Generalized Bloch Spheres for m-Qubit States. Klaus Dietz∗ Sektion Physik, LMU, Muenchen, Theresienstrasse 37, 80333 Muenchen, and 6 0 0 2 Dipartimento di Scienze Fisiche ed Astronomiche n dell’Universita di Palermo, a J via Archirafi 36, 90123 Palermo, Italy 3 February 1, 2008 1 v 3 1 0 1 Abstract 0 6 m-Qubitstates areembeddedinCl Cliffordalgebra. Theirprob- 2m 0 ability spectrum then depends on O(2m)- or O(2m+1)-invariants / h respectively. Parameter domains for O(2m(+1))-vector and -tensor p configurations, generalizing the notion of a Bloch sphere, are derived. - t n a u 1 Introduction q : v For many purposes it is useful to consider m-qubit states as vectors in a i X R-linear Hilbert space H whose basis is a set {B , i = 1...22m} of 2m ×2m i r orthonormal a trace(B ·B ) = δ , i j ij hermitian matrices: 22m H = { b B | b real}. (H) k k k k=1 X A state is either represented by a hermitian, normalized matrix or an ap- propriate coordinate vector [b ,b ,...,b ] (a formulation in an appropriate 1 2 22m ∗Permanent Address:Physics Department,University of Bonn, 53115 Bonn, Germany, email: [email protected] 1 projective space would more adequate).In [2] [3] the generators of the quan- tum invariance group SU (2m) are proposed as such a basis, a possibility which we shall discuss in the Summary. A straightforward solution for the parametrisation of a state ̺ (a density matrix) is to write the set of all states as U {̺} = ρ Λ Λ [ where ρU = {U+̺ U | U ∈ U(2m)} Λ Λ and ̺ is the diagonal matrix ̺ = diag{Λ}, Λ Λ Λ = {[λ ,λ ,...,λ ] | λ real, λ = 1, λ ≥ 0}; 1 2 2m i i i X U(2m) is the unitary group in 2m-dimensions and Λ is the probability spec- U trum generating the State ρ . This construction warrants positivity and Λ normalisation. It is however not always (or, better, almost never) conve- nient in the discussion of physical situations.∗ On the other hand writing ̺ as a vector in H confronts us with the problem of deriving conditions for the expansion coefficients † that guaranteethe expansion to yield a state. Formu- lated in this general way the problem has no obvious solution: positivity and normalisation conditions can derived by expressing the eigenvalues in terms of the expansion coefficients, i.e. finding the zeroes of the characteristic poly- nomial as functions of these parameters. As we know from Abel and Galois a solution by rational operations and radicals does not exist for quintic or higher degrees, i.e. for general 3-qubit and a fortiori for higher systems. For the 2-qubit explicit expressions are given by the Ferrari-Cardano formulae. In this paper I explicitly construct classes of states for all m whose spectra are determined by charactistic polynomials factorizing into polynomials of a given degree. The novel point in our considerations is the use of hermitian matrix representations of a Clifford algebra to construct bases in H. This particular choice of basis allows us to arrange the 22m−1 real coordinates of a m-Qubit state in multidimensional arrays which are shown to ’transform’ as O(2m) tensors. This fact implies that the probability spectrum of a m- Qubit state depends only on O(2m)-invariants, a considerable simplification ∗This is equally true for the parametrisation̺=eA/trace eA , A hermitian, which is rather clumsy e.g. when it comes to the discussion of separability conditions. (cid:0) (cid:1) †These parameters are linearly related, a matrix representation of the basis in H B ,i=1...22m given, to the matrix elements of the density