Entangled graphs: A classification of fourqubit entanglement
Abstract
We use the concept of entangled graphs with weighted edges to present a classification for fourqubit entanglement which is based neither on the LOCC nor the SLOCC. Entangled graphs, first introduced by Plesch et al. [Phys. Rev. A 67, (2003) 012322], are structures such that each qubit of a multiqubit system is represented as a vertex and an edge between two vertices denotes bipartite entanglement between the corresponding qubits. Our classification is based on the use of generalized Schmidt decomposition of pure states of multiqubit systems. We show that for every possible entangled graph one can find a pure state such that the reduced entanglement of each pair, measured by concurrence, represents the weight of the corresponding edge in the graph. We also use the concept of tripartite and quadripartite concurrences as a proper measure of global entanglement of the states. In this case a circle including the graph indicates the presence of global entanglement.
Keywords: Quantum entanglement; Entangled graphs; Fourqubit entanglement
PACS: 03.65.Ud; 03.67.Mn
1 Introduction
Entanglement, first noticed by Einstein, Podolsky, and Rosen [1] and also Schrödinger [2], is at the heart of quantum mechanics. It turns out that quantum entanglement provides a fundamental potential resource for quantum information tasks such as teleportation and cryptography [3]. However, as a resource, entanglement needs to be classified in the sense that each class do the same tasks in quantum information science.
Bipartite pure state entanglement is almost completely understand by means of Schmidt decomposition [4]. Schmidt decomposition allows one to write, by means of local unitary transformation, any pure state shared by two parties A and B in a canonical form, in the sense that all the nonlocal properties of the state are contained in the positive Schmidt coefficients. Accordingly, any twoqubit pure state can be written in the following form [4] (the subscript indicates the number of qubits)
(1) 
In this sense any twoqubit pure state is separable if , i.e. one of the Schmidt coefficients is zero, or entangled if .
For threequbit pure state, there is no a straightforward generalization of Schmidt decomposition in terms of three orthogonal product states [5]. A generalization of twoqubit Schmidt decomposition for threequbit pure states has been presented by Acín et al. [6]. They showed that for any pure threequbit state there exist a local bases which allows one to build a set of five orthogonal product states in terms of which the state can be written in a unique form as
(2)  
They also presented a canonical form for pure states of fourqubit system as [7]
(3)  
The generalized Schmidt decomposition (GSD) for pure states of a general multipartite system is presented by Carteret et al [8].
Multipartite entanglement refers to nonclassical correlations between three or more quantum particles, and characterization and quantification of them is far less understood than for bipartite entanglement. In the case of multipartite entanglement there are various aspects of entanglement and, in particular, it is no longer sufficient to ask if the particles are entangled, but one needs to know how they are entangled. There are different ways that the multipartite state can be entangled from which the bipartite entanglement, i.e. the entanglement of the reduced density matrix , is only a certain aspect of the multipartite entanglement properties of . For example for threequbit states, apart from separable and biseparable pure states, there exist also two different types of locally inequivalent genuine tripartite entangled states: the socalled GreenbergerHornZeilinger (GHZ) type [9] and W type [10, 11], with representatives and , respectively. It is well known that states belonging to GHZ and W types cannot be transformed into each other by local operations and classical communication (LOCC). Morover, for threequbit entanglement, complete classifications were derived [10, 11, 12, 13]. In [10] it has been shown that for threequbit entanglement, there are six equivalent classes under stochastic local operations and classical communication (SLOCC). However, for four or more qubits, there are infinite SLOCC classes [14], and it is therefore highly desirable to partition the infinite classes into a finite number of families. In [15], Verstraete et al. have shown that there are nine families in fourqubit entanglement. In addition, many attempts have been taken for classification of fourqubit entanglement in complete sets [16, 17, 18, 19] or in special cases [20, 21]. So, it seems that fourqubit entanglement still requires further study.
The entanglement properties of a multiqubit system may be represented mathematically in several ways. In [22], Dür has investigated the entanglement properties of multipartite systems, concentrating on the bipartite aspects of multipartite entanglement, i.e. the bipartite entanglement that is robust against the disposal of the particles. In this sense, the two qubits belonging to a multipartite system are entangled if their reduced density matrix is entangled. Dür introduces the concept of entanglement molecules, i.e. structures such that each qubit is represented by an atom while an entanglement between a pair of qubits is represented by a bound. He also shows that an arbitrary entanglement molecule can be represented by a mixed state of a multiqubit system. On the other hand, Plesch et al. [23, 24] introduce the concept of entangled graphs such that each qubit of a multipartite system is represented as a vertex and an edge between two vertices denotes bipartite entanglement between the corresponding qubits. They show that any entangled graph of qubits can be represented by a pure state from a subspace of the whole dimensional Hilbert space of N qubits.
In this paper we turn our attention to entangled graphs and associate a pure fourqubit state to each graph of order 4. This, naturally, leads to a classification of entanglement in fourqubit systems. Our classification is based on the use of GSD of pure states of multiqubit systems. We show that for every possible entangled graph one can find a pure state such that the reduced entanglement of each pair, measured by concurrence [25], represents the weight of the corresponding edge in the graph. In order to characterize the global entanglement of the state, we use the concept of tripartite and quadripartite concurrences, introduced in [26], in such a way that a circle including the graph indicates the nonzero global entanglement of the corresponding state. It is worth to mention that our classification does not contain permutations.
The paper is organized as follows: In section 2 we present the tripartite concurrence and give a classification of the tripartite pure states, Section 3 is devoted to present a classification for fourqubit pure states. We conclude the paper in section 4, with a brief conclusion.
2 Classification of threequbit entanglement
In this section we provide a classification for threequbit pure states, and associate to each possible entangled state an entangled graph (see Figure 1). Entangled graphs are structures such that each qubit is represented as a vertex and an edge between two vertices denotes bipartite entanglement between the corresponding qubits. Accordingly, an edge connecting two vertices indicates that the entanglement between the corresponding qubits is robust against the disposal of the remaining qubits. On the other hand, following [12], we use a circle including three vertices in order to indicate full tripartite entanglement. We show that for every possible entangled graph one can find a pure state such that the reduced entanglement of each pair, measured by concurrence, represents the weight of the corresponding edge in the graph. In order to characterize the full entanglement of the state we use the tripartite concurrence defined by [26]
(4) 
where the bipartite concurrence is defined as where is the reduced density matrix obtained from the whole density matrix by taking trace over the two subsystems B and C. For a general twoqubit pure state , the concurrence takes the simple form .
( , )  ( , )  ( , ) 

