ENTANGLEMENT DEGREE OF FINITE-DIMENSIONAL PAIR COHERENT STATES

Transcription

1 Journal of Russian Laser Research, Volume 34, Number 4, July, 2013 ENTANGLEMENT DEGREE OF FINITE-DIMENSIONAL PAIR COHERENT STATES F. Khashami, 1 Y. Maleki, 1 and K. Berrada Young Researcher Club, Gorgan Branch Islamic Azad University Gorgan, Iran 2 The Abdus Salam International Centre for Theoretical Physics Strada Costiera 11, Miramare, Trieste, Italy 3 College of Science, Department of Physics Al Imam Mohammad Ibn Saud Islamic University (IMSIU) Riyadh, Saudi Arabia 4 LaboratoiredePhysiqueThéorique, Faculté des Sciences Université Mohammed V-Agdal Avenue Ibn Battouta, Boîte Postale 1014, Agdal Rabat, Morocco Corresponding author Abstract ictp.it We study in detail the entanglement degree of finite-dimensional pair coherent states (PCSs) in terms of different parameters involved in the coherent states. Since these states are a type of correlated two-mode states in finite dimension, we use the D concurrence and linear entropy to quantify their amount of entanglement. We show that the maximum entanglement can be obtained for two and threedimensional (finite-dimensional) PCSs, and states with higher dimensions cannot attain this limit. We generalize the discussion to a superposition of two states of this class and give the maximum entangled states for even and odd finite-dimensional PCSs. In addition, we consider the entanglement degree of nonlinear finite-dimensional PCSs and survey the maximality condition. Finally, we discuss the entanglement for a class of mixed states defined as a statistical mixture of two pure finite-dimensional PCSs. Our observations may have important implications in exploiting these states in quantum information theory. Keywords: finite-dimensional pair coherent states, entanglement, D concurrence, linear entropy, negativity. 1. Introduction The past few years has witnessed rapid progress in theoretical aspects and experimental implementations of quantum entanglement in various branches of quantum information and communication processing such as quantum cryptography, quantum computing, quantum key distribution, and dense coding [1 5]. Entanglement is a very peculiar quantum characteristic and has become an important resource in quantum information and quantum applications; it is recognized as one of the most intriguing features Manuscript submitted by the authors in English on June 26, /13/ c 2013 Springer Science+Business Media New York

2 Volume 34, Number 4, July, 2013 Journal of Russian Laser Research of quantum mechanics [6 8]. The emergence of quantum entanglement is due to the Hilbert space structure of the quantum-mechanical-state space. In fact, the subtleness of the idea lies in the fact that, unlike the classically correlated systems, when we consider correlated systems in quantum mechanics, the state vector describing the whole system may not be written as a product of the states of each component. Such states are called entangled states [1, 2, 9 11]. Recently, various devices have been proposed and realized experimentally to generate quantum entanglement, such as beam splitters [12], cavity QED [13], trapped ions [14, 15], etc. These realizations represent the high potential of applications of quantum optics in quantum information and computation [16,17]. We know that nonclassically correlated light plays an important role in quantum information theory, both fundamentally and practically. A very useful concept in quantum optics that is widely used and applied in quantum information is the notion of coherent states [18 28]. This notion opened the possibility of using entanglement in continuous variables [29 31]. Entanglement in correlated two-mode states within the framework of twomode squeezed states [32 34] and their applications has been investigated in recent years. For instance, in [34] two-mode squeezed vacuum states have been applied to quantum dense coding. The other twomode correlated state that has been considered in the literature is the pair coherent state [35]. In [36], the notion of quantum entanglement between the two modes and the inseparability of pair coherent states in the context of Peres Horodecki criteria and entanglement measures have been studied. We note that by casting these two-mode entangled states in terms of mathematical definitions, the states are described in infinite-dimensional Hilbert space. Recently, a new type of correlated two-mode states but in finite dimensions, which is called finite-dimensional pair coherent state (PCS) and denoted by ξ,q, was introduced [37]. In addition, the nonclassical and statistical properties of these states have been studied in [37, 38]. The results that we present is the basis of a new approach to such state. In this paper, we quantify the amount of entanglement of finite-dimensional PCS and consider its entanglement properties. For this purpose, we basically use linear entropy [39] and D concurrence [40] as measures of entanglement. Our main motivation for the present work comes from the fact that such a new type of correlated two-mode states associated with PCSs can be used in quantum information applications, where the entanglement is regarded as a crucial resource. We show that, within the framework of finite-dimensional PCSs, the maximality of entanglement can be obtained for two- and three-dimensional states, namely, ξ,1 and ξ,2 can rise to the maximality of entanglement with respect to the parameter ξ. The states with higher dimensions (q 3) cannot be maximum entangled. In addition, we consider a superposition of two finite-dimensional PCSs and investigate the entanglement degree of such states. Then we obtain the maximum entangled states in the set of superposed finite-dimensional PCSs, in particular, some even and odd maximum entangled states. We show show that such superposed states, which have inherited nonclassical properties, can be useful tools in quantum entanglement processing. In addition, we consider the entanglement degree of nonlinear finite-dimensional PCS and obtain its maximality condition. Finally, we generalize the discussion to the case of mixed states and consider the statistical mixture of two pure finite-dimensional PCSs. To this aim, we use the negativity and logarithmic negativity as entanglement measures [41]. Consequently, studying the entanglement degree of finite-dimensional PCSs highlights the subtleties of applying entanglement measures to such two-mode correlated state with continuous variables. This paper is organized as follows. In Sec. 2, we give briefly the notion of finite-dimensional PCS. In Sec. 3 we consider the entanglement 389

