Quantum entanglement and generalized information processing

# Quantum entanglement and generalized information processing using entangled states with odd number of particles

Atul Kumar and Mangala Sunder Krishnan Department of Chemistry, Indian Institute of Technology Madras, Chennai 600 036, India
###### Abstract

We discuss and generalize multi-particle entanglement based on statistical correlations using Ursell-Mayer 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 five-particle 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.Hk

## 1 Introduction

Quantum entanglement is a key resource for quantum information processing (QIP) protocols [1-4]. Information processing involving multi-particle states requires entangled channels which can process the information from one remote location to another with reliability. Experimental realization of multi-particle systems and the detection of all orthogonal basis states forming a complete set of entangled states remains a challenge [5-9], nevertheless, efficient theoretical construction and characterization of different multi-particle 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 multi-particle entangled systems [10-22]. Many experiments have also been performed which provide partial experimental support to this concept [23-28]. Information processing protocols such as dense coding deal with sending classical information using an entangled quantum state as a shared resource [29-33]. Quantum information processing techniques through nuclear magnetic resonance have been considered in detail elsewhere [34-42].

In this article, we propose generalized multi-particle entangled systems for improving the efficiency of information processing. We do this by proposing particle correlations, as a direct measure of entanglement, using standard Ursell-Mayer terms which are firmly founded on the principles of many body statistical mechanics [43-47]. The approach presented here can be expanded and is applicable to statistical ensembles, and therefore, to electrons and other spin-1/2 systems as well as photons [48-50]. 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 five-particle 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 Multi-particle 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 multi-particle 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 multi-partite states have been studied extensively [51-60]. However, the same for the multi-particle states is not well established. Here, the extent of entanglement is assessed by the well established statistical mechanical formula for correlation coefficients [43-47]. Correlation measures for multi-particle systems defined using Ursell-Mayer 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 three-particle states

Correlation coefficients for two spin-1/2 particles (qubits) are defined as

 C12αβ = ⟨σ1ασ2β⟩−⟨σ1α⟩⟨σ2β⟩ (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 spin-1/2 particles, namely

 (2)

and the non-zero correlation coefficients , and have the absolute value . The maximum value of correlation between the particles indicates that the states are maximally entangled. The non-zero 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 two-particle 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 off-diagonal 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 non-zero correlation-coefficients or the state in question must be pure along with at least one non-zero correlation coefficient with maximum value [61]. The fact that the four Bell-states are pure and possess more than one non-zero correlation coefficients shows that the correlations between the particles are quantum.

Correlation coefficients for the three-particle systems are represented as

 C123αβγ = + 2⟨σ1α⟩⟨σ2β⟩⟨σ3γ⟩.

They are components of a third rank tensor. The non-zero correlation coefficients for the three-particle Greenberger-Horne-Zeilinger (GHZ) states [5], given by

 |ψ⟩(1),(2)123=1√2[|000⟩±|111⟩]123 , |ψ⟩(3),(4)123=1√2[|001⟩±|110⟩]123, |ψ⟩(5),(6)123=1√2[|010⟩±|101⟩]123 and |ψ⟩(7),(8)234=1√2[|011⟩±|100⟩]123 (5)

are either or for the coefficients , , , . The values suggest that the correlations between three particles are genuine and quantum. The three-particle 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 three-particle entangled state is W state [62], given by

 (6)

have the value for the non-zero 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

 |ζ⟩(1),(2)123 = ∣∣ϕ⟩+13⊗|0⟩2±∣∣ϕ⟩−13⊗|1⟩2√2, |ζ⟩(3),(4)123 = ∣∣ϕ⟩+13⊗|1⟩2±∣∣ϕ⟩−13⊗|0⟩2√2 , |ζ⟩(5),(6)123 = ∣∣ψ⟩+13⊗|0⟩2±∣∣ψ⟩−13⊗|1⟩2√2  and |ζ⟩(7),(8)123 = ∣∣ψ⟩+13⊗|1⟩2±∣∣ψ⟩−13⊗|0⟩2√2

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 multi-particle quantum correlations and is maximally entangled. However, if the value is not maximum but more than one non-zero correlation coefficients exists the state is non-maximally entangled. For a direct product state all the correlation coefficients are zero suggesting no genuine multi-particle 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 two-particle systems) and with 3-tangle for three-particle maximally entangled GHZ states and average value of square of the concurrence for less than maximally entangled W state [54, 62]. Concurrence for a two-particle system is defined as

 C(|ψ⟩) = ⟨ψ∣∣ψ~ψ~ψ⟩=⟨ψ|σy|ψ∗⟩=C12yy. (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 three-particle systems (ABC) such as GHZ state(s), 3-tangle is defined as

 τ=C2A(BC)−C2AB−C2AC=2(λAB1λAB2+λAC1λAC2) (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 |ψ⟩W123 ∼0.45 ∼0.45 |ψ⟩W1234 0.25 0.25 |ψ⟩W12345 0.16 ∼0.16

Table 3

state 3-tangle
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 non-zero correlation coefficients for three-particle GHZ state(s) and three-particle are in excellent argument with the value of 3-tangle whereas average value of square of the concurrence for is also a match with the value of non-zero 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 multi-particle systems and is a noble idea to study and analyze the properties of multi-particle systems. This can thus be generalized for arbitrary number of particles.

