The structure of degradable quantum channels

被引:92
作者
Cubitt, Toby S. [1 ]
Ruskai, Mary Beth [2 ]
Smith, Graeme [3 ]
机构
[1] Univ Bristol, Dept Math, Bristol BS8 1TW, Avon, England
[2] Tufts Univ, Dept Math, Medford, MA 02155 USA
[3] IBM Corp, Thomas J Watson Res Ctr, Yorktown Hts, NY 10598 USA
基金
美国国家科学基金会; 英国工程与自然科学研究理事会;
关键词
D O I
10.1063/1.2953685
中图分类号
O4 [物理学];
学科分类号
0702 ;
摘要
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 an environment that is "small" in the sense of Phi(C). Such channels include all those with qubit or qutrit output, those that map some pure state to an output with full rank, and all those which can be represented using simultaneously diagonal Kraus operators, even in a non-orthogonal basis. 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)>1/4d(B)(2). These examples can also be used to give a negative answer to the question of whether additivity of the coherent information is helpful for establishing additivity for the Holevo capacity of a pair of channels. In the case of channels with diagonal Kraus operators, we describe the subclasses that 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. (C) 2008 American Institute of Physics. [DOI: 10.1063/1.2953685]
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页数:27
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