Quantum systems with finite Hilbert space A. Vourdas University of Bradford

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1 Quantum systems with finite Hilbert space 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 Z(p) GF(p)(field): symplectic tr. 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: implications for physics Motivation: symplectic transf in discrete systems mutually unbiased bases classical/quantum information processing 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. 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 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 X d = Z d = 1 toroidal X β Z α = Z α X β ω( αβ) general displacements D(α, β) = Z α X β ω( 2 1 αβ) Heisenberg-Weyl group d = p then Z(p) field (inverses exist) Symplectic group: Sp(2, Z(p)) Wigner functions, marginal properties (with respect to various directions; phase space finite geometry) analytic representations with theta functions

4 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 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 : related to Galois multiplication rule (depends on irreducible polynomial) 3

5 (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

6 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 quantum mechanics on GF(p l ) with quantum mechanics 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 4 λ

7 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)] l. Position states X;(m λ ) X; m 0... X; m l 1 5

8 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 X; m 0 X; n 0... X; m l 1 X; n l 1 different from F momentum states in Galois systems P; m = F X; m = P; m 0... P; m l 1 m = i m i ǫ i = i m i E i momentum states in R-systems P; m = F... F X; m = P; m 0... P; m l 1

9 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. 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 )

10 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 }: cyclic group of order l. 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 position position to the power p λ momentum momentum to the power p λ 6

11 if G commutes with the Hamiltonian h: l 1 constants of motion tr[ρ(t) (λ)] = tr[ρ(0) (λ)] 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)

12 Discussion d-dimensinal Hilbert space Phase space Z(d) Z(d) Displacements, displaced parity operators prime d: symplectic tr well defined GF(p l ) H = H... H Fourier transform, displacements, etc quantum engineering of such system: l spins with j = (p 1)/2 described with the Hamiltonian h (special coupling corresponds to Galois multiplication rule) Frobenius transformations: constants of motion Rep. Prog. Phys. 67 (2004) 267 JMP 47, (2006) JPA 40, R285 (2007) J Fourier Anal Appl 14 (2008) 102 7