( , )  
In order to present our classification for threequbit states, let us first calculate the bipartite concurrence of the GSD of threequbit (2). We find , and , which reduces to if we choose . Now, let us consider all possible bipartite factorizations with nonzero concurrence of the threequbit GSD state (2). This enables us to find coefficients of the GSD which make nonzero bipartite concurrences and guarantee a weighted edge between arbitrary vertices. Moreover, in order to have nonzero global entanglement we must choose proper Schmidt coefficients such that none of the qubits can be factorized from the state. These considerations are summarized in Table 1. It follows from this table that in order to have nonzero twoqubit concurrence , we have to choose a nonzero value for the pair , but for nonzero we are allowed to choose either or to be nonzero. Now, our classification is as follows:

Fully separable: A pure state is fully separable or unentangled if it can be written as . The corresponding graph has three vertices without any edge. All bipartite entanglements as well as the global entanglement of this state are zero, i.e. and . These states can be represented by a graph of order 3, without any edge (see Fig. 1(a)). Any state which cannot be written in fully separable form is entangled and belongs to one of the following categories.

Biseparable: A pure state is biseparable if it is not fully separable and that it can be written as for one . A representative of this class is given by
(5) In this case we find for bipartite concurrences , . This state does not have global entanglement and therefore we have . The corresponding graph has three vertices with only one edge connecting two vertices and (see Fig. 1(b)).

Fully inseparable: A pure state is fully inseparable if it is not separable or biseparable. A fully inseparable state has nonzero global entanglement, i.e. . This category includes the following classes (see Figs. 1(c)(f)):

Three vertices without any edge but a circle includes the graph (Fig. 1(c)): A pure representative of this class is as follows
(6) Surprisingly, similar to fully separable states, all bipartite entanglements of this state are zero, i.e. , but the state has nonzero global entanglement, i.e. . This class corresponds to GHZ states.

Three vertices with one edge and a circle includes the graph (Fig. 1(d)): This class has the following representative
(7) In this case for bipartite concurrences we find , and for global entanglement we get .

Three vertices with two edges and a circle includes the graph (Fig. 1(e)): This class is represented by the following state
(8) The state has two nonzero bipartite concurrences as and . The nonzero global entanglement of the state is given by
. 
Three vertices with three edges and a circle includes the graph (Fig. 1(f)): The last class of this category has the following representative
(9) All bipartite concurrences of this state are nonzero as , and . The state has also a nonzero global entanglement as . This class corresponds to W states.
In [13], the authors derived a decomposition for threequbit pure states, which, as the Schmidt decomposition for bipartite states, can be easily computed. Their work led to a classification beyond the SLOCC paradigm, since there exist three classes correspond to GHZtype states. If we use tangle [27], we find that classes in Fig. 1(c)(e) are GHZtype and the class in Fig. 1(f) is Wtype.

