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 \( \mathcal{H} \) 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 \( \mathcal{K} \) under consideration is received as an output state ρ′ of the form $$\rho ' = \frac{{\sum\nolimits_j {{N_j}_\rho N_j^\dag } }}{{T{r_\rho }\sum\nolimits_j {N_j^\dag {N_j}} }},{N_j} \in \mathcal{N}$$ (5.1.1) where N1, \( \mathcal{N} \)2,… is an arbitrary finite subset of operators in a subspace \( \mathcal{N} \) of \( \mathcal{B} \)(\( \mathcal{H} \)), the algebra of all operators in \( \mathcal{H} \). We shall assume that the noise is moderate in the sense that the dimension of \( \mathcal{N} \) is ‘small’ relative to the dimension of \( \mathcal{B} \)(\( \mathcal{H} \)). We call \( \mathcal{N} \) the noise space of the channel \( \mathcal{K} \) and any element of \( \mathcal{K} \) a noise or error operator. If |ψ〉 ∈ \( \mathcal{H} \) 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 $$\rho ' = \frac{{\sum\nolimits_j {{N_j}\langle \psi |\langle \psi |N_j^\dag } }}{{\sum\nolimits_j {{N_j}\psi {^2}} }}.$$ (5.1.2) 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 \( \mathcal{N} \).