Quantum entanglement and generalized information processing using entangled states with odd number of particles
Abstract
We discuss and generalize multiparticle entanglement based on statistical correlations using UrsellMayer type of cluster coefficients. Cluster coefficients are used to distinguish different, independent entangled systems as well as those which are connected through local unitary transformations. We propose a genuinely and maximally entangled fiveparticle state for efficient information processing. The physical realization of entangled states and information processing protocols are analyzed using quantum gates and circuit diagrams. We show that direct as well as controlled communication can be achieved using the state proposed here, with certainty in the case of teleportation and with a high degree of optimity in the case of dense coding. For controlled dense coding the amount of information transferred from the sender to the receiver is always a maximum irrespective of the measurement basis used by the controller.
pacs:
03.67.Mn, 03.67.a, 03.67.Hk1 Introduction
Quantum entanglement is a key resource for quantum information processing (QIP) protocols [14]. Information processing involving multiparticle states requires entangled channels which can process the information from one remote location to another with reliability. Experimental realization of multiparticle systems and the detection of all orthogonal basis states forming a complete set of entangled states remains a challenge [59], nevertheless, efficient theoretical construction and characterization of different multiparticle entangled channels for analyzing different information protocols is an important precursor to successful design of experiments.
Quantum teleportation involving many particles has been studied theoretically using different multiparticle entangled systems [1022]. Many experiments have also been performed which provide partial experimental support to this concept [2328]. Information processing protocols such as dense coding deal with sending classical information using an entangled quantum state as a shared resource [2933]. Quantum information processing techniques through nuclear magnetic resonance have been considered in detail elsewhere [3442].
In this article, we propose generalized multiparticle entangled systems for improving the efficiency of information processing. We do this by proposing particle correlations, as a direct measure of entanglement, using standard UrsellMayer terms which are firmly founded on the principles of many body statistical mechanics [4347]. The approach presented here can be expanded and is applicable to statistical ensembles, and therefore, to electrons and other spin1/2 systems as well as photons [4850]. Statistical correlation coefficients are shown to be useful in distinguishing entangled systems belonging to different families. The properties of correlation coefficients are used to determine whether the states under study are related through local transformations or not. In section 3, we propose and discuss the properties of a fiveparticle entangled channel and generalize the quantum channel for number of particles. The quantum channel proposed in that section is used for various information processing protocols successfully. This is followed by a conclusion.
2 Multiparticle entanglement
In this section, we first review the entanglement properties of a few maximally entangled states used in the past by others and then propose multiparticle genuinely entangled states for use in information processing. A criterion is used to define the extent of correlation between particles and several examples of entangled states of many particles are considered. The entanglement properties of bipartite states and a few multipartite states have been studied extensively [5160]. However, the same for the multiparticle states is not well established. Here, the extent of entanglement is assessed by the well established statistical mechanical formula for correlation coefficients [4347]. Correlation measures for multiparticle systems defined using UrsellMayer type cluster coefficients are suggested by us as a means for generalizing the defining of degree of entanglement between many particles.
2.1 Two and threeparticle states
Correlation coefficients for two spin1/2 particles (qubits) are defined as
(1) 
