits and a subset of these qubits with the properties as in Lemma 3. Let be a witness for as in Eq. (23). Then, the white noise tolerance of with respect to is given by
(24) 
For a family of graph states on any number of qubits with , this expression implies
(25) 
For high particle numbers, the fidelity required to detect the state is given by and therefore vanishes exponentially fast.
For the proof, we refer to Sec. A.3 of the Appendix.
Note that Lemma 3 has been proven for the special case of linear cluster states in Ref. ourpaper . Moreover, entanglement criteria for Dicke states that also exhibit a white noise tolerance which converges to one have been found recently in Ref. huberdicke .
iv.3 Examples
2D cluster state — Let us consider a 2D cluster state of 16 qubits as given in Fig. 3 a). To construct a witness according to Lemma 3, one could choose as indicated by red circles. However, it would also be possible to choose qubit instead of qubit . In both cases, the white noise tolerance is .
Other graph states — Consider state No. 13, the state, of Table 1. Here, would be a valid choice.
For state No. 11, the state, is a possible choice. However, one could have also selected , or . Indeed, in the next section, we will see that all these choices can be combined to construct an even better witness, namely the witness of Appendix B which is obtained by our SDP. As mentioned before, the corresponding white noise tolerances are given in Table 2.
iv.4 Extended construction method
Although Lemma 3 can be applied to many graph states, for most graph states there exist witnesses with a higher white noise tolerance (cf. Appendix B). In this section, we provide an extended construction method that, for some states, allows one to subtract additional terms from the witnesses constructed by Lemma 3. This extended method can be applied to, e.g., the states No. 6 () and No. 11 () of Table 1 to obtain the witnesses of Eqs. (20) and (IV.1).
Lemma 5.
Given a connected graph state and subsets of its qubits that fulfill the following two conditions:

No two qubits in a set have a neighbor in common or are neighbors of each other.

Any two qubits and from two different subsets either have the same neighborhood or no common neighbor at all.
Moreover, let be the fully decomposable witnesses that one can construct from the subsets according to Lemma 3. Then,
(26) 
is a fully decomposable witness. Note that is clearly better than any of the witnesses alone.
iv.5 Examples
State No. 6 () — Consider state No. 6 of Table 1. Here, the subsets are given by and , which fulfill the conditions of Lemma 5, since . Lemma 3 then yields the two witnesses
(27)  
(28) 
Thus, performing the minimization of Eq. (26) is tantamount to subtracting the terms and from the projector witness and then adding the terms which have been subtracted twice in this way, namely . This results in the witness given in Eq. (20).
V Fully PPT witnesses
In this section, we provide analytical construction methods for fully PPT witnesses of graph states. In Sec. V.1, Lemma 6 gives a method analogous to the fully decomposable witnesses in Lemma 3. An example will be given in Sec. V.2.
As in the last section, we then provide an extended method to construct even better witnesses using the witnesses of Lemma 6. This is done in Sec. V.3, with examples in Sec. V.4. This time, however, the extension is more general and can be applied to a larger family of states. Thus, our main results of this section are Lemmata 6 and 7. Finally, we provide a witness for the 2D cluster state in Sec. V.5 which does not fit into the construction methods presented so far.
As mentioned before, fully PPT witnesses are easier to characterize, since they are fully decomposable witnesses with for all . This allows for a further generalization of the construction methods presented above — however, only resulting in fully PPT witnesses — and for the construction of a new witness for the 2D cluster state.
v.1 Arbitrary graph states
Let us first give the analogon to Lemma 3 for fully PPT witnesses.
Lemma 6.
Given a connected graph state . Let be a subset of the set of all qubits such that any two qubits in are neither neighbors of each other nor have a neighbor in common. We define . Let be the sum over all vectors of length with elements that contain at least two elements which equal , i.e., . In this case,
(29) 
is a fully PPT witness for . Here, is the number of elements in , i.e., .
v.2 Examples
2D cluster state — When applying the presented construction to the 2D cluster state of Fig. 3, one obtains the witness
(30) 
This witness has a white noise tolerance of .
v.3 Extended construction method
We can now rewrite Lemma 5 for fully PPT witnesses. Although these witnesses have a smaller white noise tolerance, they can be handled easier analytically, which enabled us to relax the premises of Lemma 5. Therefore, one can apply the new lemma to a larger class of states.
Lemma 7.
Given a connected graph state and subsets of its qubits that fulfill the following two conditions:

No set contains two qubits that have a neighbor in common.

No two qubits in are neighbors of each other.
Moreover, let be the fully PPT witnesses that one can construct from the subsets according to Lemma 6. Then,
(31) 
is a fully PPT witness.
v.4 Examples
Linear cluster state — Consider an qubit linear cluster state as shown in Fig. 2 b). We define a subset for the construction of a witness according to Lemma 6) by picking the qubits . These are marked by red circles in Fig. 2 b). Analogously, the qubits marked by a green square belong to a second subset which is used to construct a witness . Then, Lemma 7 implies that there is a witness as given in Eq. (31).
Let us present this witness for a sevenqubit cluster state. Then, and . Consequently,
(32)  
(33) 
Since the only terms that and have in common are given by , Eq. (31) can be expressed as
(34) 
A fully PPT witness for the sevenqubit linear cluster state constructed according to Lemma 6 with has a white noise tolerance of . The witness of Eq. (V.4), however, only has a tolerance of . While Lemma 7 does not allow to construct more robust witnesses for linear cluster states compared to simply using Lemma 6, it still has some advantages.
First, for many graph states, e.g. the state No. 6 () and the state No. 11 () of Table 1, Lemma 7 does provide a method to construct witnesses that are more robust than witnesses constructed via Lemma 6 alone. We note that the fully decomposable witnesses of Lemma 5 are even more robust. However, as mentioned before, the prerequisites for Lemma 5 are more strict than those for Lemma 7 and therefore, there are graph states for which the former cannot be used, but the latter applies. For example, this is the case for the 2D cluster state of 16 qubits, to which Lemma 7 can be applied, as we will see at the end of this section, but Lemma 5 can not be used as there are no two qubits with the same neighborhood.
Second, witnesses constructed according to Lemma 7 using two sets and as shown in Fig. 2 b) can be used to improve the linear cluster state witnesses of Ref. entdecstab , which results in a witness that only needs two experimental settings to be measured.
To illustrate this, we consider the sevenqubit linear cluster state and its witness of Eq. (V.4) again. The linear cluster state witness of Eq. (9) in Ref. entdecstab is given by
(35) 
Due to the form of the generators, it can be measured locally using only two settings, namely the eigenbases of and . Since , one has
(36) 
where the last equality sign defines the improved witness . This witness detects more states than and also requires only two settings, since the additional terms can be determined through the measurement of . Note that this is not in contradiction with the result of Ref. entdecstabpra stating that has the highest possible white noise tolerance amongst all stabilizer witnesses that can be measured using two settings, as only witnesses obeying for some where considered in Ref. entdecstabpra .
Note that it is possible to construct a better witness for linear cluster state of seven qubits by adding a third witness constructed for the subset . Then, the white noise tolerance increases to .
Finally, we apply the construction of Lemma 7 to the 2D cluster state of 16 qubits.
2D cluster state — Fig. 3 b) shows how to choose four subsets of qubits from a 2D cluster state made up of 16 qubits. is shown by red circles, by blue triangles, by green squares and by orange pentagons. The resulting witnesses can be combined as in Eq. (31) to yield a witness that can be rewritten as