Secret Message Transmission over Quantum Channels under Adversarial Quantum Noise: Secrecy Capacity and Super-activations

We determine the secrecy capacities of AVQCs (arbitrarily varying quantum channels). Both secrecy capacity with average error probability and with maximal error probability are derived. Both derivations are based on one common code construction. The code we construct fulfills a stringent secrecy requirement, which is called the strong code concept. We determine when the secrecy capacity is a continuous function of the system parameters and completely characterize its discontinuity points both for average error criterion and for maximal error criterion. Furthermore, we prove the phenomenon “super-activation” for secrecy capacities of AVQCs, i.e., two quantum channels both with zero secrecy capacity, which, if used together, allow secure transmission with positive capacity. We also discuss the relations between the entanglement distillation capacity, the entanglement generating capacity, and the strong subspace transmission capacity for AVQCs.

[1]  Holger Boche,et al.  Secrecy results for compound wiretap channels , 2011, Probl. Inf. Transm..

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

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

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

[5]  I. Devetak,et al.  The private classical information capacity and quantum information capacity of a quantum channel , 2003 .

[6]  Moritz Wiese,et al.  The Arbitrarily Varying Wiretap Channel - deterministic and correlated random coding capacities under the strong secrecy criterion , 2014, ArXiv.

[7]  Adam Paszkiewicz,et al.  On quantum information , 2012, ArXiv.

[8]  V. Paulsen.  Completely Bounded Maps and Operator Algebras: Completely Bounded Multilinear Maps and the Haagerup Tensor Norm , 2003 .

[9]  H. Vincent Poor,et al.  On the Continuity of the Secrecy Capacity of Compound and Arbitrarily Varying Wiretap Channels , 2014, IEEE Transactions on Information Forensics and Security.

[10]  R. Werner,et al.  A Continuity Theorem for Stinespring's Dilation , 2007, 0710.2495.

[11]  Khaled Fazel,et al.  Multi-Carrier and Spread Spectrum Systems , 2003 .

[12]  R. Ahlswede.  A Note on the Existence of the Weak Capacity for Channels with Arbitrarily Varying Channel Probability Functions and Its Relation to Shannon's Zero Error Capacity , 1970 .

[13]  Holger Boche,et al.  Classical Capacities of Averaged and Compound Quantum Channels , 2007, ArXiv.

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

[15]  D. Blackwell,et al.  The Capacities of Certain Channel Classes Under Random Coding , 1960 .

[16]  Holger Boche,et al.  Positivity, discontinuity, finite resources, nonzero error for arbitrarily varying quantum channels , 2014, 2014 IEEE International Symposium on Information Theory.

[17]  A. D. Wyner,et al.  The wire-tap channel , 1975, The Bell System Technical Journal.

[18]  A. Winter,et al.  Private capacity of quantum channels is not additive. , 2009, Physical review letters.

[19]  Rudolf Ahlswede,et al.  Classical Capacity of Classical-Quantum Arbitrarily Varying Channels , 2007, IEEE Transactions on Information Theory.

[20]  Holger Boche,et al.  Message Transmission Over Classical Quantum Channels With a Jammer With Side Information: Message Transmission Capacity and Resources , 2018, IEEE Transactions on Information Theory.

[21]  Holger Boche,et al.  Classical–quantum arbitrarily varying wiretap channel: Ahlswede dichotomy, positivity, resources, super-activation , 2013, Quantum Inf. Process..

[22]  Rudolf Ahlswede,et al.  Quantum Capacity under Adversarial Quantum Noise: Arbitrarily Varying Quantum Channels , 2010, ArXiv.

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

[24]  Holger Boche,et al.  Secrecy capacities of compound quantum wiretap channels and applications , 2013, ArXiv.

[25]  Holger Boche,et al.  Arbitrarily Varying and Compound Classical-Quantum Channels and a Note on Quantum Zero-Error Capacities , 2012, Information Theory, Combinatorics, and Search Theory.

[26]  Holger Boche,et al.  Arbitrarily small amounts of correlation for arbitrarily varying quantum channels , 2013, 2013 IEEE International Symposium on Information Theory.

[27]  Holger Boche,et al.  Classical-quantum arbitrarily varying wiretap channel: Secret message transmission under jamming attacks , 2017, 2017 IEEE International Symposium on Information Theory (ISIT).

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

[29]  Holger Boche,et al.  Classical-quantum arbitrarily varying wiretap channel: common randomness assisted code and continuity , 2016, Quantum Information Processing.

[30]  Moritz Wiese,et al.  The Arbitrarily Varying Wiretap Channel - Secret Randomness, Stability, and Super-Activation , 2016, IEEE Trans. Inf. Theory.

[31]  Holger Boche,et al.  Capacity Results for Arbitrarily Varying Wiretap Channels , 2012, Information Theory, Combinatorics, and Search Theory.

[32]  Thomas H. E. Ericson,et al.  Exponential error bounds for random codes in the arbitrarily varying channel , 1985, IEEE Trans. Inf. Theory.

[33]  Imre Csiszár,et al.  The capacity of the arbitrarily varying channel revisited: Positivity, constraints , 1988, IEEE Trans. Inf. Theory.

[34]  J. Smolin,et al.  Quantum communication with Gaussian channels of zero quantum capacity , 2011 .

[35]  Thierry Paul,et al.  Quantum computation and quantum information , 2007, Mathematical Structures in Computer Science.

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

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

[38]  Holger Boche,et al.  Secret message transmission over quantum channels under adversarial quantum noise: Secrecy capacity and super-activation , 2017, Journal of Mathematical Physics.