GHZ States, Almost-Complex Structure and Yang–Baxter Equation (I) Yong Zhangab1 and Mo-Lin Geb2 a Department of Physics, University of Utah 115 S, 1400 E, Room 201, Salt Lake City, UT 84112-0830 b Theoretical Physics Division, Chern Institute of Mathematics 7 Nankai University, Tianjin 300071, P. R. China 0 0 2 Abstract n Recentstudysuggeststhattherearenaturalconnectionsbetweenquantumin- a J formationtheoryandtheYang–Baxterequation. Inthispaper,intermsofthe generalizedalmost-complexstructureandwiththehelpofitsalgebra,wedefine 1 3 the generalized Bell matrix to yield all the GHZ states from the product base, proveitto formaunitary braidrepresentationandpresentanew type ofsolu- 1 tionofthequantumYang–Baxterequation. WealsostudyYang-Baxterization, v Hamiltonian, projectors,diagonalization,noncommutativegeometry,quantum 4 algebra and FRT dual algebra associated with this generalized Bell matrix. 4 2 1 0 7 0 / PACS numbers: 02.10.Kn, 03.65.Ud, 03.67.Lx h p Key Words: GHZ State, Yang–Baxter, Almost-Complex Structure, FRT dual - t n a u q : v i X r a [email protected] [email protected] 1 Introduction Recently, a series of papers [1, 2, 3, 4, 5, 6, 7, 8, 9, 10] have suggested there are natural and deep connections between quantum information theory [11] and the Yang–Baxterequation(YBE)[12,13]. UnitarysolutionsofthebraidedYBE(i.e., the braid group relation) [1, 2] as well as unitary solutions of the quantum Yang– Baxter equation (QYBE) [3, 4] can be often identified with universal quantum gates[14]. Yang–Baxterization [15]isexploitedtosetuptheSchro¨dingerequation determiningtheunitaryevolution ofaunitary braidgate [3,4]. Furthermore, the Werner state [16] is viewed as a rational solution of the QYBE and the isotropic state [17] with a specific parameter forms a braid representation, see [7, 8]. More interestingly, the Temperley–Lieb algebra [18] deriving a braid representation in the state model for knot theory [19] is found to present a suitable mathematical framework for a unified description of various kinds of quantum teleportation phenomena [20], see [9, 10]. The present paper is a further extension of the previous published research work[3,4,6]inwhichtheBellmatrixhasbeenrecognizedtoformaunitarybraid representation and generate all the Bell states from the product base. In this paper, a unitary braid representation also called the Bell matrix for convenience is defined to create all the Greenberger-Horne-Zeilinger states (GHZ states) from theproductbase. TheGHZstates aremaximallymultipartiteentangled states(a natural generalization of the Bell states) and play important roles in the study of quantuminformation phenomena[21,22,23]. More importantly, thisBell matrix has a form in terms of the almost-complex structure which is fundamental for complex and Ka¨hler geometry and symplectic geometry. Therefore, our paper is building heuristic connections among quantum information theory, the Yang– Baxter equation and differential geometry. We hereby summarize our main result which is new to our knowledge. 1. We define the Bell matrix to produce all the GHZ states from the product base, prove it to be a unitary braid representation, and derive the Hamil- tonian to determine the unitary evolution of the GHZ states. 