Approximate degradable quantum channels

Degradable quantum channels are an important class of completely positive trace-preserving maps. Among other properties, they offer a single-letter formula for the quantum and the private classical capacity and are characterized by the fact that the complementary channel can be obtained from the channel by applying a degrading map. In this work we introduce the concept of approximate degradable channels, which satisfy this condition up to some finite ε ≥ 0. That is, there exists a degrading map which upon composition with the channel is ε-close in the diamond norm to the complementary channel. We show that for any fixed channel the smallest such ε can be efficiently determined via a semidefinite program. Moreover, these approximate degradable channels also approximately inherit all other properties of degradable channels. As an application, we derive improved upper bounds to the quantum and private classical capacity for certain channels of interest in quantum communication.

[1]  Igor Devetak.  The private classical capacity and quantum capacity of a quantum channel , 2005, IEEE Transactions on Information Theory.

[2]  P. Shor,et al.  QUANTUM-CHANNEL CAPACITY OF VERY NOISY CHANNELS , 1997, quant-ph/9706061.

[3]  P. Shor,et al.  The Capacity of a Quantum Channel for Simultaneous Transmission of Classical and Quantum Information , 2003, quant-ph/0311131.

[4]  Igor Devetak,et al.  Capacity theorems for quantum multiple-access channels: classical-quantum and quantum-quantum capacity regions , 2008, IEEE Transactions on Information Theory.

[5]  Howard Barnum,et al.  On quantum fidelities and channel capacities , 2000, IEEE Trans. Inf. Theory.

[6]  Joseph M Renes,et al.  Structured codes improve the Bennett-Brassard-84 quantum key rate. , 2008, Physical review letters.

[7]  Schumacher,et al.  Quantum data processing and error correction. , 1996, Physical review. A, Atomic, molecular, and optical physics.

[8]  Michael J. Todd,et al.  Polynomial Algorithms for Linear Programming , 1988 .

[9]  M. Fannes.  A continuity property of the entropy density for spin lattice systems , 1973 .

[10]  Eric M. Rains.  A semidefinite program for distillable entanglement , 2001, IEEE Trans. Inf. Theory.

[11]  M. Ruskai.  Some Open Problems in Quantum Information Theory , 2007, 0708.1902.

[12]  Thomas M. Cover,et al.  Network Information Theory , 2001 .

[13]  Andreas J. Winter,et al.  Tight Uniform Continuity Bounds for Quantum Entropies: Conditional Entropy, Relative Entropy Distance and Energy Constraints , 2015, ArXiv.

[14]  W. Stinespring.  Positive functions on *-algebras , 1955 .

[15]  Guillaume Aubrun.  On Almost Randomizing Channels with a Short Kraus Decomposition , 2008, 0805.2900.

[16]  Graeme Smith,et al.  Additive extensions of a quantum channel , 2007, 2008 IEEE Information Theory Workshop.

[17]  D. Bruß,et al.  Optimal universal and state-dependent quantum cloning , 1997, quant-ph/9705038.

[18]  Alexander Schrijver,et al.  Theory of linear and integer programming , 1986, Wiley-Interscience series in discrete mathematics and optimization.

[19]  J. Smolin,et al.  Degenerate quantum codes for Pauli channels. , 2006, Physical review letters.

[20]  R. Schumann.  Quantum Information Theory , 2000, quant-ph/0010060.

[21]  Stephen P. Boyd,et al.  Convex Optimization , 2004, Algorithms and Theory of Computation Handbook.

[22]  E. Rains.  Bound on distillable entanglement , 1998, quant-ph/9809082.

[23]  Cerf,et al.  Pauli cloning of a quantum Bit , 2000, Physical review letters.

[24]  M. Fannes,et al.  Continuity of quantum conditional information , 2003, quant-ph/0312081.

[25]  Charles H. Bennett,et al.  Mixed-state entanglement and quantum error correction. , 1996, Physical review. A, Atomic, molecular, and optical physics.

[26]  Andreas J. Winter,et al.  The Quantum Capacity With Symmetric Side Channels , 2008, IEEE Transactions on Information Theory.

[27]  D. Leung,et al.  Continuity of Quantum Channel Capacities , 2008, 0810.4931.

[28]  D. Leung,et al.  Quantum and private capacities of low-noise channels , 2017, 2017 IEEE Information Theory Workshop (ITW).

[29]  Ning Cai,et al.  Quantum privacy and quantum wiretap channels , 2004, Probl. Inf. Transm..

[30]  M. Wilde.  Quantum Information Theory: Noisy Quantum Shannon Theory , 2013 .

[31]  Volkher B. Scholz,et al.  Approximate Degradable Quantum Channels , 2014, IEEE Transactions on Information Theory.

[32]  K. Audenaert.  A sharp continuity estimate for the von Neumann entropy , 2006, quant-ph/0610146.

[33]  Elliott H. Lieb,et al.  A Fundamental Property of Quantum-Mechanical Entropy , 1973 .

[34]  Schumacher,et al.  Sending entanglement through noisy quantum channels. , 1996, Physical review. A, Atomic, molecular, and optical physics.

[35]  David Elkouss,et al.  Unbounded number of channel uses may be required to detect quantum capacity , 2014, Nature Communications.

[36]  M. Ruskai,et al.  The structure of degradable quantum channels , 2008, 0802.1360.

[37]  David Elkouss,et al.  Superadditivity of private information for any number of uses of the channel , 2015, Physical review letters.

[38]  Graeme Smith.  Private classical capacity with a symmetric side channel and its application to quantum cryptography , 2007, 0705.3838.

[39]  M. Nielsen,et al.  Information transmission through a noisy quantum channel , 1997, quant-ph/9702049.

[40]  A. Winter,et al.  Quantum privacy and quantum wiretap channels , 2004 .

[41]  John Watrous,et al.  Semidefinite Programs for Completely Bounded Norms , 2009, Theory Comput..

[42]  S. Lloyd.  Capacity of the noisy quantum channel , 1996, quant-ph/9604015.

[43]  M. Wolf,et al.  Quantum capacities of channels with small environment , 2006, quant-ph/0607070.

[44]  Yingkai Ouyang,et al.  Channel covariance, twirling, contraction, and some upper bounds on the quantum capacity , 2011, Quantum Inf. Comput..

[45]  Dennis Kretschmann,et al.  The Information-Disturbance Tradeoff and the Continuity of Stinespring's Representation , 2008, IEEE Transactions on Information Theory.