Galois fields in quantum mechanics A. Vourdas University of Bradford

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1 Galois fields in quantum mechanics A. Vourdas University of Bradford Phase space methods: position and momentum in the ring Z(d) and the field GF(p l ) Finite quantum systems (eg spin) phase space: toroidal lattice Z(d) Z(d) displacements Symplectic transf: isotropy of phase space Z(p) GF(p)(field) phase space: finite geometry Galois field GF(p l ) quantum systems in GF(p l ) quantum engineering : l spins with j = (p 1)/2 coupled in a particular way Frobenius transformations and Galois group: algebraic concepts in harmonic analysis implications for physics Motivation: symplectic transf in discrete systems mutually unbiased bases classical/quantum information processing field extension in quantum mechanics and harmonic analysis 1

2 Finite quantum systems d-dimensional Hilbert space H e.g., spin j = (d 1)/2 position states X; m m Z(d) ring (field when d = p l ). Fourier transform: F = d 1/2 m,n ω(mn) X; m X; n ω(α) = exp momentum states P; m : [ i 2πα ] ; F 4 = 1 d P; m = F X; m = d 1/2 n ω(mn) X; n arbitrary state f = f m X; m = f m P; m entropic uncertainty relations position and momentum operators ˆx = m m X; m X; m ; ˆp = m m P; m P; m 2

3 in harmonic oscillator: phase space R R displacements: exp(iαˆx), exp(iβˆp) where α, β R exp( iβˆp) x = x + β exp( iβˆp) p = exp( iβp) p exp(iαˆx) x = exp(iαx) x exp(iαˆx) p = p + α position-momentum phase space: Z(d) Z(d)(toroidal lattice) Displacement operators: [ i 2πd ] ˆp X = exp ; Z = exp [i 2πd ˆx ] here α, β Z(d) X β X; m = X; m + β X β P; m = ω( mβ) P; m Z α X; m = ω(mα) X; m Z α P; m = P; m + α general displacements D(α, β) = Z α X β ω( 2 1 αβ); [D(α, β)] = D( α, β) Heisenberg-Weyl group X d = Z d = 1 toroidal X β Z α = Z α X β ω( αβ)

4 marginal properties (d odd) 1 D(α, β) = X;2 1 β X; 2 1 β d α 1 D(α, β) = P;2 1 α X; 2 1 α d β arbitrary operator Θ Θ = 1 D(α, β) W( α, β) d α,β W(α, β): Weyl function W(α, β) = tr[θd(α, β)] arbitrary operator Θ 1 D(α, β) Θ d trθ [D(α, β)] = 1 α,β In the special case Θ = s s 1 α, β; s α, β; s = 1 d α,β α, β; s D(α, β) s resolution of the identity Wigner and Weyl functions, etc

5 Symplectic transformations in Galois quantum systems Transformations: X = SXS = X κ Z λ ; κν λµ = 1 (mod d) preserve the X d = Z d = 1; Z = SZS = X µ Z ν X β Z α = Z α X β ω αβ 3 independent integer parameters. For a given triplet (κ,λ,µ) can we find integer ν such that ν = κ 1 (λµ + 1) (mod d)? Yes, if parameters in Z(p) GF(p) or GF(p l ) Galois quantum systems : position and momentum in GF(p l ) Phase space GF(p l ) GF(p l ) finite geometry. symplectic trs: Sp(2, GF(p l )) group 3

6 now Z(p) GF(p)(later GF(p l )) construct explicitly symplectic operator S S(κ, λ, µ) = S(1,0, ξ 1 )S(1, ξ 2,0)S(ξ 3,0,0) = µκ(1 + λµ) 1 where ξ 1 ξ 2 ξ 3 = λκ 1 (1 + λµ) = κ(1 + λµ) 1 S(ξ 3,0,0) = m S(1, ξ 2,0) = m S(1,0, ξ 1 ) = m X; ξ 3 m X; m ω(2 1 ξ 2 m 2 ) X; m X; m ω( 2 1 ξ 1 m 2 ) P; m P; m S(ξ 3,0,0) dilation/contraction (squeezing) transformations S(ξ 3,0,0) X; m = X; ξ 3 m S(ξ 3,0,0) P; m = P; ξ3 1 example:p = 3, ξ 3 = 2, ξ 1 3 = 2: S X;0 = X;0 ; S X;1 = X;2 ; S X;2 = X;1 S P;0 = P;0 ; S P;1 = P;2 ; S P;2 = P;1 reordering of the points

