Czech Technical University in Prague Faculty of Nuclear Sciences and Physical Engineering Břehová 7, CZ - 115 19 Prague, Czech Republic J. Tolar Symmetries of finite quantum kinematics and quantum computation Student Colloquium and School on Mathematical Physics 2015 Based on joint work with M. Korbelář
Outline 1. Introduction 2. Linear canonical transformations in classical mechanics 3. Linear canonical transformations in quantum mechanics 4. Shale s metaplectic representation 5. Simple finite-dimensional quantum kinematics 6. Classification of composite quantum kinematics 7. Symmetries of quantum kinematics Synopsis Appendix A. Symmetries of simple quantum kinematics Appendix B. Symmetries of composite quantum kinematics Appendix C. Complementarity structures
1. Introduction Non-relativistic quantum mechanics of particle systems can be divided into quantum kinematics and quantum dynamics. First the Hilbert space and the non-commuting operators of complementary observables, position and momentum, are constructed. Their algebraic properties constitute quantum kinematics, while quantum dynamics of the system is determined by the unitary group generated by the Hamiltonian expressed as a function of the position and momentum operators. This kinematical structure then remains the same for all possible quantum dynamics which are determined by the respective system Hamiltonians.
2. Linear canonical transformations in classical mechanics Symmetries in classical mechanics are the canonical transformations which leave the Hamiltonian invariant. Linear canonical transformations in the phase space R 2n form the symplectic group Sp(2n, R). For n = 1 degree of freedom it reduces to Sp(2, R) = SL(2, R) with the action on the canonical variables q, p ( ) (q, p a b ) = (q, p)a = (q, p), ad bc = 1. c d Infinitesimal linear canonical transformations are generated by quadratic polynomials in the canonical variables. Examples are the Hamiltonians of a free particle and of the harmonic oscillator. The corresponding phase trajectories in R 2 are straight lines with constant linear momentum and ellipses centered at the origin, respectively.
3. Linear canonical transformations in quantum mechanics Symmetries in quantum mechanics are represented, according to Wigner s theorem, by unitary or anti-unitary operators in the Hilbert space of a quantum system. According to the Stone von Neumann theorem, the canonical observables satisfying the canonical commutation relations [ ˆQ i, ˆP j ] = i Iδ ij are represented by essentially self-adjoint operators unitarily equivalent to the well-known operators in the Hilbert space L 2 (R n, d n q) of the Schrödinger representation. In the simplest case of the Hilbert space L 2 (R, dq) (n = 1) ˆQψ(q) = qψ(q), ˆP ψ(q) = i ψ(q) q. If any group element can be expressed as a square of another group element, as for Lie groups, then the symmetry transformations are represented by unitary operators.
4. Shale s metaplectic representation is the unitary representation of linear canonical transformations in the same Hilbert space L 2 (R n, d n q). It is a unitary irreducible representation X(A) of the double covering of Sp(2n, R) (called the metaplectic group) which for n = 1 reduces to the double covering of SL(2, R), ( ˆQ, ˆP ) = ( ˆQ, ˆP )A, det A = 1. The transformed commutators [ ˆQ, ˆP ] = i I obviously lead, via the Stone - von Neumann theorem, to unitary equivalence ˆQ = X(A) ˆQX(A) 1, ˆP = X(A) ˆP X(A) 1, or, for the exponential Weyl operators X(A)W (s, t)x(a) 1 = W ((s, t)a T ).
5. Simple finite-dimensional quantum kinematics The N-dimensional Hilbert space of an N-level system an orthonormal basis B = { 0, 1,... N 1 } and unitary operators Q N, P N are defined by the relations (Weyl, Schwinger) Q N j = ω j N j, P N j = j 1 (mod N ), where j = 0, 1,..., N 1, ω N = exp(2πi/n). B is taken as the canonical or computational basis of H N = C N and the operators P N and Q N are represented by the generalized Pauli matrices Q N = diag ( 1, ω N, ωn 2,, ) ωn 1 N
and P N = Their commutation relation 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0.... 0 0 0 0 1 1 0 0 0 0 P N Q N = ω N Q N P N (1) is an expression of their complementarity. Further, P N N = QN N = I, ωn N = 1.
