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2009

An Extreme form of Superactivation for Quantum Zero-Error Capacities

Cubitt, Toby S., Smith, Graeme

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The zero-error capacity of a channel is the rate at which it can send information perfectly, with zero probability of error, and has long been studied in classical information theory.

We show that the zero-error capacity of quantum channels exhibits an extreme form of non-additivity, one which is not possible for classical channels, or even for the usual capacities of quantum channels.
By combining probabilistic arguments with algebraic geometry, we prove that there exist channels E1 and E2 with no zero-error classical capacity whatsoever, C_0(E1) = C_0(E2) = 0, but whose joint zero-error quantum capacity is positive, Q_0(E1 x E2) >= 1.
This striking effect is an extreme from of the superactivation phenomenon, as it implies that both the classical and quantum zero-error capacities of these channels can be superactivated simultaneously, whilst being a strictly stronger property of capacities.

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C. King and M. Ruskai, “Minimal entropy of states emerging from noisy quantum channels,” IEEE Trans. Info. Theory , vol. 47, pp. 192 – 209, 2001
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Then

G. Smith and J. Yard, “Quantum communication with zero-capacity channels,” Science , vol. 321, p. 1812, 2008, (arXiv:0807.4935 [quant-ph])
Original
2008
Later among the works it cites.
T. Cubitt, A. W. Harrow, D. Leung, A. Montanaro, and A. Winter, “Counterexamples to additivity of minimum output p-rényi entropy for p close to 0,” CMP , vol. 284, p. 281, 2008, (arXiv:0712.3628 [quant-ph])
Original
2008
Later among the works it cites.
R. Duan, “Superactivation of zero-error capacity of noisy quantum channels,” arXiv:0906.2527 [quant-ph], 2009
Original
2009
Closest in time.
M. B. Hastings, “A counterexample to additivity of minimum output entropy,” Nature Physics , vol. 5, 2009, (arXiv:0809.3972 [quant-ph])
Original
2009
Closest in time.

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T. S. Cubitt, J. Chen, and A. W. Harrow, “Superactivation of the asymptotic zero-error classical capacity of a quantum channel,” arXiv:0906.2547

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R. Duan, J. Chen, and Y. Xin, “Unambiguous and zero-error classical capacity of noisy quantum channels,” (Manuscript in preparation)

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