2. We recognize the almost-complex structure in the formulation of the Bell matrix as well as its algebra in the proof for the Bell matrix satisfying the braided YBE, and exploit it to represent a new type of the solution of the QYBE. 3. We study topics associated with the generalized Bell matrix which include Yang–Baxterization, diagonalization, noncommutative geometry, quantum algebra via the RTT relation and standard FRT procedure [24, 25]. Asthefirstpaperinthisresearchproject,forsimplicity,thepresentmanuscript only focuses on thegeneralized Bell matrix of thetype 22n 22n correspondingto × the GHZ states of an even number of objects, while our result on the generalized Bell matrix of the type 22n+1 22n+1 is collected [26]. × 2 Theplanofthispaperisorganizedasfollows. Section2sketchesthedefinition of the GHZ states and represent the Bell matrix in terms of the almost complex structure. Section 3 introduces the generalized Bell matrix and show both an algebra and a interesting type of solution of the QYBE in terms of the general- ized almost-complex structure. Sections 4 and 5 briefly deal with various topics about the generalized Bell matrix: projectors, diagonalization, noncommutative geometry, quantum algebra and FRT dual algebra. Last section concludes with worthwhile problems for further research. 2 GHZ states, Bell matrix and Hamiltonian This section is devised to set up a simplest example to be appreciated by readers mostly interested in quantum information and physics, and it explains how to observe the Bell matrix from the formulation of the GHZ states (as well as the almost-complex structure from the Bell matrix) and how to obtain Hamiltonians to determine the unitary evolution of the GHZ states. 2.1 GHZ states, Bell matrix and almost-complex structure In the 2N-dimensional Hilbert space with the base denoted by the Dirac kets m ,m , ,m , m , ,m = 1, there are 2N linearly independent GHZ | 1 2 ··· Ni 1 ··· N ±2 states of N-objects having the form 1 (m ,m , ,m m , m , , m ) (1) 1 2 N 1 2 N √2 | ··· i±|− − ··· − i whicharemaximallyentangledstates inquantuminformationtheory[11]. Inthis paper, all the GHZ states are found to be generated by the Bell matrix acting on the chosen product base, Φ = m ,m , ,m , Φ = m , m , , m , (2) | ki | 1 2 ··· Ni | 2N−k+1i |− 1 − 2 ··· − Ni where 1 k 2N 1. One can take a notation similar to [27, 28], − ≤ ≤ N 1 k[m , ,m ]= 2N 1+ 2N i m (3) 1 N − − i ··· 2 − Xi=1 which has the result at N = 2, for example, 1 1 1 1 1 1 1 1 k[ , ] = 1, k[ , ] = 2, k[ , ] = 3, k[ , ]= 4, (4) 2 2 2 −2 −2 2 −2 −2 assigned to label the GHZ states of two objects (the well known Bell states). The 4 4 Bell matrix B acts on the product base 11 , 1 1 and 11 , × 4 |22i |2−2 i |−2 2i 1 1 to yield the Bell states, and it has a known form [1, 2, 3, 4, 6], |−2 −2 i 1 0 0 1 1  0 1 1 0  1 1 B = (B ) = , k,n,l,m = , , (5) 4 kn,lm 4 √2 0 1 1 0 2 −2  −   1 0 0 1   −  3 and the 8 8 Bell matrix B given by 8 × 1 0 0 0 0 0 0 1  0 1 0 0 0 0 1 0  0 0 1 0 0 1 0 0   1  0 0 0 1 1 0 0 0  B (B ) =  , 8 ≡ αl,βm 8 √2  0 0 0 1 1 0 0 0   −   0 0 1 0 0 1 0 0   −   0 1 0 0 0 0 1 0   −   1 0 0 0 0 0 0 1   −  3 1 1 3 1 1 α,β = , , , , l,m = , (6) 2 2 −2 −2 2 −2 creates the GHZ states of three objects by acting on Φ , 1 