3 Journal of Russian Laser Research Volume 34, Number 4, July, 2013 degree of finite-dimensional PCS using linear entropy, D concurrence, and concurrence as measures of entanglement. In Sec. 4, we generalize the discussion to a superposition of finite-dimensional PCSs. We devote Sec. 5 to the nonlinear finite-dimensional PCS and its entanglement properties. We give a condition under which such states can be considered as maximum entangled states. In Sec. 6, we consider the entanglement of mixed states within the framework of finite-dimensional PCSs; namely, we study the entanglement of mixed states defined as a statistical mixture of two states. We give a brief conclusion in Sec Finite-Dimensional Pair Coherent States In contrast to the definition of the usual pair coherent state that gets an infinite dimension, the finite-dimensional PCS is defined as the eigenstate of the operator (a b + ξq+1 (ab ) q ) (q!) 2 and the sum of the photon number operators for the two modes, namely, (a b + ξq+1 (ab ) q (q!) 2 ) ξ,q = ξ ξ,q, (a a + b b) ξ,q = q ξ,q, (1) whereinthiscontexttheparameterξ is a complex variable, and the parameter q is a nonnegative integer number. The state can be given as ξ,q = N q ξ n (q n)! q n, n. (2) q!n! Inthetwo-modestate, n a,n b = n a n b,where n s is the Fock state for the mode s (s = a or b). Therefore, the state q n, n is a two-mode Fock state having q n photons in mode a and n photons in mode b. Thus, when photons are created and destroyed in pairs, the sum of the numbers of photons remain constant. The normalization constant N q of this state is given by ( ) 1/2 2n (q n)! N q = ξ. (3) q!n! In this paper, we address the problem of entanglement between modes and discuss the maximality of entanglement of these correlated two-mode states given by Eq. (2). Subsequently, we use quantitative measures of entanglement, such as linear entropy and D concurrence, in order to study the entanglement of this state. 3. Entanglement of Finite-Dimensional PCSs In this section, we investigate the entanglement degree of the finite-dimensional PCSs. consider the simplest entangled coherent states of these types; namely, the state with q = 1 First, we ξ,1 = 1 ( 10 + ξ 01 ). (4) 1+ ξ 2 390

