Parameterizing Qudit States

Quantum systems with a finite number of states at all times have been a primary element of many physical models in nuclear and elementary particle physics, as well as in condensed matter physics. Today, however, due to a practical demand in the area of developing quantum technologies, a whole set of novel tasks for improving our understanding of the structure of finite-dimensional quantum systems has appeared. In the present article we will concentrate on one aspect of such studies related to the problem of explicit parameterization of state space of an $N$-level quantum system. More precisely, we will discuss the problem of a practical description of the unitary $SU(N)$-invariant counterpart of the $N$-level state space $\mathfrak{P}_N$, i.e., the unitary orbit space $\mathfrak{P}_N/SU(N)$. It will be demonstrated that the combination of well-known methods of the polynomial invariant theory and convex geometry provides useful parameterization for the elements of $\mathfrak{P}_N/SU(N)$. To illustrate the general situation, a detailed description of $\mathfrak{P}_N/SU(N)$ for low-level systems: qubit $(N=2)\,,$ qutrit $(N=3)\,,$ quatrit $(N=4)\,$ - will be given.


Introduction
Quantum mechanics is a unitary invariant probabilistic theory of finite-dimensional systems. Both basic features, the invariance and the randomness, strongly impose on the mathematical structure associated with the state space P of a quantum system. In particular, the geometrical concept of the convexity of the state space originates from the physical assumption of an ignorance about the quantum states. Furthermore, the convex structure of the state space, according to the Wigner [1] and Kadison [2] theorems about quantum symmetry realization, leads to unitary or anti-unitary invariance of the probability measures (short exposition of the interplay between these two theorems see e.g. in [3]). In turn of the action of unitary/anti-unitary transformations → = U U † sets the equivalence relation between the states , ∈ P and defines the factor space P/U . This space is a fundamental object containing all physically relevant information about a quantum system. An efficacious way to describe O[P N ] := P N /SU (N ) for an N −level quantum system is a primary motivation of the present article. The properties of O[P N ] , as a semi-algebraic variety, are reflected in the structure of the center of the enveloping algebra U(su(N )) . Hence, it is pertinent to describe O[P N ] using the algebra of real SU (N )−invariant polynomials defined over the state space P N . Following this observation in a series of our previous publications, [4][5][6][7][8], we develop description of O[P N ] using the classical invariant theory [9]. On the other hand, P N /SU (N ) is related to the co-adjoint orbits space su * (N )/SU (N ) and hence it is natural to describe P N /SU (N ) directly in terms of non-polynomial variables -the spectrum of density matrices. Below we will describe a scheme which combines these points of view and provides description of the orbit space P N /SU (N ) in terms of one second order polynomial invariant, the Bloch radius of a state and additional non-polynomial invariants, the angles corresponding to the projections of a unit (N − 2)−dimensional vector on the weight vectors of the fundamental representation of SU (N ) .
The article is organised as follows. The next section is devoted to brief statements of general results about the state space P N of N −dimensional quantum systems, including discussion of its convexity (Section 2.1) and semi-algebraic structure (Section 2.2). Particularly, the set of polynomial inequalities in an (N 2 −1)−dimensional Bloch vector and the equivalent set of inequalities in N −1 polynomial SU (N )−invariants will be presented for arbitrary N −level quantum systems. Section 3 contains information on the orbit space O[P N ] -the factor space of the state space under equivalence relation against the unitary group adjoint action. In Section 3.3.1 we introduce a new type of parameterization of a qubit, a qutrit and a quatrit based on the representation of the orbit space of a qudit as a spherical polyhedron on S N −2 . This parameterization allows us to give a simple formulation of the conception of the hierarchy of subsystems inside one another. In Section 3.3.2 we present formal elements of the suggested scheme for an arbitrary final-dimensional system. Section 4 contains a few remarks on possible applications of the introduced version of the qudit parameterization.

