Two-way quantum capacity: amplitude-damping channel

Problem

What is the two-way quantum capacity \(\mathcal{Q}_2(\mathcal A_p)\) 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 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}

Equation (2) uses \(p\) as the decay probability of the excited state. Here \(\mathcal{Q}_2\) permits adaptive local operations and unlimited two-way classical communication between uses of the channel.

Source

The question is implicit in the nonmatching achievable and converse bounds for the amplitude-damping channel reported by Pirandola, Laurenza, Ottaviani, and Banchi [PLOB17].

Progress

Pirandola, Laurenza, Ottaviani, and Banchi bounded the two-way quantum capacity using an achievable reverse-coherent-information rate from below and the channel’s squashed entanglement from above:

\begin{equation} \max_{0\le u\le1}\bigl[h_2(u)-h_2(pu)\bigr] \le \mathcal{Q}_2(\mathcal A_p) \le h_2\!\left(\frac12-\frac p4\right) -h_2\!\left(1-\frac p4\right), \tag{3} \end{equation}

where \(h_2(x):=-x\log_2x-(1-x)\log_2(1-x)\), with \(0\log_2 0:=0\). The lower rate in Eq. (3) is achievable with a final round of backward classical communication, whereas the upper rate follows from a balanced amplitude-damping squashing channel [PLOB17].
Fawzi, Shayeghi, and Ta developed a symmetry-reduced hierarchy of semidefinite programs giving strong-converse upper bounds on two-way- and PPT-assisted quantum capacity. For the qubit amplitude-damping channel, their six-copy \(D^{\#}_2\) bound improves the previously best single-copy bound throughout the parameter range displayed in their numerical study. The hierarchy tightens the converse side but still does not meet the achievable lower bound in Eq. (3) [FST22].

Comment

The bounds in Eq. (3) do not coincide in general. Determining \(\mathcal{Q}_2(\mathcal A_p)\) therefore requires either an improved two-way-assisted protocol, a tighter converse bound, or both. This problem concerns the same amplitude-damping channel as Problem 2, but Problem 2 asks for a constructive code achieving the unassisted quantum capacity \(\mathcal{Q}(\mathcal A_p)\), whereas the present problem asks for the two-way quantum capacity \(\mathcal{Q}_2(\mathcal A_p)\).

References

[PLOB17]: S. Pirandola, R. Laurenza, C. Ottaviani, and L. Banchi, “Fundamental Limits of Repeaterless Quantum Communications,” Nature Communications 8, 15043 (2017).DOI link
[FST22]: O. Fawzi, A. Shayeghi, and H. Ta, “A Hierarchy of Efficient Bounds on Quantum Capacities Exploiting Symmetry,” IEEE Transactions on Information Theory 68, 7346–7360 (2022).DOI arXiv

Two-way quantum capacity: amplitude-damping channel

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