Zero-Error Channel Capacity and Simulation Assisted by Non-Local Correlations

The theory of zero-error communication is re-examined in the broader setting of using one classical channel to simulate another exactly in the presence of various classes of nonsignalling correlations between sender and receiver i.e., shared randomness, shared entanglement and arbitrary nonsignalling correlations. When the channel being simulated is noiseless, this is zero-error coding assisted by correlations. When the resource channel is noiseless, it is the reverse problem of simulating a noisy channel exactly by a noiseless one, assisted by correlations. In both cases, separations between the power of the different classes of assisting correlations are exhibited for finite block lengths. The most striking result here is that entanglement can assist in zero-error communication. In the large block length limit, shared randomness is shown to be just as powerful as arbitrary nonsignalling correlations for exact simulation, but not for asymptotic zero-error coding. For assistance by arbitrary nonsignalling correlations, linear programming formulas for the asymptotic capacity and simulation rates are derived, the former being equal (for channels with nonzero unassisted capacity) to the feedback-assisted zero-error capacity derived by Shannon. Finally, a kind of reversibility between nonsignalling-assisted zero-error capacity and exact simulation is observed, mirroring the usual reverse Shannon theorem.

[1]  Jianxin Chen,et al.  Superactivation of the Asymptotic Zero-Error Classical Capacity of a Quantum Channel , 2009, IEEE Transactions on Information Theory.

[2]  Francisco M. de Assis,et al.  Quantum states characterization for the zero-error capacity , 2006 .

[3]  Asher Peres,et al.  Two simple proofs of the Kochen-Specker theorem , 1991 .

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

[5]  A. Cabello,et al.  Bell-Kochen-Specker theorem: A proof with 18 vectors , 1996, quant-ph/9706009.

[6]  S. Massar,et al.  Nonlocal correlations as an information-theoretic resource , 2004, quant-ph/0404097.

[7]  D. Leung,et al.  Entanglement can Increase Asymptotic Rates of Zero-Error Classical Communication over Classical Channels , 2010, Communications in Mathematical Physics.

[8]  László Lovász,et al.  On the Shannon capacity of a graph , 1979, IEEE Trans. Inf. Theory.

[9]  Toby S. Cubitt,et al.  Super-Duper-Activation of Quantum Zero-Error Capacities , 2009 .

[10]  Andreas J. Winter,et al.  Quantum Reverse Shannon Theorem , 2009, ArXiv.

[11]  Claude E. Shannon,et al.  The zero error capacity of a noisy channel , 1956, IRE Trans. Inf. Theory.

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

[13]  Runyao Duan,et al.  Super-Activation of Zero-Error Capacity of Noisy Quantum Channels , 2009, 0906.2527.

[14]  Simone Severini,et al.  Zero-Error Communication via Quantum Channels, Noncommutative Graphs, and a Quantum Lovász Number , 2010, IEEE Transactions on Information Theory.

[15]  S. Popescu,et al.  Quantum nonlocality as an axiom , 1994 .

[16]  Simone Severini,et al.  Zero-error communication via quantum channels, non-commutative graphs and a quantum Lovasz theta function , 2010, ArXiv.

[17]  Graeme Smith,et al.  An Extreme Form of Superactivation for Quantum Zero-Error Capacities , 2009, IEEE Transactions on Information Theory.

[18]  Willem H. Haemers,et al.  On Some Problems of Lovász Concerning the Shannon Capacity of a Graph , 1979, IEEE Trans. Inf. Theory.

[19]  Joel E. Cohen,et al.  Nonnegative ranks, decompositions, and factorizations of nonnegative matrices , 1993 .

[20]  Debbie W. Leung,et al.  Improving zero-error classical communication with entanglement , 2009, Physical review letters.

[21]  Runyao Duan,et al.  Zero-error communication via quantum channels and a quantum Lovász script v sign-function , 2011 .

[22]  J. Bell.  On the Problem of Hidden Variables in Quantum Mechanics , 1966 .

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

[24]  Alon Orlitsky,et al.  Zero-Error Information Theory , 1998, IEEE Trans. Inf. Theory.

[25]  Salman Beigi,et al.  Entanglement-assisted zero-error capacity is upper bounded by the Lovasz theta function , 2010, ArXiv.

[26]  Femke Bekius,et al.  The Shannon Capacity of a Graph , 2011 .