where ’s are the Pauli spin matrices for the indicated particles, {, , = }. They are components of a second rank symmetric traceless tensor. The averages are calculated for the four Bell states of two entangled spin1/2 particles, namely
(2) 
and the nonzero correlation coefficients , and have the absolute value . The maximum value of correlation between the particles indicates that the states are maximally entangled. The nonzero correlation coefficients , , for the states
(3) 
which can be obtained by doing a Hadamard operation on the particle of Bell states in Eq. (2), show that they are also maximally entangled. The value of all correlation coefficients associated with states such as are zero. It is evident because , a direct product state of particle 1 and particle 2. Also the existence of the maximum value for a single correlation coefficient alone does not ensure that a given system is maximally entangled, e.g. a twoparticle system in a mixed state with its density operator given by shows , though the two particles are not entangled. They are nevertheless correlated in the sense that measurement results for spin 1 and spin 2 are not independent of each other. However, there is no “quantum” correlations which is due to the offdiagonal components and and which is the characteristic of the entangled particles. Thus, to ensure maximum entanglement, more than one information is needed i.e. either more than one statistical data should be available with respect to nonzero correlationcoefficients or the state in question must be pure along with at least one nonzero correlation coefficient with maximum value [61]. The fact that the four Bellstates are pure and possess more than one nonzero correlation coefficients shows that the correlations between the particles are quantum.
Correlation coefficients for the threeparticle systems are represented as
They are components of a third rank tensor. The nonzero correlation coefficients for the threeparticle GreenbergerHorneZeilinger (GHZ) states [5], given by
(5) 
are either or for the coefficients , , , . The values suggest that the correlations between three particles are genuine and quantum. The threeparticle GHZ states, though maximally entangled, are not robust with respect to disposal of any of the particles i.e. tracing of any of the particles results in the disappearance of quantum correlation between the rest of the particles. The other popular threeparticle entangled state is W state [62], given by
(6) 
have the value for the nonzero correlation coefficients , , , , , , which suggests that the correlation between three particles is less than the maximum. The state is robust with respect to tracing of any of the particles. A similar calculation of correlation coefficients for a set of states such as
shows that these states are maximally entangled as well , , , . In angular momentum algebraic parlance states represented in Eq. (7) and GHZ states refer to different coupling schemes and can be locally transformed into each other. The entanglement properties of these states are similar to the GHZ states if we consider the extent of correlation between three particles. Thus, if the value of correlation coefficients associated with a particular system is maximum then it indicates that the state in question possesses genuine multiparticle quantum correlations and is maximally entangled. However, if the value is not maximum but more than one nonzero correlation coefficients exists the state is nonmaximally entangled. For a direct product state all the correlation coefficients are zero suggesting no genuine multiparticle correlation between the particles.
The criteria to measure the degree of entanglement using statistical correlations is compared with the existing criteria’s such as concurrence [52,53] (for twoparticle systems) and with 3tangle for threeparticle maximally entangled GHZ states and average value of square of the concurrence for less than maximally entangled W state [54, 62]. Concurrence for a twoparticle system is defined as
(8) 
where and is complex conjugate of . Above expression shows that the value of concurrence is equal to one of the coefficient of second rank symmetric traceless tensor representing the correlation between the particles. Table 1 summarizes the comparison between the value of concurrence and correlation coefficients obtained for Bell states.
Table 1
state  concurrence  