k 8. k | i ≤ ≤ The 2N 2N Bell matrix generating the GHZ states of N-objects from the × product base Φ , 1 k 2N, has a form in terms of the almost-complex k | i ≤ ≤ structure 3 denoted by M, 1 1 B = (11+M), B Bkl = (δkδl +Mkl) (7) √2 ij,kl ≡ ij √2 i j ij where 11 denotes the identity matrix, the lower index of B is omitted for con- 2N venience, δj is the Kronecker function of two variables i,j, which is 1 if i = j and i 0 otherwise, and the almost-complex structure M has the component formalism using the step function ǫ(i), M Mkl = ǫ(i)δ kδ l, ǫ(i) = 1,i 0; ǫ(i) = 1,i < 0, (8) ij,kl ≡ ij i− j− ≥ − which satisfies M2 = 11. In terms of the tensor product of the Pauli matrices, − the Bell matrix B and the almost complex structure M for N-objects have the forms given by B = eπ4M, M = √ 1σy (σx)⊗(N−1), (σx)⊗(N−1) = σx σx. (9) − ⊗ ⊗···⊗ N 1 − | {z } Note that there exist other interesting matrices related to the GHZ states, for example, one can have matrix entries ǫ(i)B for a new matrix. But so ij,kl far as the authors know, only the Bell matrix is found to form a unitary braid representation. 2.2 Yang–Baxterization and Hamiltonian The Bell matrix B satisfies the following characteristic equation given by 1+√ 1 1 √ 1 (B − 11)(B − − 11) = 0 (10) − √2 − √2 3The almost-complex structure is usually denoted by the symbol J in the literature and it is a linear map from a real vector space to itself satisfying J2 =−1. More details on geometry underlyingwhat we are presenting here will bediscussed elsewhere. 4 whichsuggestsithavingtwodistincteigenvalues 1 √ 1. UsingYang–Baxterization4, ± − √2 a solution of the QYBE with the Bell matrix as its asymptotic limit, is obtained to be 1 1 Rˇ(x) = B +xB 1 = (1+x)11+ (1 x)M. (11) − √2 √2 − As this solution Rˇ(x) is required to be unitary, it needs a normalization factor ρ with a real spectral parameter x, B(x)= ρ−21Rˇ(x), ρ= 1+x2, x R. (12) ∈ As the real spectral parameter x plays the role of the time variable, the Schro¨dinger equation describing the unitary evolution of a state φ (independent of x) determined by the B(x) matrix, i.e., ψ(x) = B(x)φ, has the form ∂ ∂B(x) √ 1 ψ(x) = H(x)ψ(x), H(x) √ 1 B 1(x), (13) − − ∂x ≡ − ∂x where the time-dependent Hamiltonian H(x) is given by ∂ H(x) = √ 1 (ρ−12Rˇ(x))(ρ−12Rˇ(x))−1 = √ 1ρ−1M. (14) − ∂x − − To construct the time-independent Hamiltonian, a new time variable θ instead of the spectral parameter x is introduced in the way 1 x cosθ = , sinθ = , (15) √1+x2 √1+x2 so that the Bell matrix B(x) has a new formulation as a function of θ, B(θ)= cosθB+sinθB−1 = e(π4−θ)M, (16) and hence the Schro¨dinger equation for the time evolution of ψ(θ) = B(θ)φ has the form ∂ ∂B(θ) √ 1 ψ(θ) = Hψ(θ), H √ 1 B 1(θ)= √ 1M, (17) − − ∂θ ≡ − ∂θ − − wherethetime-independentHamiltonian5H ishermitianduetotheanti-hermitian of the almost-complex structure, i.e., M = M, and the unitary evolution op- † − erator U(θ) has the form U(θ)= e Mθ. − 4See [4] or Subsection 3.1 and Subsection 4.1 for the detail. Here Yang–Baxterization is applied to the Bell matrix of the type 22n×22n, while Yang–Baxterization of the Bell matrix of the type22n+1×22n+1 is rather subtleto be presented [26]. 