7 Galois fields Z(p): field Field extension: elements of GF(p l ) α = α 0 + α 1 ǫ α l 1 ǫ l 1 ; α i Z(p) defined modulo irreducible polynomial of degree l: P(ǫ) = c 0 + c 1 ǫ c l 1 ǫ l 1 + ǫ l ; different P(ǫ), isomorphic results addition, multiplication c i Z(p) Frobenius automorphism: σ : α α p ; σ l = 1 α α p α p2... α pl 1 α pl = α Galois conjugates: α, α p,...α pl 1 Galois group:{1, σ,..., σ l 1 } = Z(l) Trace: sum of all conjugates Trα = α + α p α pl 1 ; trace of αβ can be written as Trα Z(p) Tr(αβ) = g ij α i β j ; g ij = Tr[ǫ i+j ] g ij : depends on irreducible polynomial matrix g has inverse 4

8 (1, ǫ,..., ǫ l 1 ) (E 0, E 1,..., E l 1 ) dual basis E i such that Tr(ǫ κ E λ ) = δ κλ. α = l 1 λ=0 α λ ǫ λ = l 1 λ=0 ᾱ λ E λ α λ = Tr[αE λ ]; ᾱ λ = Tr[αǫ λ ] = κ g λκ α κ trace of αβ can be written as Tr(αβ) = g ij α i β j = ᾱ i β i = α i β i exponential of α (complex valued function): χ(α) = ω(trα); ω = exp(i2π/p) additive characters: χ(α)χ(β) = χ(α + β) later in Fourier transforms: ( ) ( ) ( ) χ(αβ) = ω gij α i β j = ω ᾱi β i = ω αi β i where χ(αβ) = δ(β,0) α example: GF(9)(where p = 3, l = 2) choose P(ǫ) = ǫ 2 + ǫ + 2 ( ) 1 1 g = 1 0 off-diagonal elements: coupling between subsystems later

9 Ring [Z(p)] l versus Galois field GF(p l ) ring [Z(p)] l Z(p)... Z(p) Addition and multiplication: (α λ ) + (β λ ) = (α λ + β λ ); (α λ )(β λ ) = (α λ β λ ) (0,...,0) zero; and (1,...,1) unity additive characters ( ) ψ[(α λ )] = ω α λ and 1 p l (α λ ) ψ[(α λ )(β λ )] = δ[(β λ ),(0)] compare harmonic analysis on GF(p l ) with harmonic analysis on [Z(p)] l at this stage compare ( ) ψ[(α λ β λ )] = ω α λ β λ with the χ(αβ) = ω λ,µ ( g λµ α λ β µ = ω g ij related to Galois multiplication rule λ λ λ ) ( ) α λ β λ = ω α λ β λ 5 λ

10 Galois quantum systems tensor product of l spaces: H = H... H H is p-dimensional H is p l -dimensional e.g., l coupled spins j = (p 1)/2 in this space: Galois systems with position/momentum in GF(p l ) position states in H X; m X; m 0... X; m l 1 m = i m i ǫ i GF(p l ); m i Z(p) R-systems with position/momentum in the ring Z(p)... Z(p). Position states X;(m λ ) X; m 0... X; m l 1 6

11 Fourier transform in Galois systems: F = (p l ) 1/2 χ(mn) X; m X; n m,n = (p l ) 1/2 ω g ij m i n j m i,n j i,j X; m 0 X; n 0... X; m l 1 X; n l 1 g ij related to Galois theory Fourier transform in R-systems: F... F = (p l ) 1/2 ω m i,n i i,j m i n i different from F X; m 0 X; n 0... X; m l 1 X; n l 1

