“Pretty Strong” Converse for the Quantum Capacity of Degradable Channels

We exhibit a possible road toward a strong converse for the quantum capacity of degradable channels. In particular, we show that all degradable channels obey what we call a “pretty strong” converse: when the code rate increases above the quantum capacity, the fidelity makes a discontinuous jump from 1 to at most 1/√2, asymptotically. A similar result can be shown for the private (classical) capacity. Furthermore, we can show that if the strong converse holds for symmetric channels (which have quantum capacity zero), then degradable channels obey the strong converse. The above-mentioned asymptotic jump of the fidelity at the quantum capacity then decreases from 1 to 0.

[1]  Naresh Sharma,et al.  On the strong converses for the quantum channel capacity theorems , 2012, ArXiv.

[2]  P ? ? ? ? ? ? ? % ? ? ? ? , 1991 .

[3]  Yingkai Ouyang.  Upper bounds on the quantum capacity of some quantum channels using the coherent information of other channels , 2011 .

[4]  Rudolf Ahlswede,et al.  Beiträge zur Shannonschen Informationstheorie im Falle nichtstationärer Kanäle , 1968 .

[5]  N. Datta,et al.  The apex of the family tree of protocols: optimal rates and resource inequalities , 2011, 1103.1135.

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

[7]  Peter W. Shor,et al.  Entanglement-assisted capacity of a quantum channel and the reverse Shannon theorem , 2001, IEEE Trans. Inf. Theory.

[8]  Richard Hughes.  Quantum Key Distribution , 2004 .

[9]  Renato Renner,et al.  Security of quantum key distribution , 2005, Ausgezeichnete Informatikdissertationen.

[10]  Amir Dembo,et al.  Large Deviations Techniques and Applications , 1998 .

[11]  R. Werner,et al.  Tema con variazioni: quantum channel capacity , 2003, quant-ph/0311037.

[12]  R. Renner,et al.  The Quantum Reverse Shannon Theorem Based on One-Shot Information Theory , 2009, 0912.3805.

[13]  F. Dupuis.  The decoupling approach to quantum information theory , 2010, 1004.1641.

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

[15]  T. Dorlas,et al.  Invalidity of a strong capacity for a quantum channel with memory , 2011, 1108.4282.

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

[17]  P. Hayden,et al.  Conjugate degradability and the quantum capacity of cloning channels , 2009, 0909.3297.

[18]  R. Renner,et al.  The Decoupling Theorem , 2011 .

[19]  R. Jozsa.  Fidelity for Mixed Quantum States , 1994 .

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

[21]  P. Shor,et al.  Quantum Error-Correcting Codes Need Not Completely Reveal the Error Syndrome , 1996, quant-ph/9604006.

[22]  Jeroen van de Graaf,et al.  Cryptographic Distinguishability Measures for Quantum-Mechanical States , 1997, IEEE Trans. Inf. Theory.

[23]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[24]  Andreas J. Winter,et al.  Coding theorem and strong converse for quantum channels , 1999, IEEE Trans. Inf. Theory.

[25]  Nilanjana Datta,et al.  Strong converses for classical information transmission and hypothesis testing , 2011 .

[26]  Tomohiro Ogawa,et al.  Strong converse to the quantum channel coding theorem , 1999, IEEE Trans. Inf. Theory.

[27]  A. Uhlmann.  The "transition probability" in the state space of a ∗-algebra , 1976 .

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

[29]  Marco Tomamichel,et al.  Chain Rules for Smooth Min- and Max-Entropies , 2012, IEEE Transactions on Information Theory.

[30]  Andreas J. Winter.  Coding theorems of quantum information theory , 1999 .

[31]  Andreas J. Winter,et al.  The Quantum Reverse Shannon Theorem and Resource Tradeoffs for Simulating Quantum Channels , 2009, IEEE Transactions on Information Theory.

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

[33]  R. Ahlswede.  Elimination of correlation in random codes for arbitrarily varying channels , 1978 .

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

[35]  M. Tomamichel.  A framework for non-asymptotic quantum information theory , 2012, 1203.2142.

[36]  J. Cirac,et al.  Entanglement cost of bipartite mixed states. , 2001, Physical Review Letters.

[37]  Naresh Sharma,et al.  Fundamental bound on the reliability of quantum information transmission , 2012, Physical review letters.

[38]  Sang Joon Kim,et al.  A Mathematical Theory of Communication , 2006 .

[39]  M. Berta.  Single-shot Quantum State Merging , 2009, 0912.4495.

[40]  Graeme Smith,et al.  Quantum Communication with Zero-Capacity Channels , 2008, Science.

[41]  K. Birgitta Whaley,et al.  Lower bounds on the nonzero capacity of Pauli channels , 2008 .

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

[43]  Peter W. Shor.  Capacities of quantum channels and how to find them , 2003, Math. Program..

[44]  M. Hayashi.  Optimal sequence of POVMs in the sense of Stein's lemma in quantum hypothesis testing , 2001, quant-ph/0107004.

[45]  M. Horodecki,et al.  Irreversibility for all bound entangled states. , 2005, Physical Review Letters.

[46]  R. Renner.  Symmetry of large physical systems implies independence of subsystems , 2007 .

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

[48]  W. Wootters.  Entanglement of Formation of an Arbitrary State of Two Qubits , 1997, quant-ph/9709029.

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

[50]  Christopher King,et al.  Properties of Conjugate Channels with Applications to Additivity and Multiplicativity , 2005 .

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

[52]  Kazuoki Azuma.  WEIGHTED SUMS OF CERTAIN DEPENDENT RANDOM VARIABLES , 1967 .

[53]  Joseph M. Renes,et al.  Noisy Channel Coding via Privacy Amplification and Information Reconciliation , 2010, IEEE Transactions on Information Theory.

[54]  Marco Tomamichel,et al.  Duality Between Smooth Min- and Max-Entropies , 2009, IEEE Transactions on Information Theory.

[55]  S. Wehner,et al.  A strong converse for classical channel coding using entangled inputs. , 2009, Physical review letters.

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

[57]  Masahito Hayashi,et al.  On error exponents in quantum hypothesis testing , 2004, IEEE Transactions on Information Theory.

[58]  M. Hayashi.  Optimal sequence of quantum measurements in the sense of Stein's lemma in quantum hypothesis testing , 2002, quant-ph/0208020.

[59]  Matthias Christandl,et al.  Postselection technique for quantum channels with applications to quantum cryptography. , 2008, Physical review letters.

[60]  Tomohiro Ogawa,et al.  A New Proof of the Direct Part of Stein's Lemma in Quantum Hypothesis Testing , 2001 .

[61]  Mario Berta,et al.  Entanglement cost of quantum channels , 2012, 2012 IEEE International Symposium on Information Theory Proceedings.

[62]  Nilanjana Datta,et al.  The Quantum Capacity of Channels With Arbitrarily Correlated Noise , 2009, IEEE Transactions on Information Theory.

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