4 Volume 34, Number 4, July, 2013 Journal of Russian Laser Research In the language of quantum information, the state (4) presents a two-qubit state, and the amount of its entanglement can be measured by concurrence [42], which is defined as C = ψ σ y σ y ψ, (5) where σ y is the spin flip operator and ψ is the complex conjugate of ψ. Concurrence ranges from 0 for a separable state to 1 for a maximum entangled state (MES). Considering the state ξ,1, we can write the concurrence as C = 2 ξ 1+ ξ 2. (6) We simply see that the state (4) becomes maximum entangled when ξ = 1. It is notable that, when ξ = ±1, the finite-dimensional PCS for q = 1 reduces to the well-known maximum entangled Bell state. Recently, a novel measure of entanglement named D concurrence was proposed by Ma et al. [40]. In the case of pure states, it is defined as D(ρ) = det(i ρ 1 )= det(i ρ 2 ), (7) where ρ 1 and ρ 2 are reduced density matrixes of subsystems 1 and 2, respectively. entanglement of state ξ,1 using the D concurrence measure. We obtain We compute the D(ρ) = ξ 1+ ξ 2. (8) For the maximum entangled state, ξ = 1,andD concurrence tends to D(ρ) = 1/2. The other useful entanglement measure, which we use to quantify the degree of entanglement for the state ξ,q, is the linear entropy. For a bipartite pure state, taking the dimension d, we define the linear entropy as I lin = d ( 1 Tr (ρ 1 ) 2). (9) d 1 Linear entropy tends to 1 for MES, and to 0 for a separable state. For the state ξ,1, itisgivenby I lin = 4 ξ 2 (1 + ξ 2 ) 2. (10) In this case, the relationship between these measures reads C = D(ρ)/2 = I lin. In order to observe the influence of the amplitude parameter ξ on the entanglement behavior, we show the different measures of entanglement in terms of ξ for q = 1 in Fig. 1. We can see a different order of the entanglement for the quantum measures. In fact, when the condition ξ = ±e iθ does not hold, we find strong entanglement (the measures of entanglement tend to approach the maximum value) as long as one of the amplitude is near ±e iθ, while the other differs significantly from ±e iθ. As the amplitude ξ differs significantly from ±e iθ, the measures of entanglement become weaker and nonexistent for large enough deviation from ±e iθ. From these results, it is clear that the amplitude value can restrain the entanglement of infinite-dimensional PCSs as it gets far from ±e iθ. 391

5 Journal of Russian Laser Research Volume 34, Number 4, July, 2013 Fig. 1. Variation of the entanglement of finite-dimensional PCS ξ,q as a function of ξ for the case q =1,where concurrence is shown by the solid line and D concurrence by the dotted line (left) and the linear entropy (right). Now, we take a step further and investigate the entanglement properties of finite-dimensional PCSs for the case q = 2. In this respect, we have a three-level system with the following explicit form: or equivalently ξ,2 = N 2 The normalization factor N 2 is given by 2 ξ,2 = N 2 ( 2, 0 + ξ (2 n)! ξ n 2 n, n, (11) 2!n! 1, 1 + ξ2 2 2 ) 0, 2. (12) ( ) 1/2 N 2 = 1+ ξ ξ 4. (13) 4 The reduced density matrix of the second subsystem can be written as ( ) 1 ρ 2 = [ ] ξ 2 ξ ( ξ 2 /2) + ξ 4 / , (14) which leads to the linear entropy I lin = 3 2 (1 Tr ρ 2 2 )= 6 ξ ξ 2 + ξ 4. (15) The maximum amount of this measure occurs for ξ 2 = 2, which is the maximum entanglement condition for the state ξ,2. On the other hand, the D concurrence of the state ξ,2 is given by ( (4 + 2 ξ 2 )(4 + ξ 4 )(2 ξ 2 + ξ 4 ) 1/2 ) D = (4 + 2 ξ 2 + ξ 4 ) 3. (16) 392