### 2.2 Four particle systems

The expression for four-particle correlation coefficients is given by

 C1234αβγδ = (10) − − ⟨σ1ασ4δ⟩⟨σ2βσ3γ⟩+2⟨σ1α⟩⟨σ2β⟩⟨σ3γ⟩⟨σ4δ⟩

The non-zero correlation coefficients calculated for the four-particle GHZ states, namely

 |ψ⟩GHZ1234=1√2[|n1n2n3n4⟩±∣∣n′1n′2n′3n′4⟩] (11)

where if then and vice versa are , , , , , , , , and indicate that four-particle GHZ states possess maximum correlations. Similarly, the non-zero correlation coefficients calculated for the four-particle 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 four-particle states to be used for information processing, however, all the 16 four-particle correlation coefficients associated with the generalized Bell basis are zero suggesting that there is no genuine correlation between the four-particles. Yeo and Chua [20] proposed a four-particle entangled system ; the maximum value of non-zero correlation coefficients , , and indicates that the state is maximally and genuinely entangled. We consider here three sets of four-particle maximally entangled states, in addition to GHZ states, given by , and where

 |ϕ⟩(1)−(16)1234 = (12) ±(|1⟩1|0⟩1)⊗(|ϕ−⟩24|ψ−⟩24)⊗(|1⟩3|0⟩3)],
 |χ⟩(1)−(16)1234 = (13) ±(|1⟩1|0⟩1)⊗(|ψ−⟩24|ψ+⟩24)⊗(|1⟩3|0⟩3)],

and

 ∣∣ϕ′⟩(1)−(16)1234 = (14) ±(|1⟩1|0⟩1)⊗(|ψ+⟩24|ψ−⟩24)⊗(|1⟩3|0⟩3)].

The non-zero 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 Five-particle systems

The expression for the five-particle correlation coefficient is given in the Appendix A. The generalized five-particle GHZ states are represented as

 |ψ⟩GHZ12345=1√2[|n1n2n3n4n5⟩±∣∣n′1n′2n′3n′4n′5⟩] (15)

and are maximally correlated as shown by non-zero correlation coefficients , , , , , , , , , , , , , , and . Unlike the GHZ state, the generalized five-particle W state, , is not maximally correlated as shown by non-zero correlation coefficients , , , , , , , , , , , , , , , , , , , and . Other five-particle entangled systems to be considered are two sets of basis states given as and , where

 |Ψ⟩(1)−(32)12345 = (16) ±(|1⟩1|0⟩1)⊗⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝∣∣ψ(6)⟩234∣∣ψ(5)⟩234∣∣ψ(8)⟩234∣∣ψ(7)⟩234⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠⊗(|1⟩5|0⟩5)⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦

and

 |Φ⟩(1)−(32)12345 = (17) ±(|1⟩1|0⟩1)⊗⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝∣∣ψ(5)⟩234∣∣ψ(6)⟩234∣∣ψ(7)⟩234∣∣ψ(8)⟩234⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠⊗(|1⟩5|0⟩5)⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦.

are three-particle GHZ states and are given by Eq. (5). The non-zero correlation coefficients for the two sets are , , , and , , , , respectively and show maximum value. The extent of correlation between five-particles 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 non-maximally 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 multi-particle entanglement.
(iv) The value of non-zero 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 multi-particle 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.

1. 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 non-zero 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 non-null 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.

2. If the number of non-zero correlation coefficients corresponding to two entangled sets are not equal, then they belong to two different family of states.
(1) Three-particle GHZ state , , , and three-particle W state , , , , , , show four and seven non-zero correlation coefficients, respectively. They belong to two different families of states.
(2) Four-particle maximally entangled GHZ states , , , , , , , , represented by Eq. (11) and four-particle maximally entangled set represented by Eq. (12) , , , show nine and four non-zero correlation coefficients, respectively which indicates that these two sets belong to different families of states.
(3) The set of five-particle states represented by Eq. (17) , , , and five-particle Brown state [16] , , , ,
, possess four and six non-zero correlation coefficients, respectively and hence belong to two different families of entangled systems.

3. 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 four-particle sets represented by Eq. (12), Eq. (13) and Eq. (14), respectively and possess four non-zero correlation coefficients. Although and belong to same family, belongs to different family of states.