1  1  1  1  
1  1  1  1  
1  1  1  1  
1  1  1  1 
The average value of the square of the concurrence for less than maximally entangled generalized W states is given by . For maximally entangled threeparticle systems (ABC) such as GHZ state(s), 3tangle is defined as
(9) 
where , and , are the square roots of eigen values of and , respectively such that . These two values are calculated and compared with that of correlation coefficients obtained using criterion used by us. The results are summarized in Table 2 and Table 3, respectively.
Table 2
state  average value of square of the concurrence  value of correlation coefficients 
Table 3
state  3tangle  

1      
1      
1      
1      
1    1    1  1  +1  
1    1    1  +1  1  
1    +1    +1  +1  1  

1    +1    +1  1  +1 

1    +1    1  +1  +1 
1    +1    1  1  1  

1    1    +1  1  1 

1    1    +1  +1  +1 
Table 3 and Table 2 show that the value of nonzero correlation coefficients for threeparticle GHZ state(s) and threeparticle are in excellent argument with the value of 3tangle whereas average value of square of the concurrence for is also a match with the value of nonzero correlation coefficients obtained. This suggests that the criterion using statistical correlation coefficients to measure the degree of entanglement include all possible type of entanglement in multiparticle systems and is a noble idea to study and analyze the properties of multiparticle systems. This can thus be generalized for arbitrary number of particles.
2.2 Four particle systems
The expression for fourparticle correlation coefficients is given by
(10)  
The nonzero correlation coefficients calculated for the fourparticle GHZ states, namely
(11) 
where if then and vice versa are , , , , , , , , and indicate that fourparticle GHZ states possess maximum correlations. Similarly, the nonzero correlation coefficients calculated for the fourparticle W state, , are , , , , , , , , , , , and and show the value as indicating less than maximum correlations between particles. Rigolin [17] proposed a generalized Bell basis as a set of fourparticle states to be used for information processing, however, all the 16 fourparticle correlation coefficients associated with the generalized Bell basis are zero suggesting that there is no genuine correlation between the fourparticles. Yeo and Chua [20] proposed a fourparticle entangled system ; the maximum value of nonzero correlation coefficients , , and indicates that the state is maximally and genuinely entangled. We consider here three sets of fourparticle maximally entangled states, in addition to GHZ states, given by , and where
(12)  
(13)  
and
(14)  
The nonzero correlation coefficients calculated for the above three sets are , , and , , , , and , , , , respectively and indicate maximum entanglement. The set of states represented by Eq. (12) and Eq. (14) are cluster type of states [63] and can be transformed into each other through local transformations whereas Eq. (13) represents type of states [20] .
2.3 Fiveparticle systems
The expression for the fiveparticle correlation coefficient is given in the Appendix A. The generalized fiveparticle GHZ states are represented as
(15) 
and are maximally correlated as shown by nonzero correlation coefficients , , , , , , , , , , , , , , and . Unlike the GHZ state, the generalized fiveparticle W state, , is not maximally correlated as shown by nonzero correlation coefficients , , , , , , , , , , , , , , , , , , , and . Other fiveparticle entangled systems to be considered are two sets of basis states given as and , where
(16)  
and
(17)  
are threeparticle GHZ states and are given by Eq. (5). The nonzero correlation coefficients for the two sets are , , , and , , , , respectively and show maximum value. The extent of correlation between fiveparticles remains the same even after interchanging the particle indices.
The general expression for the particle correlation coefficient can be obtained
by solving the equations for cluster functions derived formally from
the th quantum virial coefficient.
The following summarizes the relation between correlation
coefficients and the degree of entanglement.
(i) Existence of maximum values for more than one correlation coefficient for a system under
study, indicates that the state of the system possesses genuine and maximum
entanglement.
(ii) For nonmaximally entangled states the value of correlation
coefficients lies between and .
(iii) Null results for all correlation coefficients of state suggest
that it is a direct product of fewer particle states and there
exists no genuine multiparticle entanglement.
(iv) The value of nonzero correlation coefficients remains the same for states
connected to each other by local unitary transformations.
(v) The extent of correlation remains invariant to
changing the particle indices.
2.4 Importance and properties of cluster coefficients
The criterion to use cluster coefficients as a measure of entanglement of the state under study allows one to characterize the extent of correlation of multiparticle states on the same scale irrespective of number of particles involved. A consistent description emerges for systems irrespective of the number of particles which are entangled. In this subsection, we discuss some of the properties of correlation coefficients in addition to those described in previous section.

The relation between correlation coefficients of states which differ from each other only through permutation of particle indices can be seen immediately as follows:
(1) The state [Eq. (12)] is obtained from by permuting particles 1 and 2. Hence, the nonzero correlation coefficients associated with and are , , , and , , , , respectively.
(2) Conversely, by examining two sets of equal number of correlation coefficients, we can also relate the states. For example, the two sets , , , and , , , are related to each other through particle permutations and . The first set is the only nonnull set of coefficients for the state given by Eq. (16). Hence another can be obtained by particle permutations. Thus, a family of states can be quickly enumerated. 
If the number of nonzero correlation coefficients corresponding to two entangled sets are not equal, then they belong to two different family of states.
(1) Threeparticle GHZ state , , , and threeparticle W state , , , , , , show four and seven nonzero correlation coefficients, respectively. They belong to two different families of states.
(2) Fourparticle maximally entangled GHZ states , , , , , , , , represented by Eq. (11) and fourparticle maximally entangled set represented by Eq. (12) , , , show nine and four nonzero correlation coefficients, respectively which indicates that these two sets belong to different families of states.
(3) The set of fiveparticle states represented by Eq. (17) , , , and fiveparticle Brown state [16] , , , ,
, possess four and six nonzero correlation coefficients, respectively and hence belong to two different families of entangled systems. 
Even if the number of correlation coefficients associated with different entangled states are equal the states need not have to belong to the same family. For example, the three states , and belong to maximally entangled fourparticle sets represented by Eq. (12), Eq. (13) and Eq. (14), respectively and possess four nonzero correlation coefficients. Although and belong to same family, belongs to different family of states.

