New series of multi-parametric solutions to GYBE: quantum gates and integrability

We obtain two series of spectral parameter dependent solutions to the generalized Yang-Baxter equations (GYBE), for definite types of $N_1^2\times N_2^2$ matrices with general dimensions $N_1$ and $N_2$. Appropriate extensions are presented for the inhomogeneous GYBEs. The first series of the solutions includes as particular cases the $X$-shaped trigonometric braiding matrices. For construction of the second series the colored and graded permutation operators are defined, and multi-spectral parameter Yang-Baxterization is performed. For some examples the corresponding integrable models are discussed. The unitary solutions existing in these two series can be considered as generalizations of the multipartite Bell matrices in the quantum information theory.


Introduction
The area of the quantum exactly solvable models in recent decades admitted close relationship with the disciplines of quantum computing.The generalized Yang-Baxter equations (GYBE) are investigated intensively for depicting unitary R-matrices on the braid group representations, as candidates for complex quantum gates, used in the quantum information processing.Yang-Baxter equations (YBE), initiating as star-triangle equations for solving concrete models in statistical mechanics, have acquired an important role in solvability of the many-body systems, being key equations in the theory of quantum integrability [1]- [20].
Later YBE found applications in quantum field theory, quantum topology, string theory, quantum groups.And there are emerged numerous works ( [24]- [41]), revealing the importance of YBE in the quantum information theory.Here the information is encoded in quantum states and processed by unitary evolution operators.The quantum computation process is modeled by quantum circuits, which are the sequences of the quantum gates, the quantum analogs of classical logic gates.The n-qubit quantum gates are described by unitary matrices defined on the superpositions of the tensor products of n states-qubits (the basic unit of the quantum information, quantum two-dimensional analog of the classical binary information unit -bit).As the logical gates are basic blocks, it is natural to analyze the cases with small n-s, n = 1, 2, 3 [20,24,25].For n = 2, particularly, the quantum gates can be associated with two-state unitary Ř-matrices which are the solutions of the YBE, relying on the revealed topological properties of the quantum gates -braid group symmetries [21]- [27].
In topological quantum computation, the unitary representations of the braid group can be described by braiding anyons [30,31].The p 2 × p 2 (qupit [43]) generalizations of the quantum gates as braid group representations and corresponding YBE solutions are discussed in [42,30,35] associated with the Z p parafermion.Generalized YBE, with Ř-matrices acting on the tensor product of more than two vector states, are developed in [30,32,26], showing, that braiding unitary solutions to GYBE can be used in quantum information processing.
In this paper we consider spectral parameter dependent solutions to inhomogeneous GYBE for arbitrary dimensional matrices of two types, which both are the extensions of 4 × 4 eight-vertex trigonometric solution: extension by form (the first kind -X-shaped) and by construction (the second kind).In the work [48] entire description is done of the multiparametric solutions of YBE for the simplest case of 4 × 4 matrices (see also [2,49,50]).
Here we obtain direct generalizations of all these solutions for even dimensional X-shaped [30] matrices.In this paper we present a thorough investigation for even dimensional matrices, the odd dimensional cases we describe comparably in a brief way: the main results are obtained and presented, and the detailed analysis with some peculiar solutions will be presented in the further work [55].
In Section 2 the GYB equations are analysed in the context of the integrability of 1D quantums models.The inhomogeneous extensions of GYBE are proposed, and in Subsection 2.1 we emphasize the important differences of these cases in comparison to inhomogeneous YBE.We have discussed the inhomogeneous N 2 1 × N 2 2 dimensional "check" or braiding Řmatrices defined on the space V N 1 ⊗ V N 2 .The multi-parametric extensions of the known solutions to GYBE are constructed in Section 3. The general even dimensional X-shaped solutions to GYBE, their parameterizations and the corresponding integrable models are discussed, the particular examples (for the solutions to inhomogeneous (2, 3, 1)-GYBE) are presented in Appendix.The peculiarities of the odd-dimensional solutions we discuss in Subsection 3.1.Then in Section 4 we sketch also the new perspectives given by the obtained R-matrices in the quantum information theory.In Section 5 the graded and colored representations of the braid group are considered, with corresponding Yang-Baxterization, bringing to trigonometric solutions.After appropriate re-parametrization they can considered as new second kind solutions to YBE [28].The examples of 16 × 16 solutions are discussed in details in Subsections 5.2 and 5.3.