6 Volume 34, Number 4, July, 2013 Journal of Russian Laser Research Fig. 2. Variation of the entanglement of finite-dimensional PCS ξ,q as a function of ξ for the case q =2. Linear entropy (left) and D concurrence (right). In this case, the maximality of entanglement of ξ,2 is attained at D =2/3 2/3. Figure2showsthe variation of the different measures of entanglement as a function of ξ for q = 2. We see that the behavior of the linear entropy and D concurrenceissimilartothecaseq = 1, but with a different condition of the maximality of entanglement condition ξ =2. Now we consider the general form of finite-dimensional PCSs. To quantify the entanglement of these states, we apply the linear entropy and D concurrence and investigate the maximum entanglement condition. In this form, the density matrix of the second subsystem reads We rewrite the density matrix as ρ 2 = N q 2 ρ 2 = N q 2 2n (q n)! ξ n n. (17) q!n! c q,n 2 n n, c q,n 2 2n (q n)! = ξ. (18) q!n! With this description, the normalization factor of the system can be simply expressed in terms of c q,n s as N 2 q =( q c q,n 2 ) 1. Consequently, the entanglement of the state using the linear entropy is given by ( )( I lin = q +1 ) 2 1 c q,n 4 c q,n 2. (19) q Figure 3 shows the variation of the linear entropy of the finite-dimensional PCS as a function of ξ for largevaluesoftheparameterq. From Fig. 3, we see that the linear entropy does not fulfill the maximum entanglement condition for q>2, demonstrating that the increase in the system dimension can destroy the maximum amount of the entanglement. Therefore, it would be tempting to obtain the maximum entanglement condition in the general form of the finite-dimensional PCSs and find out whether there are some other maximum entangled states in this set or not. We apply the maximality condition of the entanglement for linear entropy on the state. Indeed, we have I lin = 1, which leads to q c q,n 4 ( q c q,n 2 ) 2 = 1 q

7 Journal of Russian Laser Research Volume 34, Number 4, July, 2013 Fig. 3. Variations in the linear entropy of the finite-dimensional PCS ξ,q as a function of ξ for different values of q equal to 4 (left), 8 (middle), and 20 (right). The only solution for this equation implies that for any n and m we must have c q,n = c q,m or, in other words, c q,0 = c q,1 = = c q,q 1 = c q,q. Since c q,0 = 1, the the above condition can be written as [ ] (q n)! ξ 2n =1, n =0, 1, 2,...,q. q!n! From the above condition, for n = 1, we immediately get ξ 2 = q, and consequently for maximum entangled finite-dimensional PCSs there is a tight relation between the coherent-state parameters ξ and q. For n = q, wehave ξ 2q = q! 2, and hence q q = q! 2. Thus, the maximum entanglement condition can be satisfied only for q = 1 and q =2,with ξ 2 = q. In other words, in the context of finite-dimensional PCSs, just two- and three-level states can be maximum entangled. Let us now study the entanglement of the state in terms of the D concurrence. Using the explicit form of the reduced density matrix ρ 2, we write the entanglement of the finite-dimensional PCS in the context of D concurrence as D(ρ) = det(1 ρ 2 )= q For maximum entangled states, D(ρ) gets its maximum value D(ρ) max = (1 c q,n 2 q i=0 c q,i 2 ) 1/2. (20) ( 1 1 ) (q+1)/2. (21) q +1 Figure 4 shows variations in the D concurrence for ξ,q for various values of the parameter q > 2. We see that, for these particular states, D concurrence does not hold for the maximum entanglement. Therefore, we see that in the set of all finite-dimensional PCSs with respect to the different values of q, the maximum entanglement can be obtained only for q = 1 and q = 2. These PCSs, in the context of quantum information, can be interpreted as two-qubit entangled states and two-qutrit entangled states. For q>2, the finite-dimensional PCS, regardless of the coherent-state parameter ξ, cannot rise to the maximum entanglement. In the following section, we consider the entanglement degree of the superposed finite-dimensional PCS. 394

8 Volume 34, Number 4, July, 2013 Journal of Russian Laser Research Fig. 4. Variations in the D concurrence of the finite-dimensional PCS ξ,q as a function of ξ for different values of q equalto10(left)and13(right). 4. Entanglement of Superposed Finite-Dimensional PCS The maximum entangled states, which play an important role in different tasks of quantum information theory, can be constructed from the different superpositions of finite-dimensional PCSs, showing the potential of application of these states in different branches of quantum information and optics. In this section, we study the entanglement of a superposition of two finite-dimensional PCSs in terms of different parameters involved in the states. Such states possess inherent remarkable nonclassical properties such as sub-poissonian distribution and anticorrelation between the two modes [38]. On the basis of the superposition principle in quantum mechanics, we consider the superposition of two finite-dimensional PCS as ξ,q,θ = ξ,q + e iθ ξ,q, (22) which is reduced to the even finite-dimensional PCSs for θ = 0, and to the odd finite-dimensional PCSs for θ = π [38]. Taking the following definitions: ξ,q = N q C q,n (ξ) q n, n, ξ,q = N q C q,n ( ξ) q n, n, (23) we obtain the superposed finite-dimensional PCS as ξ,q,θ = N q [C q,n (ξ)+e iθ C q,n ( ξ)] q n, n, (24) which leads to the following normalized state: ξ,q,θ = N q [C q,n (ξ)+e iθ C q,n ( ξ)] q n, n = N q D q,n (ξ) q n, n, (25) where D q,n (ξ) =C q,n (ξ)+e iθ C q,n ( ξ), and the normalization constant is given by [ ] 1/2 N q = D q,n (ξ) 2, (26) 395

