The quantum dynamic capacity formula of a quantum channel

The dynamic capacity theorem characterizes the reliable communication rates of a quantum channel when combined with the noiseless resources of classical communication, quantum communication, and entanglement. In prior work, we proved the converse part of this theorem by making contact with many previous results in the quantum Shannon theory literature. In this work, we prove the theorem with an “ab initio” approach, using only the most basic tools in the quantum information theorist’s toolkit: the Alicki-Fannes’ inequality, the chain rule for quantum mutual information, elementary properties of quantum entropy, and the quantum data processing inequality. The result is a simplified proof of the theorem that should be more accessible to those unfamiliar with the quantum Shannon theory literature. We also demonstrate that the “quantum dynamic capacity formula” characterizes the Pareto optimal trade-off surface for the full dynamic capacity region. Additivity of this formula reduces the computation of the trade-off surface to a tractable, textbook problem in Pareto trade-off analysis, and we prove that its additivity holds for the quantum Hadamard channels and the quantum erasure channel. We then determine exact expressions for and plot the dynamic capacity region of the quantum dephasing channel, an example from the Hadamard class, and the quantum erasure channel. Quantum Shannon theory is the study of the transmission capabilities of a noisy resource when a large number of independent and identically distributed (IID) copies of the resource are available. An important task in this area of study is to determine how noiseless resources interact with a noisy quantum channel. That is, we would like to know the reliable communication rates if a sender can use noiseless resources in addition to a quantum channel to generate other noiseless resources. In prior work, we have studied one such setting, where a sender and receiver generate or consume classical communication, quantum communication, and entanglement along with the consumption of a noisy quantum channel [11, 12, 13]. The result of these efforts was a characterization of the “dynamic capacity region” of a noisy quantum channel. One of the shortcomings of the characterization of the dynamic capacity region in Refs. [11, 12, 13] is that its computation for a general quantum channel is intractable. Though, later, Brádler et al. demonstrated that the quantum Hadamard channels [14] are a natural class of channels for which the computation of one octant of the region is tractable [7] because the structure of these channels appears to be “just right” for this tractability to hold. The proof considers special quadrants of the dynamic capacity region and employs several reductio ad absurdum arguments to characterize one octant of the full region. The work of Brádler et al. showed that we can claim a complete understanding of the abilities of an entanglement-assisted quantum Hadamard channel for the transmission of classical and quantum information. The aim of the present work is two-fold: 1) to simplify the proof of the dynamic capacity theorem in Ref. [13] and 2) to show that there is one important formula to consider for any task in dynamic quantum 1The first few chapters of Yard’s thesis provide an introductory and accessible overview of the subject [22]. 1 ar X iv :1 00 4. 04 58 v1 [ qu an tph ] 3 A pr 2 01 0 Shannon theory. Our previous proof in Ref. [13] relies extensively on prior literature in quantum Shannon theory, making our ideas inaccessible to an audience unfamiliar with this increasingly “tangled web.” Here, we apply an “ab initio” approach to the proof, using only four tools from quantum information theory: the Alicki-Fannes’ inequality [1], the chain rule for quantum mutual information, elementary properties of quantum entropy, and the quantum data processing inequality [18]. As such, the proof here should be more accessible to a broader audience and more streamlined because it is “disentangled” from the complex web that the quantum Shannon theory literature has become. We also propose a new formula that characterizes any task involving classical communication, quantum communication, and entanglement in dynamic quantum Shannon theory. For this reason, we call this formula the “quantum dynamic capacity formula.” In particular, additivity of this formula implies a complete understanding of any task in dynamic quantum Shannon theory involving the three fundamental noiseless resources. We find a simplified, direct proof that additivity holds for the Hadamard class of channels and for a quantum erasure channel—thus their full dynamic capacity regions are tractable. The additivity proof for the quantum erasure channel is different from that of the Hadamard channel—it exploits the particular structure of the quantum erasure channel. We structure this work as follows. The next section reviews the minimal tools from quantum information theory necessary to understand the rest of the paper. Section 2 states the dynamic capacity theorem, and Section 3 contains a brief review of the proof of the achievability part of the dynamic capacity theorem. The main protocol for proving this part is the “classically-enhanced father protocol,” whose detailed proof we gave in Ref. [11]. Section 4 contains the converse proof, where we proceed with the minimal tools stated above. We then show in Section 6 how the quantum dynamic capacity formula characterizes the optimization task for computing the Pareto optimal trade-off surface for the dynamic capacity region. Sections 7 and 9 prove that the quantum dynamic capacity formula is additive for the Hadamard class of channels and the quantum erasure channel, respectively. We directly compute and plot the region for a qubit dephasing channel, which is a channel that falls within the Hadamard class, and the quantum erasure channel. Finally, we conclude with a brief discussion. 1 Definitions and notation We first establish some definitions and notation that we employ throughout the paper and review a few important properties of the entropy. Let Φ denote the maximally entangled state between two parties: |Φ〉 ≡ 1 √ D D ∑