Unital Quantum Channels – Convex Structure and Revivals of Birkhoff’s Theorem

The set of doubly-stochastic quantum channels and its subset of mixtures of unitaries are investigated. We provide a detailed analysis of their structure together with computable criteria for the separation of the two sets. When applied to O(d)-covariant channels this leads to a complete characterization and reveals a remarkable feature: instances of channels which are not in the convex hull of unitaries can become elements of this set by either taking two copies of them or supplementing with a completely depolarizing channel. These scenarios imply that a channel whose noise initially resists any environment-assisted attempt of correction can become perfectly correctable.

[1]  A. Balakrishnan.  Applied Functional Analysis , 1976 .

[2]  A. Jamiołkowski.  Linear transformations which preserve trace and positive semidefiniteness of operators , 1972 .

[3]  Bill Rosgen,et al.  Additivity and distinguishability of random unitary channels , 2008, 0804.1936.

[4]  Oliver Rudolph.  On extremal quantum states of composite systems with fixed marginals , 2004, quant-ph/0406021.

[5]  John A. Smolin,et al.  Entanglement of assistance and multipartite state distillation , 2005 .

[6]  Motohisa Fukuda.  Simplification of Additivity Conjecture in Quantum Information Theory , 2007, Quantum Inf. Process..

[7]  Extremal quantum states in coupled systems , 2003, quant-ph/0307182.

[8]  M. Wolf,et al.  Simplifying additivity problems using direct sum constructions , 2007, 0704.1092.

[9]  John Watrous.  Mixing doubly stochastic quantum channels with the completely depolarizing channel , 2009, Quantum Inf. Comput..

[10]  Eberhard Zeidler,et al.  Applied Functional Analysis: Main Principles and Their Applications , 1995 .

[11]  R. Werner,et al.  Counterexample to an additivity conjecture for output purity of quantum channels , 2002, quant-ph/0203003.

[12]  C. King.  Additivity for unital qubit channels , 2001, quant-ph/0103156.

[13]  Alvaro Arias,et al.  Fixed points of quantum operations , 2002 .

[14]  David P. DiVincenzo,et al.  Entanglement of Assistance , 1998, QCQC.

[15]  L. J. Landau,et al.  On Birkhoff's theorem for doubly stochastic completely positive maps of matrix algebras , 1993 .

[16]  R. Werner,et al.  Entanglement measures under symmetry , 2000, quant-ph/0010095.

[17]  Man-Duen Choi.  Completely positive linear maps on complex matrices , 1975 .

[18]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[19]  G. Vidal,et al.  Computable measure of entanglement , 2001, quant-ph/0102117.

[20]  Francesco Buscemi.  On the minimum number of unitaries needed to describe a random-unitary channel , 2006 .

[21]  D. Petz,et al.  Contractivity of positive and trace-preserving maps under Lp norms , 2006, math-ph/0601063.

[22]  Volker Mehrmann,et al.  A quaternion QR algorithm , 1989 .

[23]  Hans Maassen,et al.  The essentially commutative dilations of dynamical semigroups onMn , 1987 .

[24]  R. F. Werner,et al.  Quantum lost and found , 2002, quant-ph/0209025.

[25]  H. Faßbender,et al.  Some observations on the Youla form and conjugate-normal matrices , 2007 .

[26]  Koenraad M.R. Audenaert,et al.  On random unitary channels , 2007, 0709.0824.