The state space
The state space of a quantum system P N comes in many faces. One can discuss its mathematical structure from several points of view: as a topological set, as a measurable space, as a convex body, as a Riemannian manifold. 1 Below we concentrate mainly on a brief description of P N as a convex body realized as a semi-algebraic variety in R N 2 −1 following in general the publications [4][5][6][7][8].

The state space as a convex body
According to the Hilbert space formulation of the quantum theory, a possible state of a quantum system is associated to a self-adjoint, positive semi-definite "density operator" acting on a Hilbert space. Considering a non-relativistic N -dimensional system whose Hilbert space H is C N , the density operator can be identified with the Hermitian, unit trace, positive semi-definite N × N density matrix [14,15].
The set of all possible density matrices forms the state space P N of an N -dimensional quantum system. It is a subset of the space of complex N × N matrices: A generic non-minimal rank matrix describes the mixed state, while the singular matrices with rank( ) = 1 are associated to pure states. Since the set of N −th order Hermitian matrices has a real dimension N 2 , and due to the finite trace condition Tr( ) = 1 , the dimension of the state space is dim(P N ) = N 2 − 1 . The semi-positivity condition ≥ 0 restricts it further to a certain (N 2 − 1)−dimensional convex body. The convexity of P N is the fundamental property of the state space. The next propositions summarise results on a general pattern of the state space P N as a convex set with an interior Int(P N ) and a boundary ∂ P N [10].
• Proposition II • The boundary ∂P N consists of non-invertible matrices of all possible non-maximal ranks: 1 Here is a short and extremely subjective list of publications on these issues [10][11][12][13].
The subset of pure states F N ⊂ ∂ P N , contains N extreme boundary points P i ( ) which generate the whole P N by taking the convex combination: In (5) every extreme component P i ( ) can be related to the standard rank-one projector by a common unitary transformation U ∈ SU (N ) and transposition P i(1) interchanging the first and i-th position: For any dimension of the quantum system the subset of extreme states provides important information about the properties of all possible states, even the pure states comprise a manifold of a real dimension dim(F N ) = 2N − 2 , smaller than that dimension of the whole state space boundary dim(∂P N ) = N 2 − 2 .

The state space as a semi-algebraic variety
According to the decomposition (5), the neighbourhood of a generic point of P N (R N 2 −1 ) is locally homeomorphic to U (N )/U (1) N × D N −1 , where the component D N −1 is an (N − 1)-dimensional disc (cf. [10,13]). Following this result, below we will describe how the state space P N can be realised as a convex body in R N 2 −1 defined via a finite set of polynomial inequalities involving the Bloch vector of a state. In order to formalize the description of the state space, we consider the universal enveloping algebra U(su(N)) of the Lie algebra su(N) . Choosing the orthonormal basis λ 1 , λ 2 , . . . , λ N 2 −1 for su(N) , the density matrix will be identified with the element from U(su(N)) of the form: The analysis (see e.g. consideration in [4,6]) shows the possibility of description of the state space via polynomial constraints on the Bloch vector of an N −level quantum system.
• Proposition III • If a real (N 2 − 1)-dimensional vector ξ = (ξ 1 , ξ 2 , . . . , ξ N 2 −1 ) in (8) satisfies the following set of polynomial inequalities: where S k (ξ) are coefficients of the characteristic equation of the density matrix : then the equation (8) defines the states ∈ P N . The inequalities (9), which guarantee the semi-positivity of the density matrix, remain unaffected by unitary changes of the basis of the Lie algebra and thus the semi-algebraic set (9) can be equivalently rewritten in terms of the elements of the SU (N )-invariant polynomial ring R[P N ] SU(N) . This ring can be equivalently represented by the integrity basis in the form of homogeneous polynomials P = (t 1 , t 2 , . . . , t N ) , The useful, from a computational point of view, polynomial basis P is given by the trace invariants of the density matrix: The coefficients S k , being SU (N )-invariant polynomial functions of the density matrix elements, are expressible in terms of the trace invariants via the well-known determinant formulae: Aiming at a more economic description of P N , we pass from N 2 − 1 Bloch variables to N − 1 independent trace variables t k . The price to pay for such a simplification is the necessity to take into account additional constraints on t k which reflect the Hermicity of the density matrix. Below we give the explicit form of these constraints in terms of P = (t 1 , t 2 , . . . , t N ) .