matrix. i 2 of the parameter dependencies of these eigenvalues, indeed. This simplifica- tion leads to a complete characterisation of complete‡ sets of states which allow for an explicit construction of a parameter domain. In this way I find the set of all states (vector-states) whose parameter domain is the Bloch 2m-sphere. Furthermore a set of (bivector)-states is proposed whose novel parameter domain generalizes the notion of a Bloch sphere. Beyond these two domains the Descartes rule for the positivity of polynomial roots can be used to derive admissible parameter domains. 2 m-Qubit states imbedded in Clifford Alge- bras. An m-qubit system is controlled by m spin-degrees of freedom and hence by 22m−1 parameters (see footnote 2 on page 2). The determining anticommu- tation relation for Clifford numbers [1] (I is the unity) Γ ·Γ + Γ ·Γ = 2δ I (1) i j j i ij with i,j = 1...2m has 2m-dimensional, hermitian, traceless matrix representations Γ{m}. j From the anticommutation relations we see immediately that the products Γ := ik−1 Γ ·Γ · ... ·Γ (2) j1,j2,...,jk j1 j2 jk k = 2...m (3) aretotallyanti-symmetricintheindices[j ...j ]. Theonlysymmetricobject 1 k constructed from Clifford numbers is the unity I = Γ2 i as we see from the anticommutation relations. A product consists of at most 2m factors. Hence we have 2m 2m = 22m k k=0(cid:18) (cid:19) X ‡Completeinthe sensethatallstatesfactorizinginaspecific wayarecontainedinthis set. 3 independent products. Furthermore because of the commutation relations we have + {m} {m} {m} {m} {m} {m} trace Γ ·Γ · ... ·Γ Γ ·Γ · ... ·Γ ∼ i1 j1 k1 i2 j2 k2 (cid:18) (cid:19) (cid:16) (cid:17) δ δ ...δ , ¯i1¯i2 ¯j1¯j2 k¯1k¯2 X(cid:0) (cid:1) where the δ-function expresses pairwise equality of the · - and · -indices. 1 2 A hermitian 2m×2m-matrix requires 22m real numbers for a complete para- metrisation. Thus m-qubit states can be expanded in terms of I and the products introduced: Clifford numbers are the starting point for the con- struction of a basis in the R-linear space of hermitian matrices: this basis is construed as a Clifford algebra Cl (22m-dimensional as we have 2m seen). The important advantage to gain from this choice of basis is that now domains for parameters are determined by O(2m)-invariants. The number of parameters necessary for the specification of these domains is thus consid- erably reduced. For thedomains found inthis paper this means one invariant for the vector-state configuration (2m parameters) and two invariants for the bivector states (m(2m−1) parameters) to be constructed below for all m. I should remark that many beautiful geometric reverberations of Clifford algebras will play no rˆole here, only very elementary properties of Clifford algebras will be sketched, emphasizing practical aspects. It is in this sense that the following, hopefully selfcontained, outline of the method should be understood. To construct a basis and its matrix representation G{m} in H I proceed as follows: • The product Γ{m} := (−i)m Γ{m}.Γ{m}....Γ{m} (4) 2m+1 1 2 2m {m} obviously anti-commutes with all the Γ i = 1...2m. i • The explicitly anti-symmetric products (ε is the totally anti-symmetric symbol in 2m-dimensions) ˆ{m,k} {m,k} {m} {m} {m} Γ = F ε Γ ....