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Abstract: Degradable quantum channels are among the only channels whose quantum andprivate classical capacities are known. As such, determining the structure ofthese channels is a pressing open question in quantum information theory. Wegive a comprehensive review of what is currently known about the structure ofdegradable quantum channels, including a number of new results as well asalternate proofs of some known results. In the case of qubits, we provide acomplete characterization of all degradable channels with two dimensionaloutput, give a new proof that a qubit channel with two Kraus operators iseither degradable or anti-degradable and present a complete description ofanti-degradable unital qubit channels with a new proof.For higher output dimensions we explore the relationship between the outputand environment dimensions $d B$ and $d E$ respectively of degradablechannels. For several broad classes of channels we show that they can bemodeled with a environment that is -small- in the sense $d E \leq d B$. Perhapssurprisingly, we also present examples of degradable channels with ``large-environments, in the sense that the minimal dimension $d E > d B$. Indeed, onecan have $d E > \tfrac{1}{4} d B^2$.In the case of channels with diagonal Kraus operators, we describe thesubclass which are complements of entanglement breaking channels. We alsoobtain a number of results for channels in the convex hull of conjugations withgeneralized Pauli matrices. However, a number of open questions remain aboutthese channels and the more general case of random unitary channels.

Autor: Toby S. Cubitt, Mary-Beth Ruskai, Graeme Smith



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