YBE and generalized YBE.
The generalized Yang-Baxter equations can be defined by the following relations, as in [30,32] ( Ř ⊗ I p )(I p ⊗ Ř)( Ř ⊗ I p ) = (I p ⊗ Ř)( Ř ⊗ I p )(I p ⊗ Ř), (2.1)Here the matrix Ř is defined on the product k V , the unity operator I p acts as an identity operator on the product of the vector states p V .These equations are called GYBEs of type (d, k, p), where d = Dim[V ].Far distanced R-matrices, which have no overlap of the acting states, are commutative.Thus, these equations reproduce the braid algebra [21,22], Here there is used the "check" notation for the matrices, since in the particular case of (d, 2, 1) these equations coincide with the ordinary YBE [8] in check formulation.This is true for the homogeneous cases, when the states on which the Ř-matrix acts, are identical.
Before proceed further let us consider the generalization of the inhomogeneous YBE to the inhomogeneous GYBE at least by the following two ways.
i.The inhomogeneous GYBE -({d}, {k, k ′ }, {p, p ′ }), k + p = k ′ + p ′ can be presented in the following way: Here the system of the equations acts on the product of the identical spaces V , with the same dimension d, and the matrix R ′ acts on the product ii.The full inhomogeneous GYBE - Let us extend definition of the action space to the inhomogeneous product of the vector ) with different characteristics of the vector states (in general with different dimensions d i ), and intend the following inhomogeneous matrices in the discussion below we use Ři 1 ...i n , where the indexes 1, ..., n show the position of the vector space, while the index i n denotes the nature of the vector space The unitary solutions to GYBE, constituting the braid group representations, are connecting to the quantum computing theory [27].However, even for the inhomogeneous case, when at general d i = d j , the GYBE solutions can also be used to the construction of the integrable models, as we describe below.The spectral parameter dependent versions of GYBE are defined just as for the case of YBE, thus ensuring the connection with the integrable models:

The differences between the usual inhomogeneous YBE and the GYBE
Let us consider a graphical interpretation of GYBE, (d, k, p), which act on the product of the states k+p V d , (Fig. 1), with notation Řk for the matrix acting on the product of the states: Note, that the equations {d, 2p, p} can also be considered as usual YBE, which act on the product of the states From this point of view, the other cases k = 2p, can be considered as the equations on the space In the Fig. 1 the way of this kind of the combination of the vector spaces into two big spaces, is apparent.We see that these equations are quite different from the usual inhomogeneous YBE defined on the spaces V ′ ⊗ V ′′ ⊗ V ′ in the "check"-matrix formulation [14]: This is the usual braiding form of ordinary YB equations, the "check"-operation do not affair the form of the equations, it only permutes the upper indexes of the matrices in the same YBEs.In the Fig. 2 there are shown two type of the matrices R V ′ ⊗V ′′ , acting on the space k V , which appear in the equations (2.6) and (2.7).

Řk
If in (2.6) the Ř-matrices are acting only on the space k V , for the case (2.7) there are also matrices defined on the space 2p V .Recall, that the non-check R-matrix acts as P Ř, so in the Fig. 2 (i) it appears with the permutation of the upper indexes: R āb ab .We see, that the given difference between the usual inhomogeneous YBE and GYBE is encoded in the definition of the "check"-matrices.For the first case it acts as the usual braiding matrix, Fig.

(i).
In the second case, correspondingly, the graphical representation of the "check"-matrix is presented in the Fig. 2 (ii), and can be considered in some sense as "reflecting" matrix.
However they both satisfy the braiding relations.
For the inhomogeneous equations, the vector states, on which the R-matrices act, can differ not only by their dimensions, also by "inner" characteristics.As example, the intertwiner R-matrices, defined in the theory of the quantum groups, satisfy to the ordinary YBE [14], and in the "check" formulation commute with the generators of the quantum group.There the vector states V constitute the representations of the quantum groups and can differ one from other by the characteristics of the representations -the eigenvalues of the center of the group [14,49].An interesting example is the case of two-dimensional cyclic states of the sl q (2) group at q 4 [49], where two vector spaces are of the same dimension, but have different characteristics, which bring to various kind of invariant solutions to YBE.And one can find how to operate with the inhomogeneous R-matrices in "check" formulation for the usual Yang-Baxter equations, particularly in the works [45,49], where we have presented the solutions with the quantum group symmetry including the permuted projection operators between the representation states.