3 Classification of fourqubit entanglement
This section is devoted to provide a classification for fourqubit pure states and associate to each state an entangled graph (see Figure 2). In order to characterize the global entanglement of the state we use the quadripartite concurrence defined by [26]
(10) 
where and are defined as and , respectively.
In a similar manner as in threequbit case, let us consider all possible bipartite factorizations with nonzero pairwise concurrences of the fourqubit GSD state (3). Table 2 summarizes this in such a way that every column of the Table corresponds to a pair of vertices and and represents all possible pairs of coefficients which make nonzero concurrence between the vertices and . For a nonzero global entanglement, one can choose proper coefficients that no qubit could be factorized. This is somewhat difficult but feasible. Using these considerations, our fourqubit classification is as follows:
( , )  ( , )  ( , )  ( , )  ( , )  ( , ) 

( , )  ( , )  ( , )  ( , )  ( , )  ( , ) 
( , )  ( , )  ( , )  ( , )  ( , )  ( , ) 
( , )  ( , )  ( , )  ( , )  ( , )  ( , ) 
( , )  ( , )  ( , )  

Fully separable: A pure state is fully separable if it can be written as . All bipartite entanglements as well as the tripartite and global entanglement of this state are zero, i.e. , and . The corresponding graph has four vertices without any edge (see Fig. 2(a)). Any state which cannot be written in fully separable form, is entangled and belongs to one of the following categories.

Triseparable: A pure state is triseparable if it is not fully separable but it can be written as for . A representative for this class is given by
(11) Only one bipartite concurrence of this state is nonzero, i.e. and . This state also has a zero tripartite entanglement and a zero global entanglement . The corresponding graph of this category has four vertices with only one edge (see Fig. 2(b)).

Biseparable A pure state is biseparable if it is not triseparable and that it can be written as for one or as for one . For the former, the global entanglement is zero and the following classes are defined (see Figs 2(c)(g)):

Four vertices without any edge, but a circle includes three of vertices (Fig. 2(c)): A pure state representative of this class is as
(12) All bipartite concurrences as well as global entanglement of this state are zero, i.e. and , but the system has entanglement through three vertices 2,3,4, i.e. .

Four vertices with only one edge and a circle includes three of vertices (Fig. 2(d)): The pure state representative of this class is as
(13) This state has zero value for all but one bipartite concurrences as . Also the global entanglement of the state is zero but the entanglement through three vertices 2,3,4, is nonzero, i.e. , .

Four vertices with two edges and a circle includes three of vertices (Fig. 2(e)): The pure state representative of this class is as
(14) The nonzero bipartite concurrences of this state are , . The state also has zero global entanglement, i.e. , but nonzero tripartite entanglement .

Four vertices with three edges and a circle includes three of vertices (Fig. 2(f)) : The pure state representative of this class is as
(15) The nonzero bipartite concurrences of this state are as , , . Again this state has a zero global entanglement and a nonzero tripartite entanglement .

Four vertices with two edges and no circle (Fig. 2(g)) : The pure state representative of this class is as
(16) The nonzero bipartite concurrences of this state are as and . This state has a zero global entanglement .


Fully inseparable A pure state is fully inseparable if it is not separable, triseparable or biseparable. This category has a nonzero global entanglement, i.e. , and has the following classes (see Figs. 2(h)(r)):

Four vertices without any edge, but a circle includes the graph (Fig. 2(h)): A pure state representative of this class is as
(17) All bipartite concurrences of this state are zero, i.e. , but the state has a nonzero global entanglement . This class corresponds to GHZ states.

Four vertices with one edge and a circle includes the graph (Fig. 2(i)): This class has the following representative
(18) The only nonzero bipartite concurrence is . Also the state has a nonzero global entanglement .

Four vertices with two edges and a circle includes the graph (Fig. 2(j)): This class has the following representative
(19) In this case two nonzero bipartite concurrences are and . The state also has a nonzero global entanglement .

Four vertices with two edges and a circle includes the graph (Fig. 2(k)): This class has the following representative
(20) The state has two nonzero bipartite concurrences as and , and a nonzero global entanglement .

Four vertices with three edges and a circle includes the graph (Fig. 2(l)): This class has the following representative
(21) The state has three nonzero bipartite concurrences as , and . This state also has a nonzero global entanglement .

Four vertices with three edges and a circle includes the graph (Fig. 2(n)): This graph has the following pure state representative
(23) This state has three nonzero bipartite concurrences as , and . Again the state has a nonzero global entanglement .