12 Hamiltonian for Galois systems: h = h(ˆq, ˆP) = h(ˆq, F ˆQF ) special coupling between component systems related to off-diagonal g ij which embodies Galois theory in contrast: Hamiltonian for R-systems : h = h ( ˆQ 0, ˆP 0 ;...; ˆQ l 1, ˆP l 1 ) arbitrary coupling between component systems h special case of h quantum engineering of a Galois system: l spins with j = (p 1)/2 described with the Hamiltonian h. momentum states: P; m = F X; m = P; m 0... P; m l 1 m = i m i ǫ i = i m i E i displacement operators: as before with trace Z α n χ(αn) X; n X; n

13 GF(p l ) GF(p l ) phase space: general displacements D(α, β) = Z α X β χ( 2 1 αβ) relationship between displacement operator and displacement operators in component systems D(α, β) = D(ᾱ 0, β 0 )... D(ᾱ l 1, β l 1 ) symplectic transformations Sp(2, GF(p l )): previous formulas with trace uses Galois multiplication marginal properties of Wigner and Weyl functions with respect to (X, P) axes, also valid with respect to other axes discrete isotropy of phase space

14 Frobenius transformations positions have the Frobenius property α α p α p2... α pl 1 α pl = α implications for physics Frobenius transformations: G m X; m p X; m = m P; m p P; m GG = 1; G l = 1; [G, F] = 0 {1, G,..., G l 1 } = Z(l) Galois group (algebraic concept) in harmonic analysis G λ X; m = X; m pλ G λ P; m = P; m pλ ) G λ D(α, β)(g ) λ = D (α pλ, β pλ for α, β Z p the G commutes with D(α, β). 7

15 if G commutes with the Hamiltonian h: density submatrices (p l p l matrices) ρ λµ (t) = (λ)ρ(t) (µ) each submatrix evolves independently: ρ λµ (t) = exp(iht)ρ λµ (0)exp( iht) eigenvalues of ρ λλ (t) constant in time ρ λµ (t) with λ µ in general not normal singular values of ρ λµ (t) constant in time l 1 constants of motion tr[ρ(t) (λ)] = tr[ρ(0) (λ)]

16 example: GF(9)(where p = 3, l = 2) choose P(ǫ) = ǫ 2 + ǫ + 2 G = (0) (1) If hamiltonian commutes with G tr[ρ (1)] = 1 2 [ρ X(ǫ, ǫ) + ρ X (1 + ǫ,1 + ǫ) constant in time Notation: + ρ X (2 + ǫ,2 + ǫ) + ρ X (1 + 2ǫ,1 + 2ǫ) + ρ X (2ǫ,2ǫ) + ρ X (2 + 2ǫ,2 + 2ǫ) ρ X (2 + 2ǫ, ǫ) ρ X (ǫ,2 + 2ǫ) ρ X (2ǫ,1 + ǫ) ρ X (1 + ǫ,2ǫ) ρ X (1 + 2ǫ,2 + ǫ) ρ X (2 + ǫ,1 + 2ǫ)] ρ X (m, n) = X; m ρ X; n tr[ρ (0)] also constant in time (not independent)

17 Discussion d-dimensinal Hilbert space Phase space Z(d) Z(d) Displacements, displaced parity operators Z(p) GF(p) Phase space: finite geometry symplectic transformations well defined GF(p l ) H = H... H Fourier transform, displacements, etc similar expressions as before with Galois trace quantum engineering of such system: l spins with j = (p 1)/2 described with the Hamiltonian h (special coupling) Frobenius transformations and Galois group: algebraic concepts in harmonic analysis physically: constants of motion J. Phys. A38, 8453 (2005) J. Math. Phys. 47, (2006) Acta Appl. Math.93, 197 (2006) J. Fourier Anal. Appl. 14, 102 (2008) reviews: Rep. Prog. Phys. 67 (2004) 267 JPA 40, R285 (2007) 8