The case p ≥ k.In this section we have considered the non-trivial case, when p < k.
However, one can consider also the remaining cases.As it can be easily followed from the presented equations and the Fig. 1, at p ≥ k the generalised YB equations factorize into two similar operator equations.The operators [ Ř ⊗ I p ] and [I p ⊗ Ř] are commutative and act non-trivially on different, non-intersecting spaces: V 1 ⊗ ... ⊗ V k and V p+1 ⊗ ... ⊗ V p+k (we use the notations introduced within the item (ii)).The right-hand and left-hand sides of GYBE act as unity operators on the vector spaces V k+1 ⊗ ... ⊗ V p .Meanwhile on the product the spectral parameter dependent GYBEs (2.5) in the cases p ≥ k take the following matrix product form, with an arbitrary function F (u, v): For the particular case of {2, 2, p}, p ≥ 2, the possible solutions (besides of the ones equivalent to unity matrices) are the following ones, with independent constants {θǫ, t, q, θ i } and a , and with the difference property of the spectral parameters in GYBE (u, v) ⇒ (u − v): (2.9) When t = ±1, ǫ = θ, q = e iφ , F (u) = e iγu and ū = tan[θu], the second matrix is the well known unitary trigonometric solution to YBE with (ū, v) ⇒ ( ū−v 1−ūv ) (the second kind solution to YBE [28], with relativistic rule of the summation of the spectral parameters -"velocities"),

GYBE and the integrable models
Sometimes there is a vision, that GYBEs in general have no connection with the integrable models.When p = 2k, the Ř-matrices are inhomogeneous matrices with another structure than there are used in the case of the usual Algebraic Bethe Ansatz (ABA) [2,9].Let us define the transfer matrix on the cyclic chain with length N and with the vector states V i k on the k-th sites, by means of the Ř-matrices in GYBE, in the following way The matrices Ř are defined, as in the Fig. 2 (ii).Note, that here, in this definition, we do not have auxiliary space in it's usual sense, there are only quantum spaces which figurate in the formulas.In the usual definition of the transfer matrix, there is a trace over an auxiliary space [14], and the auxiliary space may differ from the quantum spaces by dimension, meanwhile in (2.11) the states {V ir , V kr , V jr }, for the given r, (r ∈ [1, ..., N]), have the same nature.
In the Fig. 3 the transfer matrices are depicted by the rows.For simplicity, the states V kr , by which the summation is taken in the transfer matrices, are denoted by Vr , meanwhile the states V ir,jr -by V r (though the vector spaces V r and Vr are similar).The Ř-matrices are presented by the shaded rectangles (rotated by 45 degrees in comparison to Fig. 1,2).
For simplicity below we shall write the matrices as the functions of one spectral parameter, special cases will be discussed separately.The transfer matrices commutativity, [τ (u), τ (v)] = 0, ensures the integrability in ABA.Although now the operators τ (u) and τ (v) (2.11) in the product τ (u)τ (v) are defined differing by a space shift on the product of the quantum states, the quantum conserved charges, produced by the transfer matrices (defined by means of the logarithmic differentials of the τ (u)-operators - ∂u n | u=u 0 ), are the same due to the cyclic property.The commutativity of the transfer matrices can be satisfied by solving of the generalized equations defined on the spaces (2.12) For the homogeneous case these coincide with the ordinary YBE.
The transfer matrix we can preset in the graphical form, which we have used in the work [47], see the rows in the Fig. 3.The product of the transfer matrices on the periodic N × N square lattice (having toric periodic conditions):Z N ×N = tr i N τ constitutes the partition function in 2D statistical physics with the local weights corresponding to the matrices Ř.A graphical representation of the partition function is depicted on the Fig. 3, too.We apply the checkerboard disposition of the matrices in the square lattice, as in [47].
We shall discuss below the solutions of the inhomogeneous GYBEs, concentrating our attention to a concrete specific structure of the R-matrices.