In accordance with the classical results, the Bézoutian, the matrix B = ∆ T ∆ , constructed from the Vandermonde matrix ∆ , accommodates information on the number of distinct roots (via its rank), numbers of real roots (via its signature), as well as the Hermicity condition. A real characteristic polynomial has all its roots real and distinct if and only if the Bézoutian is positive definite. For generic invertible density matricesmatrices with all eigenvalues different, the positivity of the Bézoutian reduces to the requirement Noting that the entries of the Bézoutian are simply the trace invariants: one can be convinced that the determinant of the Bézoutian is nothing else than the discriminant of the characteristic equation of the density matrix, Disc = i>j (r i − r j ) 2 , rewritten in terms of the trace polynomials 2 Hence, we arrive at the following result.
• Proposition IV • The following set of inequalities in terms of the trace SU (N )-invariants, define the same semi-algebraic variety as the inequalities (9) in N 2 − 1 Bloch coordinates do.
3 Orbit space P N /SU (N )

Parameterizing P N /SU (N ) via polynomial invariants
Proposition IV is a useful starting point for establishing a stratification of the P N under the adjoint action of the SU (N ) group. It turns out that, due to the unitary invariant character of the inequalities (17), they accommodate all nontrivial information on possible strata of unitary orbits on the state space P N . Indeed, it is easy to find the link between the description of P N given in the previous section and the well-known method developed by Abud-Sartori-Procesi-Schwarz (ASPS) for construction of the orbit space of compact Lie group [16][17][18]. The basic ingredients of this approach can be very shortly formulated as follows. Consider a compact Lie group G acting linearly on a real d-dimensional vector space V . Let R[V ] G be the corresponding ring of the G-invariant polynomials on V . Assume P = (t 1 , t 2 , . . . , t q ) is a set of homogeneous polynomials that form the integrity basis, Elements of the integrity basis define the polynomial mapping: Since the map t is constant on the orbits of G , it induces a homeomorphism of the orbit space V /G and the image X of t-mapping; V /G X [19].
In order to describe X in terms of P uniquely, it is necessary to take into account the syzygy ideal of P, i.e., Let Z ⊆ R q denote the locus of common zeros of all elements of I P , then Z is an algebraic subset of R q such that X ⊆ Z . Denoting by R[Z] the restriction of R[y 1 , y 2 , . . . , y q ] to Z , one can easily verify that R[Z] is isomorphic to the quotient R[y 1 , y 2 , . . . , y q ]/I P and thus Therefore, the subset Z essentially is determined by R[V ] G , but to describe X the further steps are required. According to [17,18], the necessary information on X is encoded in the structure of the q × q matrix with elements given by the inner products of gradients, grad(t i ) : Hence, applying the ASPS method to the construction of the orbit space P N /SU (N ) , one can prove the following proposition.
• Proposition V • The orbit space P N /SU (N ) can be identified with the semi-algebraic variety, defined as points satisfying two conditions: a) The integrity basis for SU (N )−invariant ring contains only N independent polynomials, i.e., the syzygy ideal is trivial and the integrity basis elements of R[P N ] SU(N ) are subject to only semi-positivity inequalities b) ASPS inequality Grad(z) ≥ 0 is equivalent to the semi-positivity of the Bézoutian, provided by existence of the d−tuple where χ = (1, 2, . . . , d) : The decomposition of the density matrix (5) over the extreme states explicitly displays the equivalence relation between states, Matrices with the same spectrum are unitary equivalent. Furthermore, since the eigenvalues of the density matrix r = (r 1 , r 2 , . . . , r N ) in (5) can be always disposed in a decreasing order, the orbit space P N /SU (N ) can be identified with the following ordered (N − 1)−simplex: We are now ready to combine the above stated methods of the description of the state space P N , the polynomial invariant theory and convex geometry to write down a certain parameterization of density matrices. Based on the extreme decomposition of states (5), the parameterization of the elements of P N is reduced to fixing the coordinates on the flag manifolds of SU (N ) and the simplex ∆ N of eigenvalues of density matrices. In the remaining part of the article, we will describe P N /SU (N ) in terms of the second order polynomial invariant, which is determined uniquely by the Euclidean length r of the Bloch vector, and N − 2 angles on the sphere S N −2 , whose radius in its turn is given as N −1 N r .