Γ Γ (5) i1...ik Norm i1...i2m ik+1 2m 2m+1 (cid:16) (cid:17) (−i)m+s 2m {m,k} F = Norm (2m)! k (cid:18) (cid:19) 0 when x = 0,1 s = where x = kmod(4) (6) (1 when x = 2,3 4 The limiting cases k = 1 and k = 2m are immediately seen to be Γˆ{m,1} = (−1)m+1Γ{m} i1 i1 f Γˆ{m,2m} = m ε Γ{m} i1...i2m 2m! i1...i2m 2m+1 with m (−1)2 for m even f = m m+1 ((−1) 2 for m odd • Because of the anti-commutation relations the only symmetric tensor is the scalar, i.e. the unit matrix Γˆ{m,0} = I (7) • The set of matrices G{m} = {Γˆ{m,0},Γˆ{m,1} ... Γˆ{m,2m}} is orthonor- mal in the sense of (1). • Formally speaking this gives an identification of the linear spaces k g{m,k} = span Γˆ{m,k},R and the tensor algebra R2m of R2m. In detail we write(cid:16)§ (cid:17) V Isomorphic vector spaces: scalar, R | g{m,0} = R·I 1 vector, R2m = R2m | gˆ{m,1} ^ 2 (2-)tensor (bivector), R2m | gˆ{m,2} ^ ...... 2m volume element, R2m | gˆ{m,2m} ^ §We use the slightly old fashioned notation: vector,tensor,...k-tensorinstead of vector, bivector,...k-vector 5 • Following these observations we organize the state-parameters in terms {m,0} of a scalar G and the totally anti-symmetric real arrays 0 {m,1} {m,2} {m,2m} G , G ... G i1 i1,i2 i1,i2,...,i2m (i.e. totally antisymmetric arrays of real numbers) and thus account for 2m 2m = 22m (8) k k=0(cid:18) (cid:19) X coefficients. • We write the expansion of a m-qubit state as 2m ̺{m} = G{m,k} ◦Γˆ{m,k} (9) k=0 X where ◦ indicates the contraction A◦B = A B . i1,...,ik i1,...,ik i1,...,ik P {m} {m} • An explicit construction of the representation Γ ,...,Γ tradition- 1 2m ally proceeds e.g. as follows: Starting with the Pauli matrices 0 1 0 −I 1 0 1 0 σ = , σ = , σ = , σ = , 1 1 0 2 I 0 3 0 −1 4 0 1 (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) we have the iteration scheme G{m+1} = {Γ{m,1} ×σ ,...,Γ{m,2m} ×σ ,Γ{m,0} ×σ , Γ{m,0} ×σ } (10) 1 1 2 3 • O(2m)-symmetry: To begin with it might be useful to remind the reader the machinery of rotations in classical systems. Consider a canonical, classical system with 2m degrees of freedom, i.e. with a 2m-dimensional configuration space. Infinitesimal 2m-dimensional rotations and translations gener- ated by generators J , P respectively i,j i 6 ({A ,B} denote Poisson brackets for functions defined on the phase space of the system) are defined as Infinitesimally: F −→ F +ǫ α {J ,F}+ǫ β {P ,F} 1 i,j i,j 2 i i (repeated indices are summed over) where ǫ is infinitesimal and α i,j = 1..2m i,j is an antisymmetric array of parameters the β parametrize translations. i The Lie algebra of the Euclidean Poincar´e group {J ,J } = δ J +δ J −δ J −δ J i,j k,l i,l j,k j,k i,l i,k j,l j,l i,k {J ,P } = P δ −P δ . i,j k i j,k j i,k The anticommutation relations (1) defining the Clifford algebra Cl 2m spanned by the set of totally antisymmetric products and the unity G = {I,Γ ,iΓ Γ ,...} considered above lead to an analogous algebraic i i j structure. A straightforward calculation shows (Γ := iΓ ·Γ ) i,j i j i [Γ ,Γ ] = δ Γ +δ Γ −δ Γ −δ Γ (11) i,j k,l i,l j,k k,j i,l l,j i,k i,k j,l 2 i [Γ ,Γ ] = δ Γ −δ Γ . (12) i,j k k,i j j,k i 2 These relations constitute a quantum analogue of the classical repre- sentation of the O(2m) Lie algebra¶: the Γ generate rotations, the i,j Γ translations in the Clifford algebra Cl , the array {Γ ,Γ ,...,Γ } i 2m 1 2 2m ’transforms as a vector’. The basis elements of the dual Grassmann algebra R2m can be identified with (see above) G = {G{m,0},G{m,1},...,G{m,2m}} and ’transform as tensors’. More V ¶Precisions concerning a more precise discussion of the universalcovering group are of no avail here and will not be touched. 