3 The solutions to high-dimensional GYBE emerged from the usual low-dimensional solutions to YBE Let us consider Ř-matrices, which have non vanishing elements on the diagonal and antidiagonal positions (X-shaped matrices) [30,31,32], explored in the context of the quantum information theory as the extensions of 4 × 4 braiding matrix (the braiding limits of the trigonometric matrix in Eqs.(2.9)), ensuring Bell basis.
For the general (N 1 ) 2 × (N 2 ) 2 -dimensional case, there are V {12} non-zero elements in such i.e. these Ř-matrices can describe V {12} -vertex models in 2d statistical physics.Considering the spectral parameter dependent GYB equations with such kind matrices will restrict strongly the area of the solutions.
As it is mentioned, an example of such matrices are the braid group symmetric solutions (see e.g.[11]) The matrix M itself is a solution to the rigid (without spectral parameter) GYBE, and M ± -matrices are the braid limits lim ±∞ of the mentioned trigonometric solution (3.14), The more complete trigonometric parametrization of M ±matrices we shall discuss in this paper, as extension of (2.9), with an detailed example of 8 × 8 matrix in Appendix A.1.
We construct here the most general GYBE solutions with the matrices having only diagonal and anti-diagonal non-zero elements, and it occurs, that these are natural extensions of the eight-vertex solutions to ordinary YBE for the high-dimensional matrices.And the mentioned solutions can be constructed by considering the following ansatz.
Let us consider the matrix ŘN 1 ,N 2 which acts on the tensor product of the vector spaces The inhomogeneous equations we consider in the following form: , [ ] denotes the integer part of the numbers.The X-shaped matrices have the following substructures: at N 1 = 2N 1 and N 1 = 2N 1 + 1 cases correspondingly: where the slow arrows denote diagonal and anti-diagonal (N 1 ) × (N 2 ) matrices with the following matrix elements (disposed from the left to right) Here the diagonal and anti-diagonal matrix elements are indexed within the following rules.
The values of the indexes {i, j} of the diagonal Řij ij elements are decreasing from upper left corner to the lower right position.The first index i runs from N 1 to −N 1 , and for the fixed i, the index j goes from N 2 to −N 2 .The indexes {i, j} of the anti-diagonal Řij −i−j elements are decreasing from the lower left corner to the upper right position.The first index i runs from N 1 to −N 1 , and for fixed i the index j goes from N 2 to −N 2 .
At N 1 = 2N 1 + 1, the additional central sub-matrix has the structure ( * ) 2N 2 or ( * ) 2N 2 +1 , for the cases N 2 = 2N 2 and N 2 = 2N 2 + 1, respectively: Now let us reformulate the matrices into the superpositions, rewriting the matrix elements in the following way with the matrices defined on the minimal spins 1 2 and 0. When n i , n j = 0, then the matrix constitutes an eight-vertex model defined on the spin-1 2 spaces.As the general equations (3.16) contain as separate sub-sets the YBEs for each particular matrices řn i ,n j , hence the corresponding solutions for the general inhomogeneous eight-vertex models constitute the solutions here.And since all the sectorial equations (the parts of the GYBE), which contain different řn i ,n j -matrices, connect the sub-matrices with different indexes {i, j} one with other, hence all the solutions řn i ,n j (u) with any i, j have the same nature (elliptic, trigonometric or rational).And only some variations are possible concerning to the constants (model's parameters), as we shall face in the considered examples.The exceptional cases, when there can be disconnected sub-sets of YBEs, are discussed in the Appendix A2, within the Statement III.The possible solutions one can classify as the general XYZ or free-fermionic inhomogeneous matrices which can be formulate by means of the sl(2) operators on the fundamental spin-1 2 irreps [2,3,4,48], or on the irreps of the sl q (2) quantum group at general or cyclic cases [14,4,49].
The examples, corresponding to the XXZ-model or XX-model in a transverse field, at Also we can interpret the corresponding spin-chain models with the next-neighboring interactions between the sets of 1  2 -spins of numbers N i and N i+1 , as the models with the interactions between the complex spins at the sites i, i + 1 - 2 .By means of the complex spin operators - k=1 (σ a i+1 ) k (here σ a i are the Pauli matrices defined at the i-th sites), for the solutions corresponding, as example, to the XY Zmodel's řij matrices, the appropriate 1D spin-model Hamiltonian, H ≈ d log[τ (u)] du | u=0 , can be expressed as follows: One can extend the parametrization for the all obtained solutions in this natural way, with arbitrary numbers γ i,j , Of course, there is an ambiguity, connected with the possibility of multiplying the whole Řmatrix by an overall function.In the corresponding quantum chain Hamiltonian operators these factors bring us additional diagonal summand proportional to For the even dimensional cases we can formally rewrite it in the following form: Elucidations.One can present other even-dimensional large matrices too, emerged or induced from the well analyzed eight-vertex matrices, satisfying to inhomogeneous GYBEs.
