On 1-qubit channels (Q2766221)

Let \(T\) be a quantum channel, i.e. a complete positive and trace preserving linear map on the Banach space of trace class operators of a Hilbert space. A measure of how effective the channel transmits a given density operator \(\rho\) is NEWLINE\[NEWLINE H_T(\rho) = \max \Bigl\{ \sum p_j S(T\rho \|T\rho_j) |\rho = \sum p_j\rho_j \Bigr\}, NEWLINE\]NEWLINE where \(S(.\|.)\) is the relative entropy, and the Holevo capacity is NEWLINE\[NEWLINE C(T) = \max \{ H_T(\rho)\mid 0 \leq \rho, \text{ tr} = 1 \}. NEWLINE\]NEWLINE Let S denote the von Neumann entropy, and NEWLINE\[NEWLINE E_T(\rho) = \min \Bigl\{ \pi_j \mid \rho = \sum p_j \pi_j , \pi_j \text{ pure} \Bigr\}. NEWLINE\]NEWLINE Then NEWLINE\[NEWLINE H_T(\rho) = S(T\rho) - E_T(\rho). NEWLINE\]NEWLINE In the case that \(\rho\) is the state of a bipartite system and \(T\) is the partial trace w.r.t. one component, \(E_T\) is known as the entangelement of formation. For \({\mathbb C}^2 \otimes {\mathbb C}^2\) systems there is the Wootters formula, a closed expression for \(E_T\). The author adapts the well known method to derive this formula to get a more simple expression for the Holevo capacity of one qubit (\(T\) being of rank two) channels. The result is an estimate of \(H_T(\rho)\) and a representation of \(C(T)\) as the maximum of a real function on \((0,1]\). For a special class of one qubit channels further results are derived.