2a GF-2 1次元Ising-like spin系の集団モード

晴男 吉田, 琢磨 石川

Hiroshi Mamada, Fumihiko Takano

義秋 田中, 典清 瓜生

鑑子 大野

Apostol Vourdas

A quantum system with positions and momenta in GF(pℓ) is considered. Such a system can be constructed from ℓ smaller systems, in which the positions and momenta take values in Zp, if the Hamiltonian of this ℓ-partite system is compatible with GF(pℓ). The concept of compatibility of a Hamiltonian with GF(pℓ) allows the quantum formalism in the ℓ-partite system to be expressed in terms of Galois arithmetic. Transformations of the basis in GF(pℓ) produce unitary transformations of the quantum states, which form a representation of GL(ℓ,Zp). They are used to define which subset of the general set of Hamiltonians in the ℓ-partite system is compatible with GF(pℓ).

Ching–Yi Lai, Todd A. Brun, MacWilliams Identity

• Consider the Galois Field GF (4) = {0, 1, ω, ω2 = ω} with 1 + ω + ω2 = 0 and ω3 = 1. • For g1 = ωu1 + ωv1, g2 = ωu2 + ωv2 ∈ GF (4)n, we define a symplectic inner product ∗ in GF (4)n by g1 ∗ g2 = u1 · v2 + u2 · v1 mod 2, where · is the usual binary inner product. • The orthogonal group of a subgroup C of GF (4)n with respect to ∗ is defined by C⊥ = {g ∈ GF (4) : g ∗ h = 0, ∀h ∈ C}. If C is an (n, 2n−k) additive code, then C⊥ is an (n, 2n+k) additive code.