For the matrices Ř2N 1 ×2N 1 , acting on the product state - Then the following inhomogeneous GYBEs take place, inherited just from the usual YBEs: It is apparent, that another simple definitions also are possible for the matrix Ř2N 1 ×2N 1 , induced by one matrix Ř2×2 and the diagonal matrices, which will act, e.g. on the vectors spaces The unity matrix can be replaced by diagonal operator with diagonal matrix elements -{e iγ 1 u , e iγ 2 u , ..., e iγ N u }.
These operators differ by dispositions of the matrix elements from those X-shaped solutions, for which all the sub-matrices řij , discussed in this section, coincide one with another, up to overall exponential functions.However by the resulting chain models, they are similar operators (and have the same eigenvalues), so can be connected by unitary operations.One can check, that the corresponding unitary matrices are not separable/factorisable, and thus promote to more entangling of the states, generated by evolution operators produced by X-shaped Ř-matrices [27].
So, we come to the following Statement I.The X-shaped even-dimensional matrices, being the solutions of GYBE, are simply the superpositions of the general eight-vertex solutions.The known trigonometric solutions are the particular cases.
And, as an extension regarding to another low-dimensional solutions (e.g. 9 × 9 chiral Potts models [16]- [20], general 15-vertex models [50]) we can make such Proposition.The matrices of dimensions (pN ) 2 × (pK) 2 , constructed by means of p 2 × p 2 solutions řpp to YBE, (and in matrix form reproducing via multiplicative form the shape of these matrices), via the following representations: can be considered as solutions to GYBE.Here instead of the indexes {±} used in the case of p = 2, we can use the numbers {1, .., p}.Then, the matrix indexes of the big matrix [ ŘpN ,pK ] i ′ j ′ ij are disposed in the following way:

Odd dimensional representations.
The case, when one of the spaces on which the ŘN 1 N 2 -matrix acts, has odd dimension, the situation is changed.Here there are specific equations containing the indexes 0. In comparison to even dimensional cases, there is an additional sector in the set of GYBEs, which includes the matrices řn i ,0 , ř0,n j and ř0,0 and can be formulated as follows: The matrices ř0,n i (u) and řn i ,0 (u) have the following 2 × 2-matrix form The solution of the equation (3.30), taking into account the requirement at the point u = 0 (coming from the locality of the models) can be easily represented as a function with one arbitrary constant α ř0,0 (u) = e αu . (3.33) Since the equations (3.28, 3.29) present the following scattering relations on the matrices [r n i 0 ] and [r 0n j ] (below we omit the indexes { n i 0 , 0n i }) Then the solutions are simply trigonometric functions The parameters p and θ here are arbitrary numbers (and particularly can be imaginary numbers).
Now it is natural to look for the solutions [ř ik ] ±± ±± to the remained equations (3.21) as trigonometric ones.And indeed, the solutions for [ř ij ] coincide with the inhomogeneous two-parametric solutions as in (2.9).Below we prefer to deal with the hyperbolic parameterizations, which has been also used for ordinary YBE brought in [48]: Here the parameters {γ, β, q} are independent ones, meanwhile the constants {t, t 1 , t 2 } are the signs, and the permissible configurations for being the solutions to YBE are the following ones: So, the results of this sub-section can be formulate in the following Statement II The X-shaped odd-dimensional spectral parameter dependent solutions to GYBE have trigonometric nature.
All these solutions correspond to the 1D quantum Ising model's like spin systems, i.e.
4 Multi-spectral parametrization for the quantum gate matrices In the quantum information theory the spectral parameter dependent GYBE-solutions realize time evolution U(t) of the quantum states, and it is important to obtain the unitary operator solutions [27].Note, that the two-parametric trigonometric unitary matrix in (2.9) at t = 1, q = e iα just follows from the unitary form of the YBE solution (3.36): after the following transformations of the parameters, ) .
