Symmetry, Integrability and Geometry: Methods and Applications SIGMA 7 (2011), 011, 11 pages Entanglement of Grassmannian Coherent States for Multi-Partite n-Level Systems Ghader NAJARBASHI and Yusef MALEKI Department of Physics, University of Mohaghegh Ardabili, Ardabil, 179, Iran E-mail: [email protected], [email protected] Received September 05, 2010, in final form January 19, 2011; Published online January 24, 2011 doi:10.3842/SIGMA.2011.011 Abstract. In this paper, we investigate the entanglement of multi-partite Grassmannian coherent states (GCSs) described by Grassmann numbers for n > 2 degree of nilpotency. Choosing an appropriate weight function, we show that it is possible to construct some well- known entangled pure states, consisting of GHZ, W, Bell, cluster type and bi-separable states, which are obtained by integrating over tensor product of GCSs. It is shown that for three level systems, the Grassmann creation and annihilation operators b and by together 2πi with bz form a closed deformed algebra, i.e., SUq(2) with q = e 3 , which is useful to construct entangled qutrit-states. The same argument holds for three level squeezed states. Moreover combining the Grassmann and bosonic coherent states we construct maximal entangled super coherent states. Key words: entanglement; Grassmannian variables; coherent states 2010 Mathematics Subject Classification: 81R30; 15A75; 81P40 1 Introduction Quantum entanglement has been recognized as the vital resource for the applications of quan- tum information and quantum computation. The emergence of entanglement is a fundamental difference between classical and quantum composite systems. Consequently The question of entanglement of composite systems has been intensively studied in last years [1,2]. In the same direction, the growth of research in theoretical physics and quantum optics has revealed the importance of the coherent states and hence the entanglement of the bosonic, su(2) and su(1,1) coherent states has been widely investigated [3,4,5,6,7,8,9, 10, 11]. On the other hand, studying the entanglement properties of Grassmannian coherent states (GCSs) which is defined as a eigenstate of the annihilation operator with Grassmannian eigenvalue [12, 13, 14], remains as a challenging problem of quantum information theory [15, 16, 17, 18]. In [19] we have investigated the relation between entanglement and fermionic coherent states [20]. Aim of the present work is to generalize our previous attempt in [19] to multi-level GCS. Choosing some appropriate weight functions, we show that it is possible to construct some entangled pure states, consisting of GHZ, W, Bell, cluster type and bi-separable states, by tensor product of GCSs. It is shown that for three level systems, the Grassmann creation and y annihilation operators b and b together with bz form a closed deformed algebra, i.e., SUq(2) 2πi with q = e 3 . Based on this algebra and corresponding GCS we construct some entangled qutrit-states. Similar discussion is made for three level Grassmannian squeezed states. Moreover combining the Grassmannian and bosonic coherent states we construct maximal entangled super coherent state. The paper is organized as follows. In Section2, the GCS for n level system is introduced. In Section3, the entangled GCS is studied and explicit examples of multi-qubit entangled states 2 G. Najarbashi and Y. Maleki such as Bell, W, GHZ, bi-separable and cluster type states are constructed. Moreover, entang- led qutrit states based on coherent and squeezed states are presented and then the discussion is generalized for entangled multi-qudit states. Finally combining Grassmannian and bosonic coherent states we construct maximal entangled super coherent state. A brief conclusion is given in Section4. 2 Grassmannian coherent states Grassmann variables and their applications have been discussed in [12, 20, 21, 22, 23, 24, 25]. Here we review the properties which are particularly appealing for our purposes. Grassmann algebra is generated by variables like θis which have the following properties n 2πi θiθj = qθjθi; i; j = 1; 2; : : : ; i < j; θi = 0; q = e n : Likewise, for the Hermitian conjugate of θ, θy = θ¯, we have ¯ ¯ ¯ ¯ ¯n θiθj = qθjθi; i < j; θi = 0: One has the Berezin's rule of integration as follows Z Z k ¯¯k dθθ = dθθ = δk;n−1; where k is a positive integer. Moreover we have the relations θdθ¯ = qdθθ;¯ θdθ¯ = qdθθ;¯ θdθ =qdθθ; ¯ θd¯ θ¯ =qd ¯ θ¯θ;¯ dθdθ¯ =qd ¯ θdθ;¯ θθ¯ =q ¯θθ:¯ The quantization relations between number states fjmi; m = 0; 1; 2; : : : ; n − 1g and Grassmann variables θ, θ¯ are m−1 m−1 θjmi = q jmiθ; θhmj =q ¯ hmjθ: By definition, GCS denoted by jθin, is the eigenstate of annihilation operator b with eigenvalue θ that is bjθin = θjθin; where θ is a complex Grassmann variable with the following relations y [θ; b ]q = [b; θ]q = 0; where [A; B]q := AB − qBA. One may tempt to define the annihilation operator b as n−1 X p b = m + 1jmihm + 1j: m=0 Hence jθin can be define as below m(m+1) n−1 2 X q¯ m (byθ) jθin = p θ jmi =: e j0i: (1) m=0 m! One may regard D(θ) := e(byθ) as the displacement operator for GCS. Entanglement of Grassmannian Coherent States for Multi-Partite n-Level Systems 3 3 Entangled Grassmann coherent states Consider a linear combination of tensor product of Grassmannian coherent states as follows X j i = fi1;i2;:::;in jθi1 ijθi2 i · · · jθin i: i1;i2;:::;in Some times, if we take a proper form of the tensor product of Grassmannian coherent states, it is possible to find an appropriate weight function to make the right hand side of the following equation to be a maximally entangled state like Bell, cluster type, GHZ and W states, i.e., Z dθi1 dθi2 ··· dθin w(θi1 ; : : : ; θin )j i = jγi; (2) where w(θi1 ; : : : ; θin ) is a proper weight function, and the state jγi; is maximally entangled state. We note that the weight functions is not unique and of course for a given state there may not be such a function at all. Here there is no need to normalize GCSs since it can be included in the weight functions. 3.1 Multi-qubit states Two level Grassmannian coherent state can be written as jθi = j0i − θj1i: At first, we consider simple cases that yield maximally entangled Bell states as follows Z ∓1 1 dθ p [jθij ± θi − j − θij ∓ θi] = p (j01i ± j10i) = jΨ±i; 2 2 2 Z ±1 ¯ 1 dθdθ¯ p e±θθ jθ¯ijθi = p (j00i ± j11i) = jΦ±i: 2 2 One gets the general form of the W state as follows Z −1 1 dθ p jθijθi · · · jθi = p (j100 ::: 0i + j010 ::: 0i + ··· + j0 ::: 001i) = jW(n)i: n | {z } n n times Likewise, we can construct the general form of GHZ state as follows Z 1 (n) dθ1dθ2 ··· dθnwjθnijθn−1i · · · jθ1i = p (j00 ::: 0i + j11 ::: 1i) = jGHZ i; 2 where 1 [ n ] w = p (−1) 2 + θnθn−1 ··· θ1 : 2 Note that unlike W(n), we can not construct GHZ(n) using just one Grassmann variable for n > 2. The amount of entanglement of these states can be evaluated using the purity which 2 P 2 is defined as p(j i) = n i tr ρi − 1, where ρi is reduced density matrix of qubit i [26]. For (n) (n) n−2 2 (n) GHZ state the purity is zero and for W state reads ( n ) , which implies that GHZ state is maximally entangled while in the limit n ! 1 the purity goes to 1 for W(n) hence it become separable. The other set of important entangled states are cluster type states [27]. Here we consider one example of these states as follows 1 jCLUSTER±i = (±|0000i + j0011i + j1100i ∓ j1111i): 2 4 G. Najarbashi and Y. Maleki If we take j i = jθ1ijθ2ijθ3ijθ4i, in equation (2), then we get Z ± dθ1 ··· dθ4w(CLUSTER±)jθ1ijθ2ijθ3ijθ4i = jCLUSTER i; where 1 w ± = (±θ θ θ θ + θ θ + θ θ ∓ 1): (CLUSTER ) 2 4 3 2 1 2 1 4 3 Taking the same method, one can construct the other cluster type states with appropriate weight functions. 3.2 Multi-qutrit states and SUq(2) deformed algebra Multi-qutrit states are three level systems. Therefore, we take the following bases to describe these states 0 1 1 0 0 1 0 0 1 j0i ≡ @ 0 A ; j1i ≡ @ 1 A ; j2i ≡ @ 0 A : 0 0 1 The associated Grassmann numbers satisfy θ3 = 0. So, the following quantization relations hold θj0i =q ¯j0iθ; h0jθ¯ = qθ¯h0j; θj1i = j1iθ; h1jθ¯ = θ¯h1j; θj2i = qj2iθ; h2jθ¯ =q ¯θ¯h2j; 2πi where q = e 3 . Now, we are going to derive an algebra under which three level GCSs can be constructed. For this purpose, consider the explicit form of the three level annihilation and y creation operators b, b and bz as follows p p y y y y b := j0ih1j + 2j1ih2j; b := j1ih0j + 2j2ih1j; bz := [b; b ]q := bb − qb b: Using the commutation relation between operators b and bz, we get p 2 2 [bz; b]q = 1 − 2q + q j0ih1j + 2 − q + 2q 2j1ih2j: (3) If the right hand side of the above equation is taken to be proportional to b, we must have 1 − 2q + q2 = 2 − q + 2q2; or equivalently 2 2πi (1 + q + q ) = 0 ) q = e 3 : (4) Thus the equation (3) reduces to [bz; b]q = −3qb: y Developing the same method to the commutator of b and bz the condition (4) is obtained again. Hence we have y y y [b; b ]q = bz; [bz; b]q = −3qb; [b ; bz]q = −3qb : Entanglement of Grassmannian Coherent States for Multi-Partite n-Level Systems 5 As we see this algebra is closed and it is reminiscent of the SU(2) algebra, and we call it deformed SUq(2) algebra.