The finite Heisenberg group of order N 3 (the Pauli group) is generated by ω N, Q N and P N Π N = { ω l N Qi N P j N l, i, j = 0, 1, 2,..., N 1}. Geometric interpretation: The cyclic group Z N = {0, 1,... N 1} is a configuration space for an N-level quantum system, the finite phase space is Z N Z N. Elements of Z N label the vectors of the basis B = { 0, 1,... N 1 } with the physical interpretation that j is the (normalized) eigenvector of position at j Z N.
The action of Z N on Z N via addition modulo N is represented by unitary operators U(k) = PN k. Their action on vectors j from basis B is given by U(k) j = PN k j = j k (mod N ) Quantum kinematics of a simple N-level system can then be expressed in the form U(j)V (ρ) = ω jρ N V (ρ)u(j) or P j N Qρ N = ωjρ N Qρ N P j N, where the unitary operators V (ρ) = Q ρ N, ρ = 0, 1,..., N 1. The finite Weyl system consists of finite Weyl unitary operators W (ρ, j) = ω jρ/2 V (ρ)u(j) = ω jρ/2u(j)v (ρ). N N If N is odd, the factors ω jρ/2 N are well-defined on Z N Z N.
The finite Weyl operators satisfy the composition law for a projective representation of the Abelian group Z N Z N of translations on the finite phase space Z N Z N, W (ρ, j)w (ρ, j ) = ω (ρ j ρj )/2 N W (ρ + ρ, j + j ). According to Schwinger, the finite Weyl system consisting of N 2 operators W (ρ, j) provides an orthogonal operator basis in the space of all linear operators in C N. The operators W (ρ, j)/ N are orthogonal with respect to the Hilbert-Schmidt inner product Tr(W (ρ, j)w (ρ, j ) ) = Nδ ρρ δ jj and satisfy the completeness relation ρ,j W (ρ, j)w (ρ, j ) = N 2 1.
6. Classification of composite quantum kinematics The cyclic group Z N = {0, 1,... N 1} is a configuration space for N-dimensional quantum kinematics of a single N-level system. According to the well-known rules of quantum mechanics, finite-dimensional quantum mechanics on Z N can be extended in a straightforward way to arbitrary finite direct products Z n1 Z nk as configuration spaces. The Hilbert space of the corresponding composite system consisting of k n i -level component subsystems is the tensor product H n1 H nk of dimension N = n 1... n k, where n 1,..., n k N. If such an bottom up composition starts with component single systems, one would immediately ask: When are the composite quantum kinematics equivalent?
Clearly, the composite Hilbert spaces are isomorphic if and only if they have equal dimensions. However, composite quantum kinematics in dimension N are in general not equivalent! Thus a criterion of equivalence is needed. It follows from a theorem describing inequivalent decompositions of the configuration space for given N: Structure theorem for finite Abelian groups Let G be a finite Abelian group. Then G is isomorphic with the direct product Z N1 Z Nf of a finite number of cyclic groups for integers N 1,..., N f greater than 1, each of which is a power of a prime, i.e. N k = p r k k, where the primes p k need not be mutually different.
In the Structure theorem, the integers N k = p r k k elementary divisors of G. are called the Two finite Abelian groups are isomorphic if and only if they have the same elementary divisor decomposition. If G is the finite Abelian group G = Z N, the Chinese Remainder Theorem gives the isomorphism with the unique decomposition where N = p r 1 1... p r f f G = Z N = ZN1 Z Nf, N k = p r k k, and the primes p k > 1are distinct.
The cyclic groups involved in the Structure theorem are the configuration spaces of the smallest components. They describe Schwinger s independent quantum degrees of freedom, because the Hilbert space of the corresponding quantum system is decomposed into the tensor product of N j -level Hilbert spaces H = H 1 H f, with dim H j = N j = p r j j, dim H = N 1... N f. Also the finite Weyl system is the tensor product of unitary groups W 1 W f. In this way a classification of all quantum kinematics in dimension N is obtained: the factors of the above tensor product can be called the elementary building blocks forming a finite quantum system, each endowed with its simple quantum kinematics. Note that these factors belong to the types G = Z p, Z p 2,..., Z p r,..., where p runs through the set of all prime numbers > 1.