7 precisely we have L ∈ O(2m) 7−→ U (L) = e−4iαi,jΓi,j (13) O(2m)-transformations {m,1} {m,1} G 7−→ L G (14) i i,k k {m,2} {m,2} G 7−→ L L G (15) i,j i,i1 j,j1 i1,j1 etc induce transformations Γ 7−→ U (L)Γ U (L)−1 = L−1 Γ (16) i i i,k k Γ Γ 7−→ U (L)Γ Γ U (L)−(cid:0)1 = (cid:1)L−1 L−1 Γ Γ (17) i j i j i,i1 k,k1 i1 k1 etc. (cid:0) (cid:1) (cid:0) (cid:1) (18) Configurations parametrized by one of the tensors G{m,k} have some comfortable (and profitable) properties. For instance the coefficients of the characteristic polynomials are expected to depend on O(2m)- invariants built from these tensors. Furthermore the probability spec- tra will exhibit degeneracy patterns corresponding to the rank of the tensors G{m,k}, parameter ranges corresponding to physical states will be determined by universal polynomials in terms of these invariants. Thefollowingsectionsaredevotedtodetaileddicussionsoftheseobservations for the cases of m=2,3-qubits. General results for m-qubits will be derived. 3 O (2m)-Tensor Configurations In this chapter I introduce some nomenclature which derives from similar objects ocurring in the Dirac theory of relativistic Fermions. The iteration scheme (10)provides us with explicit bases forClifford algebras Cl . 2m The coordinates representing a m-Qubit introduced in equation (H) of the Introduction are organized in • scalar G{m,0}, G{m,0} = 1 because of state normalisation • vector G{m,1}, • 2,3-tensor G{m,2,3}, and 8 • pseudoscalar G{m,2m}, • pseudovector G{m,2m−1}, • pseudotensor G{m,2m−(2,3)} components. k • m = 1 The 2-Clifford algebra is spanned by ∗∗ Γˆ{1,0} = σ scalar 4 Γˆ{1,1} = {σ , σ } vector (20) 1 2 Γˆ{1,2} = σ pseudoscalar 3 A qubit state is then written as (G{m,o} = 1 because of normalisation) 2m 1 ̺ = G{1,0}Γˆ{1,0} +G{1,1} ◦Γˆ{1,1} + G{1,2} Γˆ{1,2} (21) 2 (cid:16) (cid:17) • m = 2 The Clifford algebra is now spanned by 0 0 0 1 0 0 0 −i 0 0 1 0 0 0 i 0 Γˆ{2,1} =   Γˆ{2,1} =   1 0 1 0 0 2 0 −i 0 0 1 0 0 0 i 0 0 0          (22) 0 0 i 0 1 0 0 0 0 0 0 i 0 1 0 0 Γˆ{2,1} =   Γˆ{2,1} =  . 3 −i 0 0 0 4 0 0 −1 0  0 −i 0 0 0 0 0 −1     kHere we followthe nomenclatureof Dirac theory(generalizedfor m6=2)for relativis- tic fermions choosing a euclidean Majorana representation for Γˆ{m,k} generated by the iteration scheme (10). ∗∗We could have chosen Γˆ{1,0} =σ4 Γˆ{1,1} ={σ2, σ3}, or {σ3,σ1} (19) Γˆ{1,2} =σ1, or σ2 as well . Both basis are connected by an O(2) rotation by π/4. 9 We write ̺ = (23) 1 G{2,0}Γˆ{2,0} + G{2,1} ◦Γˆ{2,1} + G{2,2} ◦Γˆ{2,2} + G{2,3} ◦Γˆ{2,3} 4 (cid:16) + G{2,4} Γˆ{2,4} (cid:17) The iteration algorithm (10) straightforwardly provides analogous represen- tations for m > 2. 4 Probability Spectra for Tensor Configura- tions and Their Degeneracies. In this section we explicitly determine the m = 1,2,3 probability spectra of the vector and tensor configurations by calculating the roots of the charac- teristic polynomial P{m} := Determinant ̺{m} −λI ; parameter domains generalizing Bloch sphe(cid:0)res are obta(cid:1)ined by requiring that the spectrum obtained be a probability distribution. Degeneracies of m- qubit tensor spectra are shown to follow simple patterns. Because of the normalisation condition a ’tensor configuration’ always reads as 1 ̺ = I +G{m,ktensor} ◦Γˆ{m,ktensor} (24) ktensor 2m m (cid:16) (cid:17) We find • Vector configurations: k = 1 tensor The probability spectra are 2m−1-fold degenerate, i.e. built up by one doublet repeated 2m−1-times. The doublet is found to be 1 λ = 1±kG{m,1}k (25) 2m (cid:0) (cid:1) where the absolute value of G{m,1} is simply the vector norm 1/2 2m 2 kG{m,1}k = G{m,1} (26) i ! Xi=1 (cid:16) (cid:17) Remarks: 10