The action of that matrix on the direct product states |± ⊗ |± ≡ | ± ± , brings to the entangled states In contrast to the known homogeneous case, when ǫ = θ (or γ = β), here there are two kind of entangled paired states, with the independent "concurrences" -degrees of the entanglement The higher generalizations for these matrices ŘN 1 N 2 , in case, if N i = 2 d i , can be considered as complex gates on the tensor products of two-qubit states n |± , n = d 1 + d 2 , so called n-qubit states, for obtaining entangled states (such as maximally entangled -Greenberger-Horne-Zeilinger (GHZ) states or linear cluster states [29,33,46,38]).
If the phase α = 0 and the parameter θ is time-dependent, then the evaluation of two sub-sets of the vector states {| ± ± | and | ± ∓ } can be described by two independent oscillations with different frequencies, constituting so Lissajous figures in the corresponding parametric spaces.Meanwhile, there are different visions, how to choose the time-parameter.
The Hamiltonian operator follows just from the Shrödinger equation, , with time-parameter u, give time-dependent Hamiltonian (with time-dependent eigenvalues and time-independent eigenvectors) (e.g.[38]).The trigonometric parametrization [38,28] gives time-independent Hamiltonian operator equivalent to operator M, with additional deformation parameter q [38,27,6].However, one can take into account that here (4.40) there are two independent spectral parameters, and it is possible to define their time dependence by two arbitrary different functions, say as Another Hamiltonian operator is defined by the authors of the work [29], taking the deformation parameter as time-dependent, and the usual spectral parameter as time-independent.
In the case of two-parametric solutions, considered in this paper, if to take time is connected with the parameter q, then H is not changed in comparison with the discussions in [29], and the Berry's phases and Berry's sphere for the eigen-states of the Hamiltonian operators are described in the same way.

The graded permutation matrices and multi-parametric generalizations
Let us define the following graded permutation matrices: P g and P τ g , Here we use together with the ordinary permutation matrix P (of disposition) of the states, also the colored permutation P τ , realizing the permutation of the states' "colors" (types), which for the case N = 2 means: The sign-gradation of the matrix elements could be explained in a consistent way, introducing the graded vector spaces (super-spaces) and graded tensor products (see, as example, the work [44] and the references therein).Then the basic definitions of the ABA need re-consideration, such as YB equations, R-matrices, algebraic relations with monodromy matrices, Lax opeators, transfer matrices.However, YBEs in the "check" formulations, and hence the GYBEs, remain unchanged.But in this paper we use the "grading" term only regarding to the "check" matrices, without notion of the (anti-)commutation features of the vector states.Formally, we exploit below the notion "parity" used for the super-spaces, but the fractional numbers in the last sub-section hints that the spaces here may have richer structure (can include para-fermions).The deep understanding of the underlying symmetries we shall leave for further investigation [55], giving here only the recept of the construction of the large matrices.
Note, that at N = 2 the two-parametric 4 × 4-matrix (3.36) consists of the sum of the above graded permutation operators.which means that we have graded and colored representations of the braiding algebra.
Using the described relations let us present the following multi-spectral parametric solution to the GYBE, for the homogeneous spaces -also to the usual YBE: Ř(u; u ′ ) = e u P g + e −u P † g + e u ′ P τ g + e −u ′ P τ † g , (5.47) The operation † here denotes the hermitian conjugation.The spectral parameters {u, u ′ } can be presented by means of two constant parameters and one spectral parameter: {u, u ′ } → {αu, α ′ u}.The generalization of these matrices for the X-shaped matrices ŘNK , when the large matrices are the combinations of 4×4-matrices, is described in Section 3. As for the case of 4 × 4 matrix (2.9), the GYB equations (d, k, p ≥ k), with d k = NK, give the general Xshaped ŘNK trigonometric solution with independent parameters {u, u ′ , ...} of number N K.
However the braiding solutions to GYBEs (d, k, p < k) bring to restrictions on the spectral parameters.And the resulting (N) 2 × (K) 2 matrices depend on two spectral parameters (see the examples in the Appendix A.1), and also on additional overall parameters (α ij u): Another generalization, developed for the homogeneous matrices of N × N dimensions, we present in the following subsection.

