Quantum Information and Quantum Computation
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Quantum capacity of a qubit Pauli channel

What is the unassisted quantum capacity of a general qubit Pauli channel? Let \(\mathbf p=(p_I,p_X,p_Y,p_Z)\) be a probability vector, with \(p_i\geq0\) for \(i\in\{I,X,Y,Z\}\) and \(\sum_{i\in\{I,X,Y,Z\}}p_i=1\). Define

\begin{equation} \Lambda_{\mathbf p}(\rho) =p_I\rho+p_XX\rho X+p_YY\rho Y+p_ZZ\rho Z, \tag{1} \end{equation}

where \(\rho\) is a qubit density operator and \(X,Y,Z\) are the Pauli matrices. The unassisted quantum capacity \(\mathcal Q(\Lambda_{\mathbf p})\) is the supremum of asymptotic qubit transmission rates achievable with vanishing error over independent channel uses. Arbitrary block encodings and joint decoding are allowed, without classical communication assistance or preshared entanglement. Equivalently,

\begin{equation} \mathcal Q(\Lambda_{\mathbf p}) =\sup_{n\geq1}\frac1n\max_{\rho_{A^n}} I_c\!\left(\rho_{A^n},\Lambda_{\mathbf p}^{\otimes n}\right), \tag{2} \end{equation}

where \(\rho_{A^n}\) ranges over all states of \(n\) input qubits [Dev05], Proposition 7. The coherent information in Eq. (2) is

\begin{equation} I_c(\rho,\mathcal N) =S(\mathcal N(\rho))-S(\mathcal N^c(\rho)), \tag{3} \end{equation}

where \(\mathcal N^c\) is a complementary channel and \(S(\sigma)=-\operatorname{Tr}(\sigma\log_2\sigma)\) is the von Neumann entropy used in Eq. (3). Determine \(\mathcal Q(\Lambda_{\mathbf p})\) in Eq. (2) for the channel of Eq. (1) throughout the probability simplex, including the exact boundary between zero and positive capacity.

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Entanglement cost of an amplitude-damping-channel Choi state

What is the entanglement cost of the Choi state of the qubit amplitude-damping channel

\begin{equation} \mathcal A_p(\rho)=A_0\rho A_0^\dagger+A_1\rho A_1^\dagger, \qquad 0\le p\le1? \tag{1} \end{equation}

The Kraus operators in Eq. (1) are

\begin{equation} \begin{aligned} A_0&=\lvert0\rangle\!\langle0\rvert +\sqrt{1-p}\,\lvert1\rangle\!\langle1\rvert =\begin{pmatrix}1&0\\0&\sqrt{1-p}\end{pmatrix},\\ A_1&=\sqrt p\,\lvert0\rangle\!\langle1\rvert =\begin{pmatrix}0&\sqrt p\\0&0\end{pmatrix}. \end{aligned} \tag{2} \end{equation}

In Eq. (2), \(p\) is the decay probability of the excited state. Let \(\lvert\Phi^+\rangle_{RA}=(\lvert00\rangle+\lvert11\rangle)/\sqrt2\). The normalized Choi state of the channel in Eq. (1) is

\begin{equation} \begin{aligned} \omega_p^{RB} &:=(\operatorname{id}_R\otimes\mathcal A_p) (\lvert\Phi^+\rangle\!\langle\Phi^+\rvert_{RA})\\ &=\frac12\Bigl[ \lvert00\rangle\!\langle00\rvert +\sqrt{1-p}\bigl(\lvert00\rangle\!\langle11\rvert +\lvert11\rangle\!\langle00\rvert\bigr) +(1-p)\lvert11\rangle\!\langle11\rvert +p\lvert10\rangle\!\langle10\rvert \Bigr]. \end{aligned} \tag{3} \end{equation}

Here the first and second entries in each ket in Eq. (3) label \(R\) and \(B\), respectively. Thus the question is to determine \(E_C(\omega_p)\), the asymptotic number of ebits per copy required to prepare many copies of \(\omega_p\) by local operations and classical communication.

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