9 Journal of Russian Laser Research Volume 34, Number 4, July, 2013 such that D q,n (ξ) 2 = C q,n (ξ) 2 + C q,n ( ξ) 2 + e iθ C q,n (ξ)c q,n( ξ)+e iθ C q,n(ξ)c q,n ( ξ). (27) In this case, the density matrix of the second subsystem is given as ρ 2 = N 2 q C q,n (ξ) 2 [1 + cos(θ)( 1) n ] n n = N 2 q and the linear entropy of the state reads I lin = q +1 q { 1 D q,n (ξ) 2 n n, (28) q D q,n(ξ) 4 [ q D q,n(ξ) 2 ] 2 }. (29) In order to find the maximum entangled stats, we first consider the condition for maximum entanglement in θ = 0 or, namely, the even finite-dimensional PCSs denoted by ξ,q +. Following the approach taken in the previous section, the maximality criterion in this respect reads D q,n (ξ) 2 =1 = ξ 2n [ (q n)! q!n! ] =1, n =0, 2,...,2k q. Investigating the answers by putting values n =1, 2, 3,...,weseethatn cannot be more that 2, and consequently the above equation has solutions only for q =2, 3, and 4. For other cases, the maximum entangled state seems to exist. The maximum entangled states associated to q = 2,q = 3,andq = 4 correspond to ξ 4 =4, ξ 4 = 12, and ξ 4 = 24, respectively. From the above results, the superposition of two finite-dimensional PCSs allows us to obtain some new maximum entangled states. Let us now consider the maximality of entanglement in the odd finite-dimensional PCS, which corresponds to θ = π in our superposed states and is denoted by ξ,q. Maximizing the linear entropy, we reduce the maximality condition of the state ξ,q to [ ] (q n)! ξ 2n = ξ 2, n =1, 3,...,2k +1 q. (30) q!n! 2 One may check that for the odd finite-dimensional PCS, we can have maximum entangled states for q =3, q =4,andq =5ifwesettheparameter ξ as ξ 4 = 12, ξ 4 = 36, and ξ 4 = 72, respectively. Thus, we have obtained three different maximum entangled odd finite-dimensional PCS with this description. When θ 0,π, the maximality condition reads ξ 2n [ (q n)! q!n! ] (1 + cosθ( 1) n )=1+cosθ, n =0, 1, 2,...,q. (31) If we take n = 1, the condition for the parameter ξ in terms of q and θ is ξ 2 = q 1+cosθ 1 cos θ qα. In the limit q = 1, the state becomes maximum entangled if ξ 2 =(1+cosθ)/(1 cos θ). 396