Qubit, qutrit and quatrit
In order to demonstrate the main idea of the parameterization, we start with its exemplification by considering three the lowest-level systems, qubit, qutrit and quatrit and afterwards the general case of an N −level system will be briefly outlined.
QUBIT • A two-level system, the qubit, is described by a three-dimensional Bloch vector ξ = {ξ 1 , ξ 2 , ξ 3 }: The qubit state with the spectrum r = {r 1 , r 2 } ∈ ∆ 1 is characterized by only one independent second order SU (2)−invariant polynomial t 2 = r 2 1 + r 2 2 . Introducing the length of the qubit Bloch vector, r = ξ 2 1 + ξ 2 2 + ξ 2 3 , we see that 3 Hence, the eigenvalues of the qubit density matrix (24) can be parameterized as It will be explained later that the coincidence of the constants µ 1 = 1/2 and µ 2 = −1/2 in (25) with the standard weights of the fundamental SU (2) representation, when the diagonal Pauli matrix σ 3 is used for the Cartan element of su(2) algebra, is not accidental. Below we will give a generalization of (25) for the qudit, an arbitrary N −level system. With this aim in mind, it is sapiential to start with considering the N = 3 and N = 4 cases.
QUTRIT • We assume that a generic qutrit state (N = 3) has the spectrum r = {r 1 , r 2 , r 3 } from the simplex ∆ 2 and thus is an eight-dimensional object. According to the normalization chosen in (8), it is characterized by the 8-dimensional Bloch vector ξ = (ξ 1 , ξ 2 , . . . , ξ 8 ) , A qutrit has two independent SU (3) trace invariant polynomials, the first one, t 2 = r 2 1 + r 2 2 + r 2 3 , is expressible via the Euclidean length of the Bloch vector, and the third order polynomial invariant, t 3 = r 3 1 + r 3 2 + r 3 3 , which rewritten in terms of eight components of the Bloch vectors reads: Now we want to rewrite (28) in terms of the Bloch vector of a length r and an additional SU (3) invariant. Having this in mind, it is convenient to pass to new coordinates linked to the structure of the Cartan subalgebra of su (3) . Choosing the latter as the span of the diagonal SU(3) Gell-Mann matrices and noting that the state (26) is SU (3)-equivalent to the diagonal state: In terms of new variables (r, ϕ) the expression (28) for the SU (3)−polynomial invariant t 3 simplifies, and the image of the ordered simplex ∆ 2 in the (I 3 , I 8 )−plane under the mapping (30) is given by the triangle ABC : depicted in Figure 1. The polar form of the invariants (30) prompts us to introduce a unit 2-vector n = ( cos( ϕ 3 ) , sin( ϕ 3 ) ) and represent the qutrit eigenvalues as with the aid of the weights of the fundamental SU (3) representation: Gathering all together, we convinced that the representation (32) is nothing else than the well-known trigonometric form of the roots of the 3-rd order characteristic equation of the qutrit density matrix: It is in order to present a 3-dimensional geometric picture associated to the parameterization (32). The three drawings in Figure 2 with different values of r show that (32) are the parametric form of the arc of the red circle which is the intersection ∆ 2 ∩ S 1 ( 2 3 r) . Consider an intersection of a qutrit simplex ∆ 2 with 2-sphere r 2 1 + r 2 2 + r 2 3 = 1 3 + 2 3 r 2 . The intersection depends on the value of a qutrit Bloch vector. For r = 0 the sphere and the simplex ∆ 2 intersect at point C = ( 1 3 , 1 3 , 1 3 ) , while for 0 < r < 1 the intersection is an arc C r of a circle on the plane r 1 + r 2 + r 3 = 1 of the radius 2 3 r centered at point C( 1 3 , 1 3 , 1 3 ) . The intersection for r = 1 takes place at B (1, 0, 0) . The ordering of eigenvalues 1 ≥ r 1 ≥ r 2 ≥ r 3 ≥ 0 determines the length of arc C r . For any r , the arc C r is described by (34), the depicted curve in the Figure corresponds to the fixed value r = 1/4. Furthermore, varying r within the interval r ∈ [0, 1] , provides the slices covering the whole simplex ∆ 2 = [0, π] × C r .