The extent of correlation between particles remains invariant under standard local unitary transformations. The set of states represented by Eq. (7) can be obtained by applying Hadamard operation to the second particle of threeparticle GHZ states (Eq. (5)) and possesses the same degree of correlation as that of GHZ states.

Doing a Hadamard operation would not affect the ycomponent of the correlation coefficient but would convert xcomponent to zcomponent and vice versa. Thus, the nonzero correlation coefficients for the set of states represented by Eq. (2) and Eq. (3) (which differ from each other by a Hadamard transformation on particle 2) are , , and , , , respectively. This is a trivial example. However, as the number of particles in an entangled set increases, the number of ways of doing transformations also increases and hence this scheme is useful for nontrivial, multiple local transformations as shown below.
(1) The correlation coefficients, , , , and , , , , correspond to the entangled states and , respectively. The set of coefficients can be transformed into each other by doing Hadamard transformations on , and particles and permuting particles 1 and 2.
(2) The fiveparticle maximally entangled states, namely and can be converted into one another by doing local transformations as revealed by their correlation coefficients, namely , , , , , , , , , and , , , , , , , , , , respectively. Thus by doing three Hadamard operations on particle two, three and four can be locally transformed to .
3 Generalized information processing
In this section we propose a maximally and genuinely entangled fiveparticle state and describe different information processing protocols using the state. In the past, multiparticle entangled channels involving odd number of particles have been proposed with the use of a controller to assist the sender for successful and optimal information transfer [10, 13, 14]. We show that one can eliminate the intermediate observer controlling the process such that information processing is successful in all the measurement outcomes performed by the sender. The formation of the state proposed here ensures efficient information transfer between two or more users in the communication protocol.
3.1 Direct teleportation
The fiveparticle maximally entangled set proposed here is given by
where are given by
(19) 
The set represented above is same as fourparticle entangled set given in Eq. (13), however, the order in which the states are represented is different. The set proposed here shows values for the nonzero correlation coefficients , , , . Depending on the discussions of previous section and due to the absence of ’y’ in the ’s the fiveparticle entangled set proposed above belongs to a different family of states with respect to those represented by Eq. (16) and Eq. (17).
In order to communicate an arbitrary twoparticle information to Bob i.e. , Alice must share any one of the fiveparticle entangled state given by Eq. (18) with Bob such that particles 3, 4 and 5 are with Alice and particles 6 and 7 are with Bob. Thus, using as the quantum channel shared between Alice and Bob where , Alice can communicate her unknown message with Bob by interacting her particles 1 and 2 with her share of entangled particles 3, 4 and 5 so that
(20) 
Eq. (20) can be reexpressed in form of Alice’s projection basis given by Eq. (17) as
(21) 
where and . For Alice’s measurement outcomes and , Bob’s particles are instantaneously projected on to the state Alice wanted to communicate, with total probability of 1/16, however, for all other measurement outcomes of Alice, Bob require only single qubit transformations to recover the message successfully. The preparation of above set of states and teleportation of an arbitrary twoparticle state are represented in Fig. (1) and Fig. (2), respectively.
3.2 Controlled teleportation
For controlled teleportation, the quantum state is shared between Alice, Charlie and Bob such that the particles 3 and 4 are with Alice, particles 5 and 6 are with Bob and particle 7 is with Charlie. Alice projects her four particles on to the basis set given by Eq. (13) so that Eq. (20) becomes
(22) 
where and . For example, if Alice’s measurement outcome is , the combined state of Bob’s and Charlie’s particles is given by
(23)  
For Charlie’s outcome of , Bob’s particles are in the state identical to the one communicated by Alice, however, for his outcome , Bob needs to do a and operation on the particle to complete the process successfully. Again, for all the outcomes of Alice and Charlie, Bob can recover the message with single qubit unitary transformations, if needed. The above processes can be generalized for the case of number of particles as follows.
The generalized entangled basis set corresponding to Eq. (18) is
(24)  
where ’s are ordered in
the same way as in Eq. (19) and
’s are
particle generalization of Eq. (13), namely
(25)  