4. 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 three-particle GHZ states (Eq. (5)) and possesses the same degree of correlation as that of GHZ states.

5. Doing a Hadamard operation would not affect the y-component of the correlation coefficient but would convert x-component to z-component and vice versa. Thus, the non-zero 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 five-particle 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 five-particle state and describe different information processing protocols using the state. In the past, multi-particle 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 five-particle maximally entangled set proposed here is given by

 |φ⟩(1)−(32)12345= 1√2⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝∣∣χ(1)⟩1234∣∣χ(2)⟩1234∣∣χ(3)⟩1234∣∣χ(4)⟩1234∣∣χ(5)⟩1234∣∣χ(6)⟩1234∣∣χ(7)⟩1234∣∣χ(8)⟩1234⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠⊗(|0⟩5|1⟩5)±⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝∣∣χ(9)⟩1234∣∣χ(10)⟩1234∣∣χ(11)⟩1234∣∣χ(12)⟩1234∣∣χ(13)⟩1234∣∣χ(14)⟩1234∣∣χ(15)⟩1234∣∣χ(16)⟩1234⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠⊗(|1⟩5|0⟩5)⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦

where are given by

 |χ⟩(1),(2)1234 = |0⟩1⊗∣∣ϕ⟩+24⊗|0⟩3±|1⟩1⊗∣∣ψ⟩−24⊗|1⟩3√2, |χ⟩(3),(4)1234 = |0⟩1⊗∣∣ϕ⟩−24⊗|0⟩3±|1⟩1⊗∣∣ψ⟩+24⊗|1⟩3√2, |χ⟩(5),(6)1234 = |1⟩1⊗∣∣ϕ⟩+24⊗|0⟩3±|0⟩1⊗∣∣ψ⟩−24⊗|1⟩3√2, |χ⟩(7),(8)1234 = |1⟩1⊗∣∣ϕ⟩−24⊗|0⟩3±|0⟩1⊗∣∣ψ⟩+24⊗|1⟩3√2, |χ⟩(9),(10)1234 = |0⟩1⊗∣∣ϕ⟩+24⊗|1⟩3∓|1⟩1⊗∣∣ψ⟩−24⊗|0⟩3√2, |χ⟩(11),(12)1234 = |0⟩1⊗∣∣ϕ⟩−24⊗|1⟩3∓|1⟩1⊗∣∣ψ⟩+24⊗|0⟩3√2, |χ⟩(13),14)1234 = |1⟩1⊗∣∣ϕ⟩+24⊗|1⟩3∓|0⟩1⊗∣∣ψ⟩−24⊗|0⟩3√2  and |χ⟩(15),(16)1234 = |1⟩1⊗∣∣ϕ⟩−24⊗|1⟩3∓|0⟩1⊗∣∣ψ⟩+24⊗|0⟩3√2. (19)

The set represented above is same as four-particle 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 non-zero correlation coefficients , , , . Depending on the discussions of previous section and due to the absence of ’y’ in the ’s the five-particle 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 two-particle information to Bob i.e. , Alice must share any one of the five-particle 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

 |ψ⟩1234567 = |ϕ⟩12⊗|φ⟩(10)34567. (20)

Eq. (20) can be re-expressed in form of Alice’s projection basis given by Eq. (17) as

 |ψ⟩1234567=14√2∑i,j|Φ⟩(i)12345⊗|ϕ⟩(j)67 (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 two-particle 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

 |ψ⟩1234567=14∑i,j|χ⟩(i)1234⊗|ψ⟩(j)567 (22)

where and . For example, if Alice’s measurement outcome is , the combined state of Bob’s and Charlie’s particles is given by

 |ψ⟩567 = 1√2[a|00⟩56+b|01⟩56+c|10⟩56+d|11⟩56]|0⟩7 (23) + 1√2[−a|10⟩56−b|11⟩56+c|00⟩56+d|01⟩56]|1⟩7.

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

 |φ⟩(1)−(22N+1)12...2N(2N+1) = 1√2⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝∣∣χ(1)⟩12...(2N−1)2N∣∣χ(2)⟩12...(2N−1)2N⋮∣∣χ(22N−1−1)⟩12...(2N−1)2N∣∣χ(22N−1)⟩12...(2N−1)2N⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠⊗(|0⟩2N+1|1⟩2N+1) (24)

where ’s are ordered in the same way as in Eq. (19) and
’s are -particle generalization of Eq. (13), namely

 |χ⟩(1)−(22N)12..(2N−1)2N = (25) ±(|1⟩1|0⟩1)⊗⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝∣∣χ(22N−3+1)⟩23…2N∣∣χ(22N−3+2)⟩23…2N⋮∣∣χ(22N−2−1)⟩23…2N∣∣χ