Qutrit Boundary
The introduced parameterization is very useful for analyzing the structure of qutrit boundary states. The qutrit space P 3 admits decomposition P 3 = P 3,3 ∪ P 3,2 ∪ P 3,1 into 8d-component of maximal rank-3, 7d-component of rank-2 and extreme pure states. Every component of (35) can be associated with the corresponding domains in the orbit space ∂O[P 3 ] . Particularly, the boundary ∂O[P 3 ] consists of two components and is described as follows: • Qubit inside Qutrit • For a chosen decreasing order of the qutrit eigenvalues, r 1 ≥ r 2 ≥ r 3 , the rank-2 states belong to the edge ∆ 3 , given by equation r 3 = 0 , which in the parameterization (34) reads: rank-2 states : Considering (36) as a polar equation for a plane curve, we find that the rank-2 states P 3,2 can be associated to the part of a 3-order plane curve. Indeed, rewriting (36) in Cartesian coordinates x = r cos ϕ , y = r sin ϕ , we identify this curve with the famous Maclaurin trisectrix with a special choice of a = 1 2 . For the boundary states (36), the equations (34) reduce to where These expressions for non-vanishing eigenvalues of a qutrit indicate the existence of a "qubit inside qutrit" whose effective radius is r * 2⊂3 . Since the radius of the Bloch vector of rank-2 states associated to a qubit in qutrit lies in the interval 1 2 ≤ r < 1 , the length of its Bloch vector, r * 2⊂3 , takes the same values as a single isolated qubit, 0 ≤ r * 2⊂3 < 1 . • Orbit space of pure states of qutrit • The boundary ∂O[P 3,1 ] corresponding to all pure states P 3,1 is attainable by SU (3) transformation from the point, r = 1 for ϕ = π .
QUATRIT • Now, following the qutrit case, consider a 4-level system, the quatrit, whose mixed state is described by the Bloch vector ξ = {ξ 1 , ξ 2 , . . . , ξ 15 }, The integrity basis for a quatrit ring of SU (4)−invariant polynomials R[ξ 1 , . . . , ξ 15 ] SU(4) consists of three polynomials R[t 2 , t 3 , t 4 ] . Using the compact notations (see details in Appendix A), they can be represented in terms of the Casimir invariants of su(4) algebra in the following form: From the expressions (40) one can see that apart from the length r of the Bloch vector, there are two independent parameters required to unambiguously characterize the quatrit eigenvalues. To find them, let us proceed as in the qutrit case. Consider the diagonal form corresponding to a quatrit state: The coefficients I 3 , I 8 and I 15 in (41) are invariants under the adjoint SU (4) transformations of . By equivalence relation (41), the quatrit state space is projected to the following convex body: The 2-dimensional slice I 15 = 1/3 of this body corresponds to rank-3 states, see Figure 3. In terms of new invariants, all states with a given length of Bloch vector r belong to a 2-sphere: I 2 3 +I 2 8 +I 2 15 = r 2 . Hence, the corresponding spherical angles ϕ and θ of these invariants, can be used as two additional parameters needed for the parameterization of a quatrit eigenvalues.