Our results are summarized in the Classification Theorem For a given finite Abelian group G as configuration space there is a unique class of unitarily equivalent finite Weyl systems W in a finite dimensional Hilbert space H, W = W 1 W f. Given a composite dimension N, then, according to the above theorems, it is sufficient to take all inequivalent choices of N k = p r k k with not necessarily distinct primes p k > 1 to obtain all inequivalent quantum kinematics in the Hilbert space of given dimension N.
Mathematicians often prefer an equivalent invariant factor decomposition. In that approach a finite Abelian group G is uniquely determined by an ordered finite list of integers n 1 n 2 n s greater than 1 determining the invariant factors Z ni of G such that n i+1 divides n i and N = n 1 n 2... n s. For instance, if N = 180, the full list of non-isomorphic Abelian groups of order 180 consists of four groups Z 180, Z 90 Z 2, Z 60 Z 3, Z 30 Z 6. In these examples the corresponding elementary divisor decompositions are 2 2.3 2.5, 2.2.3 2.5, 2 2.3.3.5, 2.2.3.3.5 and the corresponding configuration spaces decompose into elementary configuration spaces Z 2 2 Z 3 2 Z 5, Z 2 Z 2 Z 3 2 Z 5, Z 2 2 Z 3 Z 3 Z 5, Z 2 Z 2 Z 3 Z 3 Z 5.
All elementary divisor decompositions for given N can be enumerated starting with the factorization of N = p r 1 1... p r f f with mutually distinct primes p 1,..., p f. First one finds all permissible lists for groups of orders p r i i for each i. For a prime power p r i i the problem of determining all permissible lists is equivalent to finding all partitions of the exponent r i, and does not depend on p i. Recall that the number of partitions of a natural number r is called Bell s number B(r). Then the total number of groups of order N is equal to the product of Bell s numbers B(r 1 )B(r 2 )... B(r f ).
In physics, the dimensions of the constituent Hilbert spaces are already fixed by the list of k physical subsystems and numbers n i of their levels. Then the configuration space G is isomorphic to the direct product of cyclic groups of the respective orders. In order to determine the isomorphism class of the corresponding quantum kinematics, the elementary divisor decomposition is indispensable. For example, the elementary divisor decompositions of two composite systems with the same N = 180 are clearly inequivalent: (1) H = H 20 H 9 with G = Z 2 2 Z 3 2 Z 5, (2) H = H 30 H 6 with G = Z 2 Z 2 Z 3 Z 3 Z 5. At first sight it would seem that the quantum degrees of freedom do not play any significant role in finite quantum mechanics. However, as shown below, the study of the symmetries of quantum kinematics forces us to take them seriously.
7. Symmetries of quantum kinematics Our main recent contribution concerned detailed description of symmetries of finite Weyl-Heisenberg groups for composite quantum systems with arbitrary dimensions of subsystems. The symmetry groups are called there Sp [n1,...,n k ], where the indices denote (arbitrary) dimensions of the constituent Hilbert spaces. In quantum computation these groups are related to so-called Clifford groups. [1] Korbelář M and Tolar J 2012 Symmetries of finite Heisenberg groups for multipartite systems J. Phys. A: Math. Theor. 45 285305 (18pp); arxiv: 1210.0328 [quant-ph] [2] Korbelář M and Tolar J 2010 Symmetries of the finite Heisenberg group for composite systems J. Phys. A: Math. Theor. 43 375302 (15pp); arxiv: 1006.0328 [quant-ph]
For simplicity let a bipartite system be created by coupling two single multi-level subsystems with arbitrary dimensions n, m, i.e. G = Z n Z m and H = H n H m. The corresponding finite Weyl-Heisenberg group is embedded in GL(N, C), N = nm. Via inner automorphisms it induces an Abelian subgroup in Int(GL(N, C)). The normalizer of this Abelian subgroup in the group of inner automorphisms of GL(N, C) contains all inner automorphisms transforming the phase space into itself, hence necessarily contains P (n,m) as an Abelian semidirect factor. The true symmetry group is then given by the quotient group of the normalizer with respect to this Abelian subgroup. We shall see that in special cases the symmetry groups are reducible to standard types SL and Sp, but in general these standard types do not exhaust the obtained class of symmetry groups.