The second generalization: colored and graded permutation matrices
The basic trigonometric matrix Ř (3.36) consists of two YBE solutions Ř0 and Řτ , with different spectral parameters: The graded permutation matrix P g in as one of the basic components of such decomposition can be extended to N × N matrices , with an undefined parity p 12 .
The next summand Řτ consists of the colored graded permutation operators.If the vector space V has the states of the number N, i.e. has N colors, then it is possible to construct the colored permutation operators P τ ij with the corresponding τ -permutations of the states: {i 1 , ...i N } → {i τ 1 , ..., i τ N }, P τ = P • I τ : This means, that the permutation operator permutates not only the disposition of the states in the tensor product, but also the colors of the states.For the simple N = 2 case the there is one such possibility (5.44): One can verify, that together with the ordinary permutation operators, the general colored i 2 also satisfy to the braid group relations.
(5.54)This is the consequence of the fact, that the colored permutations just are connecting with the representations of the symmetric group S N (including the subgroup of permutations) and the partition algebra [23,26].Now let us denote the different colored permutations for N-dimensional vector spaces as τ i , P τ i P τ j = I τ i+j .Note, that the permutation operation of the colors can be either a full permutation, which means that V τ i = V i for each state, or a partial permutation for the remaining cases.Here we discuss only the simple case, when the colored permutations are identically invertible, i.e. (P τ i ) 2 = I, ensuring unitary Řτ i -operators.The grading operation can be formally introduced in the standard way, by means of the parities: Here we do not discuss the question of the gradation of the representation states, as well the question of underlying symmetries (super-symmetries), as it needs separate deep consideration.We can see below, that for the parities there are permissible not only integer numbers, but also fractional values, i.e. in general case, the factors (−1) p 12 just are phases (we are interested mainly with the unitary operators).And for general N also, the matrices constructed by the Yang-Baxerizations, as in the formulas (5.51) and (5.53), are the solutions to YBE (or GYBE).Now we can generalize the superposition form of the solution (5.50).The possible trigonometric solutions must be: ... (5.57) The constraints followed from the GYBE give the relations only on the grading rules and also on the parameters α a , α b , ..., which in some cases coincide one with another.There can be also situation, that for the given combination of the colored matrices {a, ..., a ′ } the corresponding matrix Řa...a ′ will not have appropriate set of the parameters and parities satisfying to YB or GYBE equations.We can extend our research for the cases, when the colored permutations satisfy to P τ 1 ...P τ k = I.Then in order to check that the unitary operators, constructed by linear superpositions of these operators, satisfy the braid relations (or GYBE), it must be analyzed the appropriate sequence of the colored multiple braiding relations.
At N = 2 the solution can contain two arbitrary spectral parameters, i.e two independent parameters α a,b .For the general case of N the situation is much more complicated, there are different possibilities of the gradings, but less rich possibilities for the multi-parametric parameterizations.
The examples of the solutions with the full permutation operators for 4 2 × 4 2 -matrix one can find in the next Subsections.We shall see, that the permissible parities are different for different kind of GYBEs.
In this particular case the whole set of the colored g-permutations consists of the following four full permutation operators (P e ) 2 = 1, acting on the four-dimensional space with the vectors states |v i , i = 1, 2, 3, 4, and equipped with corresponding parities p e : [P b g ] For the homogeneous case the parity matrices (−1) pe , e = 0, a, b, c, are just the signs, and here they are chosen in a rather general form, only keeping the requirement that the diagonal matrix elements must be 1, (5.62) The matrices P e g satisfy to constant YBE (or GYBE).The simplest spectral dependent matrices with the constructed permutation matrices are the following ones where the sign parameters are not fixed (they are arbitrary ones).The next matrices are Ře ′ e ′′ (u, v) = R e ′ (u) + R e ′′ (v).For such double cases {e ′ , e ′′ } the GYBE put definite constraints on the parameters [p e ] j 1 j 2 i 1 i 2 , (in most cases the second indexes {i 2 , j 2 } in [p e ] j 1 j 2 i 1 i 2 for the given permutation matrices can be omitted).