10 Volume 34, Number 4, July, 2013 Journal of Russian Laser Research For q = 2, the maximum entangled state corresponds to ξ 4 = 4 and cos θ = 0, leading to the state ξ,2 ±i ξ,2. In the case q>2, the first three equations for n =1, 2, and 3 read ξ 2 = q 1+cosθ 1 cos θ qα, ξ 4 =2q(q 1), ξ 6 =6αq(q 1)(q 2). (32) All states with q>2 satisfy at least the above three relations to be maximum entangled in the class of superposedstates. Solvingtheaboveequationsgiveusauniquevaluetotheintegerq as q =3,and consequently, the states with higher values of q could not be maximum entangled. Moreover, the above maximum entanglement relations give cos θ =(2 3)/(2 + 3). 5. Entanglement of Nonlinear Finite-Dimensional PCSs In this section, we investigate the entanglement of nonlinear finite-dimensional PCSs and apply the maximum entanglement condition of linear entropy on these sates. The nonlinear finite-dimensional PCSs is defined as ξ,q = N q ξ n (q n)! f 1 (q n)! q n, n, (33) q!n! f 1 (q)!f 2 (n)! where the parameter ξ is a complex variable, the parameter q is an integer, and f(n)! = f(n)f(n 1)...f(1)f(0) with f(0) = 1. Here, we are interested in considering the effect of nonlinearity on the entanglement of the finite-dimensional PCSs. Using the linear entropy, we write the amount of entanglement of nonlinear finite-dimensional PCSs as ( I lin = q +1 1 Nq 4 q [ ] (q n)! 2 [ ] ) ξ 4n f1 (q n)! 4. (34) q!n! f 1 (q)!f 2 (n)! For the maximum entangled states, we obtain that [ ][ ] (q n)! ξ 2n f1 (q n)! 2 =1, q!n! f 1 (q)!f 2 (n)! n =0, 1,...,q. (35) Taking n = 1, we obtain a fixed value for ξ in terms of the integer q as ξ 2 = q(f 1 (q)f 2 (1)) 2. Consequently, the condition of maximum entanglement reads [ qf 1 (q)f 2 (1)] n q!n! = (q n)! [ f1 (q)!f 2 (n)! f 1 (q n)! ]. (36) At n = q, we obtain [ qf 1 (q)f 2 (1)] q = q! 2 (f 1 (q)!f 2 (q)!). (37) Taking q = 1, we obtain that the nonlinear finite-dimensional PCS is maximum entangled if ξ = f 1 (1)f 2 (1). (38) The nonlinear finite-dimensional PCSs are able to create maximum entangled two-qubit states. These states are useful to generate and measure the entanglement not only for theoretical purposes but also for practical purposes due to their experimental accessibility. 397

11 Journal of Russian Laser Research Volume 34, Number 4, July, Entanglement of Mixed Finite-Dimensional PCSs Inthecaseofmixedstates,thequantumstateofabipartitesystemmustberepresentednotbya bracket as in the case of pure states, but by a matrix called the density matrix and denoted by ρ in quantum mechanics. In this scenario, quantum states are generally represented by the density matrix, which is denoted by ρ. We note that the density matrix is always decomposable into a mixture of density operators of a set of pure states as n ρ = p i ψ i ψ i, i where { ψ i } is a set of normalized pure states, and the corresponding probabilities p i s are positive and real numbers with n i p i =1. One of the difficulties in the quantification and characterization of mixed-state entanglement is linked to the fact that the entanglement of a superposition of pure bipartite states cannot be simply expressed as a function of the entanglement of the individual states in the superposition. This is because entanglement mostly depends on the coherence among the states in the superposition. It is therefore somewhat surprising that there exist tight lower and upper bounds on the entanglement of a superposition of states in terms of the entanglement of the individual states in the superposition. We consider a mixture of two finite-dimensional PCSs defined as a statistical mixture of two pure states ξ,q and ξ,q. The system density matrix is given by wherethesetofpurestatestakestheform which leads to the state ξ,q = ρ = p 1 ξ,q ξ,q + p 2 ξ,q ξ,q, (39) c qn q n, n, ξ,q = ρ = n,m=0 c qn q n, n, (40) α nm q n, n q m, m (41) where α nm = p 1 c qn c qm + p 2 c qnc qm. (42) We can quantify the amount of entanglement of this system based on the negativity or the logarithmic negativity measure. The negativity is a quantitative measure with the definition based on the partial transpose of the bipartite density matrix. According to the Peres criterion, if the partial transpose of the bipartite density matrix has at least one negative eigenvalue, the state becomes inseparable. Thus, the negativity of the state (39), defined in the context of the trace norm of ρ PT and denoted by ρ PT,is given by N (ρ) = ρ PT 1/2, (43) where ρ PT is the partial transpose with respect to system 2, and denotes the trace norm. The partial transpose of ρ is ρ PT = α nm q n, m q m, n. (44) n,m=0 398