The special case of n = m, N = n 2, corresponds to the symmetry group Sp [n,n] = Sp(4, Zn ). If N = nm, n, m coprime, the symmetry group is Sp [n,m] = SL(2, Zn ) SL(2, Z m ) = SL(2, Z nm ). Further, if d = gcd(n, m), n = ad, m = bd, the finite configuration space can be further decomposed under the condition that a, b are both coprime to d, G = Z n Z m = Z ad Z bd = (Za Z d ) (Z b Z d ). Thus the symmetry group is reduced to the direct product Sp [n,m] = Sp[a,b] Sp(4, Z d ) For instance, if n = 15, m = 12, then d = 3 is coprime to both a = 5 and b = 4, and also a and b are coprime, hence the symmetry group is reduced to the standard types SL and Sp, Sp [n,m] = SL(2, Za ) SL(2, Z b ) Sp(4, Z d ).
However, the general situation is more complicated. For instance, let n = 180 and m = 150. Then d = 30, a = 180/30 = 6 and b = 150/30 = 5, hence a divides d and also b divides d. In this case reduction to standard groups Sp and SL is not possible. One has to break down the composite system consisting of two single systems into its elementary building blocks. We decompose each of the finite configuration spaces Z 180 Z 150 = (Z 2 2 Z 3 2 Z 5 ) (Z 2 Z 3 Z 5 2), and take notice of coprime factors 2.2 2, 3.3 2 and 5.5 2 leading to the factorization of the symmetry group in agreement with the elementary divisor decomposition Sp [180,150] = Sp[2,2 2 ] Sp [3,3 2 ] Sp [5,5 2 ]. One sees that the symmetry groups like Sp [p k,p l ] with indices given by different powers of the same prime p deserve to be added to the standard types Sp and SL.
Final remarks Mathematical physics yields general classifications and the physicist elaborates which of the offered possibilities is realized for a concrete physical system. For instance, in non-relativistic quantum mechanics in infinitedimensional Hilbert space, the Stone-von Neumannn theorem is of prime importance. Similarly, in the finite-dimensional Hilbert spaces this theorem holds for single systems For composite systems the obtained classification of finite quantum kinematics should be considered as very important for distinguishing inequivalent finite quantum systems. However, try to consult some textbooks I bet that you will not find any reference to it.
For given N, all inequivalent quantum kinematics generate the same matrix algebra M N (C). The general algebraic approach thus seemingly does not support our classification. However, it turns out that our classification is obtained if fine gradings are considered, and these gradings embody the complementarity required in quantum kinematics.
Synopsis Quantum kinematics in Hilbert spaces of finite dimension N is studied from the point of view of number theory. The fundamental theorem describing all finite discrete abelian groups of order N as direct products of cyclic groups, whose orders are powers of not necessarily distinct primes contained in the prime decomposition of N, leads to a classification of inequivalent finite Weyl systems and finite Weyl-Heisenberg groups. The relation between the mathematical formalism and physical realizations of finite quantum systems is discussed from this perspective. The corresponding symmetries normalizers of the Weyl-Heisenberg groups in unitary groups have been fully described. They are related to so called Clifford groups in quantum information. Special attention is paid to the elementary building blocks of finite quantum systems - quantal degrees of freedom. It is shown that the symmetries can be reduced in accordance with the elementary divisor decomposition to direct products of finite groups of the types SL(2, Z n ), Sp(2k, Z n ), and Sp [p k,p l,... ].