5The Hamiltonian used in our previous published papers [3, 4] has an additional numerical factor 1 compared to the time-independent Hamiltonian (17). This factor 1 is very important 2 2 whenwerecognizetheactionofthefourdimensionalunitaryevolutionoperatorexp1θM onthe 2 productbasetobeequivalenttoaproductoftwounitaryrotations ofWignerfunctionsforthe Bell states |11i±|−1−1i and |1−1i±|−11i, respectively. Note that no boundary conditions 22 2 2 2 2 2 2 havebeen imposed on the Schr¨odingerequations (13) and (17). 5 3 Generalized Bell matrix and YBE This section proves thegeneralized6 Bell matrixB of the type 22n 22n to form a × unitary braid representation with the help of the algebra generated by the gener- e alized almost-complex structure M, and presents an interesting type of solution oftheQYBEintermsofM whichmaybenotwellnoticedbeforeintheliterature. f f 3.1 YBE and Yang–Baxterization In this paper, the braid group representation σ-matrix and the QYBE solution Rˇ(x)-matrix are d2 d2 matrices acting on V V where V is a d-dimensional × ⊗ complex vector space. As σ and Rˇ act on the tensor product V V , they are i i+1 ⊗ denoted by σ and Rˇ , respectively. i i The generators σ of the braid group B satisfy the algebraic relation called i n the braid group relation, σ σ σ = σ σ σ , 1 i n 1, i i+1 i i+1 i i+1 ≤ ≤ − σ σ = σ σ , i j > 1. (18) i j j i | − | while the quantum Yang–Baxter equation (QYBE) has the form Rˇ (x)Rˇ (xy)Rˇ (y) = Rˇ (y)Rˇ (xy)Rˇ (x) (19) i i+1 i i+1 i i+1 with the spectral parameters x and y. In addition, the component formalism the QYBE (or the braid group relation) can be shown in terms of matrix entries, Rˇ(x)i′j′ Rˇ(xy)k′k2Rˇ(y)i2j2 = Rˇ(y)j′k′ Rˇ(xy)i2i′Rˇ(x)j2k2. (20) i1j1 j′k1 i′k′ j1k1 i1j′ i′k′ In view of the fact that Rˇ(x = 0) forms a braid representation, the braid grouprelationisalsocalled thebraidedYBE.Concerningrelations betweenbraid representations and x-dependent solutions of the QYBE (19), the procedure of constructingtheRˇ(x)-matrix fromagiven braidrepresentationσ-matrix iscalled Baxterization [15] or Yang–Baxterization. For a braid representation σ with two distincteigenvaluesλ andλ ,thecorrespondingRˇ(x)-matrixobtainedviaYang– 1 2 Baxterization has the form Rˇ(x) = σ+xλ λ σ 1 (21) 1 2 − which has been exploited in Subsection 2.2, see (11). 3.2 Unitary generalized Bell matrix as a solution of YBE The generalized Bell matrix B has the form in terms of the generalized almost- complex structure M with deformation parameters q , ij e f 1 Bkl = (δkδl +Mkl), Mkl = ǫ(i)q δ kδ l, (22) ij √2 i j ij ij ij i− j− e f f 6Here “generalized” means that theobject has deformation parameters. 6 where q q = 1 is required for M2 = 11 and the step function ǫ(i) has the ij i j −− − properties given by f ǫ(i)ǫ(i) = 1, ǫ(i)ǫ( i) = 1, ǫ(i) = 1, (23) − − ± Let B be labeled by familiar indices by the angular momentum theory in quantum mechanics, e (BJ1J2)bν, µ,ν = J ,J 1, , J , a,b= J ,J 1, , J , (24) µa 1 1 1 2 2 2 − ··· − − ··· − e whereBJJ denotes thegeneralized Bell matrix B associated with theGHZ states of an even number of objects, for example, e e 11 33 77 B4 = B22, B16 = B22, B64 = B22, (25) e e e e e e but the same type of generalized Bell matrix may be labeled differently, for ex- 13 31 ample, both B22 and B22 are the type of B8. In the following, we focus on the