12 Volume 34, Number 4, July, 2013 Journal of Russian Laser Research Since α nm = α mn, the eigenvalues of the partial transposed density matrix reads and also Therefore, the negativity of the system is given by as N(ρ) = λ nn = α nn, 0 n q, λ nm = ± α nm, 0 n<m q. n,m=0 p 1 c qn c qm + p 2 c qnc qm, n < m. (45) The other important measure is the logarithmic negativity, which is defined in terms of the negativity E N (ρ) =log 2 ρ PT, (46) and for the mixed state of a statistical mixture of two pure states it reads E N (ρ) =log 2 (1 + 2 n,m=0 p 1 c qn c qm + p 2 c qnc qm ), n < m. (47) Thus, the measures of the entanglement of mixed states in this case depend on the parameters involved in the finite-dimensional PCSs and probabilities. In this context, the entanglement behavior keeps the same shape as in the case of pure states, but with different numerical values, so the maximum entanglement for the mixed state can be detected [43]. The condition for separability and inseparability of mixed states can be obtained from the properties of a statistical mixture of the mixed state. The mixed state ρ via entangled finite-dimensional PCSs is said to be separable (N =0orE N = 0) if it can be written as a convex sum of separable pure states, i.e., ρ is the reduced density operator of the subsystem ρ 1 (respectively, ρ 2 ), given by ρ 1 =Tr 2 ρ (ρ 2 =Tr 1 ρ). The state ρ is entangled if it cannot be represented as a mixture of separable pure states (N 0or E N 0). From the above results, one can investigate and control the degree of entanglement for a large class of manifold bipartite PSC mixed states, including qubits, qutrits, and qudits, by a proper choice of the parameters q and ξ involved in the pure states. 7. Conclusions We investigated the entanglement degree of finite-dimensional PCSs. The finite-dimensional PCSs are a type of correlated two-mode states in finite dimension that possess some remarkable nonclassical effects. We used the D concurrence and linear entropy to quantify the entanglement. We showed that the maximum entanglement could be obtained for two- and three-dimensional dimensional PCSs, but the states with higher dimensions could not be maximum entangled. In other words, ξ,1 and ξ,2 can become maximum entangled with respect to the parameter ξ. The states with higher dimensions (q 3) do not become maximum entangled for all different values of the parameter ξ. We generalized the discussion to a superposition of two states of this type, and, in particular, the maximum entangled states for even and odd finite-dimensional pair coherent states were achieved. The results show that such 399

13 Journal of Russian Laser Research Volume 34, Number 4, July, 2013 superposed states can be useful tools in quantum information processing. Furthermore, the entanglement degree of nonlinear finite-dimensional PCSs is considered, and the maximality condition for the entanglement is surveyed. Finally, we generalized the discussion to the case of mixed states by considering a statistical mixture of two pure finite-dimensional PCSs. The negativity and logarithmic negativity are used as entanglement measures. It is worth mentioning that studying the entanglement degree of finite-dimensional PCSs highlights the subtleties of applying entanglement measures to such two-mode correlated states with continuous variables. Realistic quantum systems are not closed; therefore, it is important to study the robustness of the entanglement for finite-dimensional PCSs when the system loses its coherence due to interactions with the environment. An important future investigation will be the study of the effects of finite-temperature Markovian and non-markovian environments on the dynamics of the entanglement. Also, it will be important to study the multipartite entanglement in the context of finite-dimensional PCSs, which contributes to a better understanding of the entanglement behavior in these systems, and to consider the possible applications in various quantum information processing and transmission tasks. Acknowledgments It is a pleasure for Y.M. to acknowledge helpful discussions with G. Najarbashi, for which he is deeply grateful. References 1. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press (2002). 2. D. Petz, Quantum Information Theory and Quantum Statistics, Springer Verlag, Berlin, Heidelberg (2008). 3. T. D. Ladd, F. Jelezko, R. Laflamme, et al., Nature, 464, 45 (2010). 4. S. P. Walborn, P. H. Souto Ribeiro, L. Davidovich, et al., Nature, 440, 1022 (2006). 5. O. Guhne and G. Toth, Phys. Rep., 474, 1 (2009). 6. C. H. Bennett, G. Brassard, C. Crepeau, et al., Phys.Rev.Lett., 70, 1895 (1993). 7. C. H. Bennett and S. J. Wiesner, Phys.Rev.Lett., 69, 2881 (1992). 8. A. K. Ekert, Phys.Rev.Lett., 67, 661 (1991). 9. M. A. Nielsen, Phys.Rev.Lett., 93, (2004). 10. C. M. Dawson, H. L. Haselgrove, and M. A. Nielsen, Phys.Rev.Lett., 96, (2006). 11. M. Horodecki, P. Horodecki, and R. Horodecki, Phys. Lett. A, 223, 1 (1996). 12. M. S. Kim, W. Son, V. Bu zek, and P. L. Knight, Phys.Rev.A, 65, (2002). 13. Z. X. Juan, X. Hui, F. M. Fa, and Z. K. Cheng, Chin. Phys. B, 19, (2010). 14. K. A. Brickman and C. Monroe, Rep. Prog. Phys., 73, (2010). 15. P. C. Haljan, P. J. Lee, K. A. Brickman, et al., Phys.Rev.A, 72, (2005). 16. M. Fox, Quantum Optics: An Introduction, 1st ed. Oxford University Press (2006). 17. C. Gerry and P. Knight, Introductory Quantum Optics, Cambridge University Press (2005). 18. G. Najarbashi and Y. Maleki, Int. J. Theor. Phys., 50, 2601 (2011). 19. K. Berrada, Opt. Commun., 285, 2227 (2012). 400