Acknowledgement Partial support by the Ministry of Education of Czech Republic (from the research plan RVO:68407700) is gratefully acknowledged. Thank you for your attention
Appendix A. Symmetries of simple quantum kinematics Consider those inner automorphisms acting on elements of Π N which induce permutations of cosets in Π N /Z(Π N ): Ad X (ω l N Qi N P j N ) = Xωl N Qi N P j N X 1, where X are unitary matrices from GL(N, C). These inner automorphisms are equivalent if, for all (i, j) Z N Z N, they define the same transformation of cosets in Π N /Z(Π N ): Ad Y Ad X Y Q i P j Y 1 = XQ i P j X 1. The group P N = Π N /Z(Π N ) has two generators, the cosets Q and P (or Ad QN and Ad PN, respectively). Hence if Ad Y induces a permutation of cosets in Π N /Z(Π N ), then there must exist elements a, b, c, d Z N such that Y QY 1 = Q a P b and Y P Y 1 = Q c P d.
It follows that to each equivalence class of inner automorphisms Ad Y a quadruple (a, b, c, d) of elements in Z N is assigned. Theorem (HPPT 2002) For integer N 2 there is an isomorphism Φ between the set of equivalence classes of inner automorphisms Ad Y which induce permutations of cosets and the group SL(2, Z N ) of 2 2 matrices with determinant equal to 1 (mod N), ( ) a b Φ(Ad Y ) =, a, b, c, d Z c d N ; the action of these automorphisms on Π N /Z(Π N ) is given by the right action of SL(2, Z N ) on elements (i, j) of the phase space P N = Z N Z N, ( ) (i, j a b ) = (i, j). c d
Appendix B. Symmetries of composite quantum kinematics Let the Hilbert space of a composite system be the tensor product H n1 H nk of dimension N = n 1... n k, where n 1,..., n k N. For the composite system, quantum phase space is an Abelian subgroup of Int(GL(N, C)) defined by P (n1,...,n k ) = {Ad M 1 M k M i Π ni }. Its generating elements are the inner automorphisms where (for i = 1,..., k) e j := Ad Aj for j = 1,..., 2k, A 2i 1 := I n1 n i 1 P ni I ni+1 n k, A 2i := I n1 n i 1 Q ni I ni+1 n k.
The normalizer of P (n1,...,n k ) in Int(GL(n 1 n k, C)) will be denoted N (P (n1,...,n k ) ) := N Int(GL(n 1 n k,c)) (P (n 1,...,n k ) ), We need also the normalizer of P n in Int(GL(n, C)), and N (P n ) := N Int(GL(n,C)) (P n ), N (P n1 ) N (P nk ) := {Ad M1 M k M i N (P ni )} Int(GL(N, C)). Further,. N (P n1 ) N (P nk ) N (P (n1,...,n k ) )
Now the symmetry group Sp [n1,...,n k ] is defined in several steps. First let S [n1,...,n k ] blocks be a set consisting of k k matrices H of 2 2 H ij = n i gcd(n i, n j ) A ij where A ij M 2 (Z ni ) for i, j = 1,..., k. Then S [n1,...,n k ] is (with the usual matrix multiplication) a monoid.
Next, for a matrix H S [n1,...,n k ], we define its adjoint H S [n1,...,n k ] by (H ) ij = n i gcd(n i, n j ) AT ji. Further, we need a skew-symmetric matrix J = diag(j 2,..., J 2 ) S [n1,...,n k ] where J 2 = ( 0 1 1 0 ). Then the symmetry group is defined as Sp [n1,...,n k ] := {H S [n 1,...,n k ] H JH = J} and is a finite subgroup of the monoid S [n1,...,n k ].
Our first theorem states the group isomorphism: Theorem 1 (KT 2012) N (P (n1,...,n k ) )/P (n 1,...,n k ) = Sp [n1,...,n k ].
Our second theorem describes the generating elements of the normalizer: Theorem 2 (KT 2012) The normalizer N (P (n1,...,n k )) is generated by where (for 1 i < j k) N (P n1 ) N (P nk ) and {Ad Rij }, and R ij = I n1 n i 1 diag(i ni+1 n j, T ij,..., T n i 1 ij ) I nj+1 n k T ij = I ni+1 n j 1 Q n j gcd(n i,n j ) n j.
