The structure of degradable quantum channels

Toby S. Cubitt; Mary-Beth Ruskai; Graeme Smith

doi:10.1063/1.2953685

paper

The structure of degradable quantum channels

arXiv:0802.1360 · doi:10.1063/1.2953685

Abstract

Degradable quantum channels are among the only channels whose quantum and private classical capacities are known. As such, determining the structure of these channels is a pressing open question in quantum information theory. We give a comprehensive review of what is currently known about the structure of degradable quantum channels, including a number of new results as well as alternate proofs of some known results. In the case of qubits, we provide a complete characterization of all degradable channels with two dimensional output, give a new proof that a qubit channel with two Kraus operators is either degradable or anti-degradable and present a complete description of anti-degradable unital qubit channels with a new proof. For higher output dimensions we explore the relationship between the output and environment dimensions ( $d_{B}$ and $d_{E}$ respectively) of degradable channels. For several broad classes of channels we show that they can be modeled with a environment that is "small" in the sense $d_{E} \leq d_{B}$ . Perhaps surprisingly, we also present examples of degradable channels with ``large'' environments, in the sense that the minimal dimension $d_{E} > d_{B}$ . Indeed, one can have $d_{E} > \frac{1}{4} d_{B}^{2}$ . In the case of channels with diagonal Kraus operators, we describe the subclass which are complements of entanglement breaking channels. We also obtain a number of results for channels in the convex hull of conjugations with generalized Pauli matrices. However, a number of open questions remain about these channels and the more general case of random unitary channels.

42 pages, 3 figures, Web and paper abstract differ; (v2 contains only minor typo corrections)

The structure of degradable quantum channels

Abstract

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