generalized Bell matrix of the type BJJ e e e denoted by B, while we submit our result on the generalized Bell matrix of the e type BJ1J2, J = J to [26]. e1 6 2 In the proof for BJJ forming a braid representation (18) in terms of its com- e ponentformalism(22),deformationparametersq arefoundtosatisfyequations, ij e q q = q q , i ,j ,k = J,J 1, , J, i1j1 −i1−j1 j1k1 −j1−k1 1 1 1 − ··· − q = q q q , q = q q q , (26) j1k1 i1j1 j1k1 i1j1 i1j1 j1k1 i1 j1 j1 k1 − − − − where no summation is imposed between same lower indices and which can be simplified by q q = 1. Furthermore, the unitarity of B leads to a con- i1j1 i1 j1 − − straint on the generalized almost-complex structure M, namely, e M M T = M 1 = M q qf = 1, (27) † ∗ − i∗j ij ≡ − ⇒ f f f f where the symbol denotes the complex conjugation and the symbol T denotes ∗ the transpose operation. As J is ahalf-integer, weobtain solutions forequations (26)and(27)interms of independent (J + 1) number of angle parameters ϕ , ϕ , , ϕ , 2 J J−1 ··· 12 qlm = eiϕl+2ϕm, ϕ l = ϕl, 0 l J, (28) − − ≤ ≤ where the method of separation of variables has been used since one can choose ql = eiϕl and then qlm = qlqm. For example, deformation parameters in the unitary generalized Bell matrix 11 B22 are calculated to be e q = eiϕ, q = e iϕ, q = q = 1, (29) 11 1 1 − 1 1 11 22 −2−2 2−2 −22 7 which are the same as those presented [3, 4, 6], and deformation parameters of 33 the generalized Bell matrix B22 have the form, q33 = eiϕ1, q3e1 = eiϕ1+2ϕ2, q3 1 = eiϕ1−2ϕ2, q3 3 = 1, 22 22 2−2 2−2 q13 = eiϕ1+2ϕ2, q11 = eiϕ2, q1 1 = 1, q1 3 = eiϕ2−2ϕ1. (30) 22 22 2−2 2−2 In the 22n-dimensional7 vector space, the generalized almost-complex struc- ture M is found in this paper to satisfy algebraic relations, f M2 = 11, M M = M M , i 1 i i i 1 − ± − ± M M = M M , i j 2, i,j N, (31) fi j j if f f f | − | ≥ ∈ whichdefinesanalgebrfaofbviousflydfifferentfromtheTemperley–Liebalgebra[18] or the symmetric group algebra and where deformation parameters q satisfy ij q q = 1, q q = q q . (32) ij i j ij ij jk j k −− − − With the help of this algebra (31), the generalized Bell matrix B can be easily proved to satisfy the braided YBE (18) in the way e B B B = 2M +2M +M M +M M = B B B . (33) i i+1 i i i+1 i i+1 i+1 i i+1 i i+1 e e e f f f f f f e e e Additionally, the generalized almost-complex structure M and the permuta- tion operator P satisfy the following algebraic relation f P M P = P M P , P = ij ji, (34) i i+1 i i+1 i i | ih | Xij f f which is underlying algebraic relations of the virtual braid group, i.e., the braid B and permutation P forming a unitary virtual braid representation, see [7, 8]. e 3.3 New type of solution of QYBE via parameterization Similar to the formalism of the rational solution of the QYBE (19), Rˇ (u) = 11+uP, P2 = 11 (35) rational where P is a permutation matrix, we obtain a solution of the QYBE in terms of the generalized almost-complex structure, Rˇ(u) = 11+uM (36) satisfying the following equationeof Yang–Baxtfer type, u+v u+v Rˇ (u)Rˇ ( )Rˇ (v) = Rˇ (v)Rˇ ( )Rˇ (u), (37) i i+1 i i+1 i i+1 1+uv 1+uv 7Here we have22n =(2J+1)2, for example, n=1,J = 1 and n=2,J = 3, see (25). 