Corollary (KT 2012) If n 1 = = n k = n, i.e. Sp [n,...,n] = Sp2k (Z n ). N = n k, the symmetry group is These cases are of particular interest, since they uncover symplectic symmetry of k-partite systems composed of subsystems with the same dimensions. This circumstance was found, to our knowledge, first by PST 2006 for k = 2 under additional assumption that n = p is prime, leading to Sp(4, F p ) over the field F p. We have generalized this result (KT 2010) to bipartite systems with arbitrary n (non-prime) leading to Sp(4, Z n ) over the modular ring, and also to multipartite systems (KT 2012). The corresponding result has independently been obtained by G. Han who studied symmetries of the tensored Pauli grading of sl(n k, C).
References Korbelář M and Tolar J 2012 Symmetries of finite Heisenberg groups for multipartite systems J. Phys. A: Math. Theor. 45 285305 (18pp); arxiv: 1210.0328 [quant-ph] Korbelář M and Tolar J 2010 Symmetries of the finite Heisenberg group for composite systems J. Phys. A: Math. Theor. 43 375302 (15pp); arxiv: 1006.0328 [quant-ph] Han G 2010 The symmetries of the fine gradings of sl(n k, C) associated with direct product of Pauli groups J. Math. Phys. 51 092104 (15 pages) Pelantová E, Svobodová M and Tremblay J 2006 Fine grading of sl(p 2, C) generated by tensor product of generalized Pauli matrices and its symmetries J. Math. Phys. 47 5341 5357
Appendix C. Complementarity structures The quantal notion of complementarity concerns very specific relation among quantum observables. Consider two almost identical definitions. Definition 1. Two observables A and B of a quantum system with Hilbert space of finite dimension N are called complementary, if their eigenvalues are non-degenerate and any two normalized eigenvectors u i of A and v j of B satisfy u i v j = 1 N. Then in an eigenstate u i of A all eigenvalues b 1,..., b N of B are measured with equal probabilities, and vice versa. This means that exact knowledge of the measured value of A implies maximal uncertainty to any measured value of B. For the next definition note that the (non-degenerate) eigenvalues a i of A and b j of B are in fact irrelevant, since only the corresponding orthonormal bases u i and v j are involved.
Definition 2. Two orthonormal bases in an N-dimensional complex Hilbert space { u i i = 1, 2,..., N} and { v j j = 1, 2,..., N } are called mutually unbiased if inner products between all possible pairs of vectors taken from distinct bases have the same magnitude 1/ N, u i v j = 1 N for all i, j {1, 2,..., N}. In the sense of these definitions one may call two measurements to be mutually unbiased, if the bases composed of the eigenstates of their observables (with non-degenerate spectra) are mutually unbiased. If the system is prepared in any eigenstate of A, then the transition probabilities to all eigenstates of the complementary observable B are the same (equal to 1/N). Our basic unitary operators Q N and P N exactly satisfy this criterion of complementarity. Further, the system of N 2 unitary Weyl- Schwinger operators formed a complete orthonormal basis in the linear space M N (C) of N N complex matrices with respect to the Hilbert-Schmidt inner product.
From the algebraic point of view, finite quantum systems in quantum information theory are given by full matrix algebras M N (C). In this part M N (C) is equipped with two additional structures: 1. the Hilbert-Schmidt inner product defined by the standard trace (defining also the Hilbert-Schmidt norm), 2. fine gradings defined by MAD-groups of commuting inner automorphisms of M N (C). We shall see that, as a result, so-called Pauli gradings emerge and they deserve to be called complementarity structures in M N (C). Definition 3. A grading of an associative algebra A is a direct sum decomposition of A as a vector space Γ : A = α A α satisfying the property x A α, y A β xy A γ.
