Quantum Error Correction

Abstract

We look at the problem of communication of states of a finite level quantum system through a channel disturbed by noise. Such a quantum system is described by a finite dimensional complex Hilbert space ℋ $$ \mathcal{H} $$ and its states are density operators, in other words, positive hermitian operators of unit trace. We assume that any such state ρ, when transmitted through the channel K $$ \mathcal{K} $$ under consideration is received as an output state ρ′ of the form (5.1.1) ρ'= ∑ j N j ρ N j † T r ρ ∑ j N j † N j , N j ∈N $$\rho ' = \frac{{\sum\nolimits_j {{N_j}_\rho N_j^\dag } }}{{T{r_\rho }\sum\nolimits_j {N_j^\dag {N_j}} }},{N_j} \in \mathcal{N}$$ where N1, N $$ \mathcal{N} $$ 2,… is an arbitrary finite subset of operators in a subspace N $$ \mathcal{N} $$ of ℬ $$ \mathcal{B} $$ ( ℋ $$ \mathcal{H} $$ ), the algebra of all operators in ℋ $$ \mathcal{H} $$ . We shall assume that the noise is moderate in the sense that the dimension of N $$ \mathcal{N} $$ is ‘small’ relative to the dimension of ℬ $$ \mathcal{B} $$ ( ℋ $$ \mathcal{H} $$ ). We call N $$ \mathcal{N} $$ the noise space of the channel K $$ \mathcal{K} $$ and any element of K $$ \mathcal{K} $$ a noise or error operator. If |ψ〉 ∈ ℋ $$ \mathcal{H} $$ is a unit vector and ρ = |ψ〉 〈ψ|, the corresponding pure state density operator then the output state ρ′ of (5.1.1) assumes the form (5.1.2) ρ'= ∑ j N j 〈ψ|〈ψ| N j † ∑ j ‖ N j ψ ‖ 2 . $$\rho ' = \frac{{\sum\nolimits_j {{N_j}\langle \psi |\langle \psi |N_j^\dag } }}{{\sum\nolimits_j {{N_j}\psi {^2}} }}.$$ It is to be noted that this output state need not be pure. If the same input state ρ is transmitted repeatedly the noise operators {N j } in (5.1.1) can differ for different transmissions and the output states can be different. However, the noise operators come from the same noise space N $$ \mathcal{N} $$ .