2 2 8 which has been exploited [4] and where new spectral parameters u,v are related to original spectral parameters x,y in the way 1 x 1 y 1 xy u+v u= − , v = − , − = . (38) 1+x 1+y 1+xy 1+uv Via a further parametrization of spectral parameters u,v in terms of angle variables Θ ,Θ , 1 2 u+v u = √ 1tanΘ , v = √ 1tanΘ , = √ 1tan(Θ +Θ ), (39) 1 2 1 2 − − − − 1+uv − − the modified Yang–Baxter equation (37) has the ordinary form Rˇ (Θ )Rˇ (Θ +Θ )Rˇ (Θ ) =Rˇ (Θ )Rˇ (Θ +Θ )Rˇ (Θ ). (40) i 1 i+1 1 2 i 2 i+1 2 i 1 2 i+1 1 with the solution given by Rˇ(Θ)= 11 √ 1tanΘM, or Rˇ(Θ) = 11+tanhΘM. (41) ′ ′ − − Note thatephysical models underlfying this typeeof solution of QYBEfin terms of the almost-complex structure will be discussed and submitted elsewhere. 4 Projectors, diagonalization and geometry This section and the next one are aimed at introducing several selective topics directly using the generalized Bell matrix and the generalized almost-complex structure, for example, associated noncommutative geometry, quantum algebra and FRT dual algebra. 4.1 Projectors and Yang–Baxterization In terms of M, two projectors P and P are defined by + − f 1 e e 1 P = (1+√ 1M), P = (1 √ 1M) (42) + 2 − − 2 − − e f e f satisfying basic properties of two mutually orthogonal projectors, P +P = 11, P2 = P , P P = 0. (43) + + − ± ± − ThegeneralizedBelelmatreixB hasetwodiestinceteiegenvaluese±iπ4 anditsatisfies the same characteristic equation as (10), e (B λ 11)(B λ+11) =0, λ+ = e−iπ4, λ = eiπ4. (44) − − − − With the projeectors P aend eigenvalues λ , the generalized Bell matrix and its ± ± inverse have the forms e B = λ P +λ P , B 1 = λ P +λ P . (45) + + − + + − − − − e e e e e e 9 Using Yang–Baxterization [4], the Rˇ(x)-matrix as a solution of the QYBE (19) has a form similar to (11), Rˇ(x) = (λ +λ x)P +(λ +λ x)P = B +xB 1, (46) + + + − − − − e e e e e and hencethe correspondingSchrodinger equation also has asimilar form to (13) or (17) except that the Hamiltonian is determined by M instead of M. f 4.2 Diagonalization of the generalized Bell matrix The diagonalization of the generalized Bell matrix B can be performed by a unitary matrix D via the following unitary transformation, e 1 DBD = Diag(1+√ 1, ,1 √ 1) (47) † √2 − ··· − − e where the diagonal matrix Diag has the same number of matrix entries 1+√ 1 − as 1 √ 1. Assume D to have the form by a Hermitian unitary matrix N, − − 1 D = (11+√ 1N), N = N, N2 = 11 (48) † √2 − and then this matrix N is found to satisfy an additional condition, NM = MN = Diag(1, 1, ,1, 1), (49) − − ··· − f f where the diagonal matrix Diag has the same number of matrix entries 1 as 1 − but the ordering between 1 and 1 is not fixed. − After some algebra, one type of formalism of the matrix N is given by Nkl = f(i)q δ kδ l, f(i)f(i) = 1, f(i)= f( i) = f (i) (50) ij ij i− j− − ∗ where q are the same as unitary deformation parameters q in the generalized ij ij Bell matrix B. This matrix N brings about the diagonalization form of B, e 1 e (DBD )mn = (1+√ 1f(i)ǫ( i))δmδn. (51) † ij √2 − − i j e in which setting f(i)= ǫ( i), i > 0 and f(i) = ǫ(i), i < 0 leads to − 1 DBD = Diag(1+√ 1, ,1+√ 1,1 √ 1, ,1 √ 1). (52) † √2 − ··· − − − ··· − − e 2N−1 2N−1 | {z } | {z } In the four dimensional case, for example, the Bell matrix B is diagonalized 4 in the way, 1 D4B4D4† = √2Diag(1−√−1,1+√−1,1−√−1,1+√−1), N4 = −σy⊗σy, (53) 10