Four vertices with four edges and a circle includes the graph (Fig. 2(o)): This class has the following representative
(24) This state has four nonzero bipartite concurrences as , , and . The state also has a nonzero global entanglement .

Four vertices with five edges and a circle includes the graph (Fig. 2(q)): This class has the following representative
(26) In this case we find five nonzero bipartite concurrences as , , , and . The state also has a nonzero global entanglement . It is obvious that if , the state describes Fig. 2(p).

Four vertices with six edges and a circle includes the graph (Fig. 2(r)): This class has the following representative
(27) All bipartite concurrences of this state are nonzero and given by , , , , and . This state also has a nonzero global entanglement . This class corresponds to W states.

4 Conclusions
We have used the concept of entangled graphs with weighted edges and have presented a classification of threequbit and fourqubit entanglement. For pure states, each qubit of a multiqubit system is represented by a vertex and an edge between two vertices denotes bipartite entanglement between the corresponding qubits. The nonzero concurrence of the twoqubit reduced density matrix indicates that the entanglement of the state is robust against the disposal of remaining particles and have represented by an edge connecting the corresponding vertices. In this classification, we have used the generalized Schmidt decomposition. It is shown that for every possible entangled graph, one can find a pure state in minimal form, i.e. with minimum number of coefficients, such that the reduced entanglement of each pair, measured by concurrence, represents the weight of the corresponding edge in the graph. By using the concept of tripartite and quadripartite concurrences, we have characterized the global entanglement of the tripartite and quadripartite pure states, respectively. The presence of global entanglement of the pure state is represented by a circle including the corresponding graph. To sum up, we have classified entanglement according to all possible entanglement structure in terms of entangled graphs without considering permutations. It is obvious that there are many states in each class, considering permutations. Moreover, by computing bipartite and global entanglements, it is possible to decide which class an arbitrary state belongs to. We think this paradigm would be extendable for Nqubit entanglement. In addition, it is worth to mention that our approach is convenient to write a pure state for an arbitrary entangled graph without considering classification, which would be fruitful for certain tasks in quantum information science.
Acknowledgements
This work was partially performed at the Department of Physics, University of Isfahan.
References
References
 A. Einstein, B. Podolsky, N. Rosen, Phys. Rev. 47, 777 (1935)
 E. Schrödinger, Naturwissenschaften 23, 807 (1935)
 S.M. Barnett, Quantum Information (Oxford University Press, Oxford, 2009)
 E. Schmidt, Math. Annalen. 63, 433 (1906)
 A. Peres, Phys. Lett. A 202, 16 (1995)
 A. Acín et al., Phys. Rev. Lett. 85, 1560 (2000)
 A. Acín et al., J. Phys. A 34, 6725 (2001)
 H.A. Carteret, A. Higuchi, A. Sudbery, J. Math. Phys. 41, 7932 (2000)
 D.M. Greenberger, M. Horn, A. Zeilinger, Bell’s Theorem, Quantum Theory and Conceptions of the Universe, edited by M. Kafatos, pp. 6972 (Dordrecht, Holland: Kluwer Academic Publishers, 1989)
 W. Dür, G. Vidal, J.I. Cirac, Phys. Rev. A 62, 062314 (2000)
 A. Acín et al., Phys. Rev. Lett. 87, 040401 (2001)
 C. Sabín, G. GarcíaAlcaine, Eur. Phys. J. D 48, 435 (2008)
 J.I. de Vicente et al., Phys. Rev. Lett. 108, 060501 (2012)
 X. Li, D. Li, Phys. Rev. A 88, 022306 (2013)
 F. Verstraete et al., Phys. Rev. A 65, 052112 (2002)
 L. Lamata at al., Phys. Rev. A 75, 022318 (2007)
 Y. Cao, A.M. Wang, Eur. Phys. J. D 44, 159 (2007)
 D. Li et al., Quant. Inf. Comput. 9, 0778 (2009)
 L. Borsten et al., Phys. Rev. Lett. 105, 100507 (2010)
 E. Jung, D.K. Park, Quant. Info. Comput. 14, 0937 (2014)
 D.K. Park, Phys. Rev. A 89, 052326 (2014)
 W. Dür, Phys. Rev. A 63, 020303(R) (2001)
 M. Plesch, V. Bužek, Phys. Rev. A 67, 012322 (2003)
 M. Plesch et al., J. Phys. A 37, 1843 (2004)
 W.K. Wootters, Phys. Rev. Lett. 80, 2245 (1998)
 P.J. Love et al., Quant. Inf. Process. 6, 187 (2007)
 V. Coffman, J. Kundu, W.K. Wootters, Phys. Rev. A 61, 052306 (2000)