Let us construct linear superposition of the operators Ře (u), applicable in the integrable models: Ř4×4 (α 0 , α a , α b , α c ; u) = Ř0 (α 0 u) + Řa (α a u) + Řb (α b u) + Řc (α c u), Ř4×4 (0) = I. (5.64)The checking of the YBEs, which turn into the constraints on the parities p e for such linear combination of the permutation matrices, shows that there are many possibilities for the solutions of p e -s.The choice of the signs or phases are different in different configurations of the GYBEs.As example, the elements of the p e -matrices are different for the solutions to (2,4,2) and (2,4,1).One set of the solutions, which we consider in this section, has rather symmetric constraints on p e , which look like as: These matrices, taking into account the normalizing factor , are unitary for independent values of the parameters α e , but however the GYBEs (YBEs) put the restrictions on their values.Then one can directly check that the following operators are solutions to YBE.
Exceptional solutions e ρ e Ře (u), e ρ e = 0.The next possible 16 × 16 solutions can be considered by partial sums of the matrices Ře (u), e = {0, a, b, c}.By direct derivation it is easy to verify, that each matrix Ře (u) is a solution to YBE.Moreover, matrices with the non-vanishing elements, disposed so, as in the matrices Ře (u), are the solutions to YBE with any constants or functions.That is, the generalized matrices with any functions [r e (u)] Let us present some exact solutions and the corresponding four-dimensional spin-spin Hamlitonian operators.

α e = α
For the homogeneous solutions, all parameters α e equal one to another: α e = α.We fix α = 1.We can represent this large matrix in the following form, by means of 4 ×4-operators: by Z 4 -operators Mx,y,z , ( M4 = 1), and by Z 2 -operators Mx,y,z , ( M2 = 1) (below I n is the n × n unity operator, and the signs ε 2 x,y,z = 1 reproduce the solutions for the parity matrices permissible by the relations (5.65): (5.71) The matrix (5.66) corresponds to the case with the signs ε x,y,z = 1.We see, that the all matrices Mi have only two eigenvalues, so the matrices are constructed rather by the generators of Z 2 ⊗ Z 2 (or sl 2 × sl 2 ) and not Z 4 .The matrices M , M can be presented by means of the tensor product of the Pauli matrices σ x = ( 0 We see, that M -operators are commutative, and the operators i M reproduce the algebra of sl 2 .Then the corresponding quantum spin-chain Hamiltonian operator for the matrix Ř44 (u) reads as Let us represent the expression by means of the spin operators for one of the next unitary matrices (5.67, 5.66), with non vanishing parameters α 0 For these cases the parities may have fractional values, i.e.
[p e ] j 1 j 2 i The second versions of the solutions, which are constructed by means of the graded and colored permutation operators, also admit trigonometric Yang-Baxterization, and here the GYBEs become the equations on the parity-matrices.The possibility of the choosing of the parities, we see, that is not unique.The second kind extensions can also be formulated in terms of the spin interactions.Apparently, such Hamiltonian operators do not possess ordinary sl(2)-symmetry, but however can have symmetries of the algebra's deformations.Such questions we are addressing to the next works.In the same time the topological properties of the underlying integrable models could be investigated, especially the obtained fractional parities may be related to the anyon (parafermion) statistic, considering in topological quantum computation [24,25,53].Also the connection with the higher dimensional integrable models needs to be considered [52,54].
And another novelty of our research is the multi-spectral parameter Yang-Baxterization of unitary X-shaped series of the rigid GYBE solutions M ± (extensions of the known quantum Bell type matrices [38]), examples are presented in Appendix A.1.For the usual two-qubit gates the evolution of the states by such unitary transformations is briefly discussed in Section 4, which brings to two kind of entanglement's degrees, relying on two independent spectral-parameters.Note, that the spectral-parameter dependent Bell matrices just are the multi-parametric solutions of separable GYBEs (d, k, p), p ≥ k (2.9), the equations with k > p put restrictions on the possible sets of the independent parameters, see Appendix A.1.
Acknowledgements.The work was supported by the Science Committee of RA, in the frames of research projects 20TTAT-QTa009, 20TTWS-1C035 and 21AG-1C024.

. 41 )
This choice shows apparently the possibility of independent time-evolutions for two couples of states [| + + , | − − ] and [| + − , | − + ].For the case (4.41) at t = 2n + 1 (n is an integer number) the first states are entirely decoupled, meanwhile the second couple is maximally entangled (for t = 2n there is an opposite situation).