Note that if Γ is a grading of the matrix algebra M n (C), then Γ is also a grading of the matrix Lie algebra gl(n, C). Similarly, as is well known for Lie algebras, linear subspaces A α can be defined as eigen-spaces of automorphisms of A. For an automorphism g of A, g(xy) = g(x)g(y) holds for all x, y A. Now if g(x) = λ α x defines the subspace A α and g(y) = λ β y defines the subspace A β, then g(xy) = g(x)g(y) = λ α λ β xy defines the subspace A γ with λ γ = λ α λ β. In this way g defines the grading decomposition A = α Ker(g λ α ). Further commuting automorphisms may refine the grading. Definition 4. For M GL(N, C), Ad M Int(M N (C)) denotes the inner automorphism of M N (C) induced by the operator M GL(N, C), i.e. Ad M (X) = MXM 1 for X M N (C). Fine gradings are those gradings which cannot be further refined. Since the commuting inner automorphisms form an Abelian subgroup of Int(M N (C)), fine gradings of M N (C) are obtained using the maximal Abelian groups of diagonalizable automorphisms
as subgroups of Int(M N (C)) to be shortly called the MADgroups. Definition 5. We define P N as the group P N = {Ad Q i N P j N (i, j) Z N Z N }. It is an Abelian subgroup of Int(M N (C)) and is generated by two commuting automorphisms Ad QN, Ad PN, each of order N, P N =< Ad QN, Ad PN >. A geometric view is sometimes useful that P N is isomorphic to the quantum phase space identified with the Abelian group Z N Z N.
All MAD-groups for M N (C) are given in the following Theorem Any MAD-group contained in Int(M N (C)) is conjugated to one and only one of the groups of the form P N1 P N2... P Nf D(m), where N i = p r i i are powers of primes, N = mn 1 N 2... N f and D(m) is the image in Int(M N (C)) of the group of m m complex diagonal matrices under the adjoint action. If m = 1, the corresponding fine grading is the Pauli grading and it decomposes M N (C) into N 2 one-dimensional subspaces spanned by the Weyl-Schwinger operators! For illustration, we give the list of MAD-groups in low dimensions: n = 2: P 2 D(1), D(2)
n = 3: P 3 D(1), D(3) n = 4: P 4 D(1), P 2 P 2 D(1), P 2 D(2), D(4) n = 5: P 5 D(1), D(5) n = 6: P 3 P 2 D(1), P 3 D(2), P 2 D(3), D(6) n = 7: P 7 D(1), D(7) n = 8: P 8 D(1), P 4 P 2 D(1), P 2 P 2 P 2 D(1), P 4 D(2), P 2 P 2 D(2), P 2 D(4), D(4)
A part of the obtained MAD-groups containing the trivial diagonal subgroup D(1) induce exactly all our Weyl-Schwinger decompositions. However, there are still fine gradings induced by D(m), m = 2, 3.... They include partial or complete decompositions which have the form of the Cartan root decompositions of Lie algebras sl(m, C) extended by the unit matrix. They contain the Abelian Cartan subalgebra of dimension m 1, the unit matrix and one-dimensional root subspaces spanned by nilpotent matrices. Concerning physical interpretation of these Cartan parts of the decompositions, one can speculate that they may reflect extra degrees of freedom corresponding to some internal symmetries. Leaving these decompositions aside, we are left with the Pauli gradings which decompose M N (C) in direct sums of N 2 one-dimensional subspaces. For these Pauli decompositions with given N one should realize that M N (C) encompasses all operators of any quantum system with N-dimensional Hilbert space.
In this way we have got a new view on the relation between the general mathematical formalism and physical realizations of finite quantum systems. The apparent contradiction that M N (C) represents any N-dimensional quantum system and at the same time there is a multitude of inequivalent quantum kinematics for given N, is simply resolved: from the physical point of view the same algebra M N (C) is the operator algebra not only for a single N-level system, but also for all other members of the set of inequivalent quantum kinematics for this N. They just correspond to different physical realizations of composite quantum systems. Of course, each such system has its preferred quantum operators. Note that from the mathematical point of view M N1 (C) M Nf (C) is isomorphic with M N (C).