Let us now, in accordance with (43), introduce the unit 3-vector n = (sin θ cos(ϕ/3) , sin θ sin(ϕ/3) , cos θ) and parameterize 4-tuple of the eigenvalues of the density matrix r = (r 1 , r 2 , r 3 , r 4 ) via the following projections: where 3-vectors µ 1 , µ 2 , µ 3 and µ 4 denote the weights of the fundamental SU (4) . Explicitly the weights read: Note that the weights µ i are normalised in a way leading to a unit norm of the simple roots of algebra su(4) and obey relations: Using these expressions, we arrive at the following parameterization of a quatrit eigenvalues: To ensure the chosen ordering of the eigenvalues r i ∈ ∆ 3 , the Bloch radius should vary in the interval r ∈ [0, 1] and angles ϕ, θ be defined over the domains: A geometric interpretation of (47)-(48), in full analogy with the qutrit case, is described in Figure 4. • Qubit inside Qutrit inside Quatrit • In ∆ 3 the rank-2 boundary component O[P 4,2 ] is comprised from points on a line given by its intersection with two hypersurfaces: Following in complete analogy with the rank-3 states, we arrive at a "matryoshka" structure with "effective qubit inside qutrit which in turn is inside quatrit". The Bloch radius of this effective qubit is given by the Bloch radius of a quatrit: Note that for rank-2 states r ∈ [ 1 √ 3 , 1] and hence 0 < r * 2⊂3⊂4 < 1 .

Generalization to N -level system
Now after examining the main features of the introduced parameterization for a qutrit and quatrit, we are ready to give a straightforward generalization to the case of an arbitrary N −level system. With this aim, we will use the Cartan subalgebra of SU (N ) as span of the following diagonal N × N Gell-Mann matrices: The corresponding weights of the fundamental SU (N ) representation are . . .
It is easy to verify that the following relations are true: Taking into account these observations, one can write down the following parameterization for the roots r of the Hermitian N × N matrix: where n ∈ S N −2 (1) and parameter r provides the fulfilment of the correspondence with a value of the second order invariant, Writing the traceless part of the density matrix as the expansion over the Cartan subalgebra H of su(N ) , we see that N − 2 angles of the unit norm vector n (65) are related to the invariants I 2 3 , I 2 8 , . . . I 2 N 2 −1 , whose values are constrained by the Bloch radius r : Finally, it is worth to give geometric arguments which emphasise the introduced parameterization (65) of qudit eigenvalues. With this goal consider the intersection S N −1 (R) ∩ Σ N −1 of (N − 1)−sphere of radius R and hyperplane Σ N −1 : N i r i = 1 in R N . Let us describe the hyperplane in parametric form, with parameters s 1 , s 2 , . . . , s N −1 : where N −vector d fixes the point P ∈ Σ N −1 and the basis vectors (Darboux frame) obey conditions: d · e (α) = 0 , e (α) · e (β) = δ αβ , α, β = 1, 2, . . . , N − 1 .
Using this parameterization, the equation for (N − 1)−sphere is reduced to the constraint we arrive at the representation (65) with the radius of intersection sphere R N −2 = N −1 N r . Passing from hyperplane Σ N −1 to its subset, the simplex ∆ N −1 , we note that S N −1 (R) ∩ ∆ N −1 will be determined uniquely for every chosen order of eigenvalues and value of r . For an arbitrary N , a special analysis is required to write down explicitly S N −1 (R) ∩ ∆ N −1 . Here we only note that the intersection is given by one out of all possible tillings of S N −2 by the spherical polyhedra. For N = 3 such polyhedron degenerates to an arc of a circle, whereas for N = 4 the intersection will be given by two types of polyhedra, either a spherical triangle, or a spherical quadrilateral, depending on the value of the Bloch radius r .

