Spin code

Description

Encodes \(K\)-dimensional Hilbert space into a tensor-product or direct sum of factors, with each factor spanned by states of a quantum mechanical spin or, more generally, an irreducible representation of a Lie group.

In the simplest case of a spin half (i.e., a qubit), the canonical states \(|^J_m\rangle\) of a single \(2J+1\)-dimensional factor are labeled by total angular momentum \(J\) (either integer or half-integer) and its \(z\)-axis projection \(m\). There can be multiple factors of the same size, as in the case of atomic or molecular state spaces, and the number of factors can be infinite. In contrast to other qudit codes, spin codes are closely associated with the angular momentum Lie algebra and/or \(SU(2)\), \(SO(3)\), or more general Lie groups.