14 Volume 34, Number 4, July, 2013 Journal of Russian Laser Research 20. K. Berrada, M. El Baz, and Y. Hassouni, J. Stat. Phys., 142, 510 (2011). 21. K. Berrada, M. El Baz, and Y. Hassouni, Phys. Lett. A, 375, 298 (2011). 22. K. Berrada and Y. Hassouni, Eur.Phys.J.D, 61, 513 (2010). 23. K. Berrada, Y. Hassouni, and H. Eleuch, Commun.Theor.Phys., 56, 679 (2011). 24. K. Berrada, A. Chafik, H. Eleuch, and Y. Hassouni, Int. J. Mod. Phys. B, 23, 2021 (2009). 25. K. Berrada, M. El Baz, H. Eleuch, and Y. Hassouni, Int. J. Mod. Phys. C, 21, 291 (2010). 26. S. J. van Enk, Phys.Rev.A, 72, (2005) S. J. van Enk and O. Hirota, Phys.Rev.A, 64, (2001). 28. H. Fu, X. Wang, and A. I. Solomon, Phys. Lett. A, 291, 73 (2001). 29. H. Chen and J. Zhang, Phys.Rev.A, 75, (2007). 30. J. Zhang, C. Xie, and K. Peng, Phys.Rev.Lett., 95, (2005). 31. X.Su,A.Tan,X.Jia,etal.,Phys.Rev.Lett., 98, (2007). 32. M. Kim, J. Mod. Opt., 50, 1809 (2003). 33. K. Jensen, W. Wasilewski, H. Krauter, et al., Nature Phys., 7, 13 (2011). 34. M. Ban, J. Opt. B, 1, L9 (1999). 35. G. S. Agarwal, Phys.Rev.Lett., 57, 827 (1986). 36. G. S. Agarwal and A. Biswas, J. Opt. B: Quantum Semiclass. Opt., 7, 350 (2005). 37. A.-S. F. Obada and E. M. Khalil, Opt. Commun., 260, 19 (2006). 38. X. G. Meng, J. S. Wang, and B. L. Liang, Optik, 122, 2021 (2011). 39. N. A. Peters, T. C. Wei, and P. G. Kwiat, Phys.Rev.A, 70, (2004). 40. Z. H. Ma, W. G. Yuan, M. L. Bao, and X. D. Zhang, Quanum Inf. Comput., 11, 0070 (2011). 41. G. Vidal and R. F. Werner, Phys.Rev.A, 65, (2002). 42. W. K. Wootters, Phys.Rev.Lett., 80, 2245 (1998). 43. K. Berrada, A. Mohammadzade, S. Abdel-Khalek, et al., Physica E, 45, 21 (2012). 401