Concluding remarks
Since the introduction of the concept of mixed quantum states, the problem of an efficient parameterization of density matrices in terms of independent variables became one of the important tasks of numerous studies. Starting with the famous Bloch vector parameterization [20], several alternative types of "coordinates" for points of quantum states have been suggested [21][22][23][24][25][26][27][28][29]. According to the generalization of Bloch vector parameterization, initially introduced for a 2-level system, the Bloch vector for an N −level system is a real (N 2 − 1)−dimensional vector. However, owing to the unitary symmetry of an isolated quantum system, those N 2 − 1 parameters can be divided into two special subsets. The first subset is given by N − 1 unitary invariant parameters, and the second one is compiled from the coordinates on a certain flag manifold constructed from the SU (N ) group. Introduction of the coordinates on both subsets has a long history. A description of the former set of SU (N )−invariant parameters is related to the classical problem of determination of roots of a polynomial equation, while the latter corresponds to a description of the homogeneous spaces of SU (N ) group 5 .
In the present article we have discussed the first part of the problem of parameterization of N × N density matrices and proposed a general form of parameterization of N −tuple of its eigenvalues in terms of a length r of the Bloch vector and N − 2 angles on sphere S N −2 ( N −1 N r). We expect that this parameterization will be useful from a computational point of view in many physical applications, including models of elementary particles. Particularly, in forthcoming publications it will be used for the evaluation of very recently introduced indicators of quantumness/classicality of quantum states which are based on the potential of the Wigner quasidistributions to attain negative values [34][35][36].

A Constructing Casimir invariants for su(N ) algebra
In this Appendix we collect few notions and formulae explaining the construction of the polynomial Casimir invariants on the Lie algebra g = su(N ) of the group G = SU (N ) .
Consider algebra g = N 2 −1 i ξ i λ i , spanned by the orthonormal basis {λ i } with the multiplication rule defined via the symmetric d ijk and anti-symmetric f ijk structure constants. Let {ω i } be the dual basis in g * , i.e., ω i (λ j ) = δ i j , and introduce the G−invariant symmetric tensor S of order r: The G−invariance of tensor S means that Using the tensor S , one can construct the elements of the enveloping algebra U(g) : which turns out to belong to the center of U(g) , i.e., [C r , λ i ] = 0 , for all generators λ i . Having in mind the solution to the invariance equations (74), one can build the polynomials in N 2 − 1 real variables ξ = (ξ 1 , ξ 2 , . . . ξ N 2 −1 ) : which are invariant under the adjoint SU (N )−transformations: p( # » Ad g (ξ)) = p( ξ ) .
It can be proved that the symmetric tensors k (r) defined in the given basis of algebra as k (r) satisfy the invariance equation (74) and form the basis for the polynomial ring of G−invariants. The tensors k (r) admit decomposition with the aid of the lowest symmetric invariants tensors, δ ij and d ijk . Particularly, the following combinations are valid candidates for the basis: As an example, for N −level system the G−invariant polynomials up to order six read: In the equation (80) the Casimir invariants are represented in a dense vectorial notation using the auxiliary (N 2 − 1)− dimensional vector defined via the symmetrical structure constants d ijk of the algebra su(N ) :

B Polynomial SU (N )−invariants on P N
In this section the explicit formulae for polynomial invariants for a quatrit will be given in terms of the suggested parameterization of density matrices.
Since the traceless part of the density matrices, − 1 N I N = (N −1) 2N g , belongs to the algebra su(N ) , all trace polynomials t k can be expanded over the su(N ) Casimir invariants. The corresponding decomposition of independent polynomials for the quatrit (N = 4) read: In order to derive the explicit form of polynomials C 2 and C 3 , the knowledge of components of the symmetric structure tensor d is needed. It is convenient at first to express the invariants for diagonal states, characterized by I 3 , I 8 and I 15 , and afterwards rewrite them for generic states using