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arXiv:1205.5851 · Physical Review A

Zero-error classical capacity of qubit channels cannot be superactivated

Jeong-Hoon Park, Soojoon Lee

Published May 25, 2012• 8 citations
arXivHTMLPDF

Abstract

It was shown [Cubitt et al., IEEE Trans. Inf. Theory 57, 8114 (2011)] that there exist quantum channels where a single use cannot transmit classical information perfectly yet two uses can. This phenomenon is called the superactivation of the zero-error classical capacity, which does not occur in classical channels. In this paper, it is shown that qubit channels cannot generate the superactivation.

DOI
10.1103/physreva.85.052321

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It was shown [Cubitt et al., IEEE Trans. Inf. Theory 57, 8114 (2011)] that there exist quantum channels where a single use cannot transmit classical information perfectly yet two uses can. This phenomenon is called the superactivation of the zero-error classical capacity, which does not occur in classical channels. In this paper, it is shown that qubit channels cannot generate the superactivation.
The zero-error classical capacity of a quantum channel is the asymptotic rate at which it can be used to send classical bits perfectly, so that they can be decoded with zero probability of error. The study of zero-error capacities dates right back to Shannon and the early days of information theory. We show that there exist pairs of quantum channels, neither of which individually have any zero-error capacity whatsoever (even if arbitrarily many uses of the channels are available), but such that access to even a single copy of both channels allows classical information to be sent perfectly reliably. In other words, we prove that the zero-error classical capacity can be superactivated. This result is the first example of superactivation of a classical capacity of a quantum channel. We further strengthen this result to show that there exist pairs of channels, neither of which have any zero-error classical capacity (as before), yet for which access to one copy of the joint channel even allows far more delicate quantum information to be transmitted perfectly. This subsumes the first result, and also implies that the quantum zero-error capacity can be superactivated. But it is strictly stronger than either of these. Indeed, this is the strongest conceivable form of superactivation, and nothing similar is possible for standard Shannon capacities of quantum channels or for zero-error capacities of classical channels.
The zero-error classical capacity of a quantum channel is the asymptotic rate at which it can be used to send classical bits perfectly so that they can be decoded with zero probability of error. We show that there exist pairs of quantum channels, neither of which individually have any zero-error capacity whatsoever (even if arbitrarily many uses of the channels are available), but such that access to even a single copy of both channels allows classical information to be sent perfectly reliably. In other words, we prove that the zero-error classical capacity can be superactivated. This result is the first example of superactivation of a classical capacity of a quantum channel.
The zero-error classical capacity of a quantum channel is the asymptotic rate at which it can be used to send classical bits perfectly so that they can be decoded with zero probability of error. We show that there exist pairs of quantum channels, neither of which individually have any zero-error capacity whatsoever (even if arbitrarily many uses of the channels are available), but such that access to even a single copy of both channels allows classical information to be sent perfectly reliably. In other words, we prove that the zero-error classical capacity can be superactivated. This result is the first example of superactivation of a classical capacity of a quantum channel.
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 nonadditivity, 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 and with no zero-error classical capacity whatsoever, , but whose joint zero-error quantum capacity is positive, . This striking effect is an extreme form of the superactivation phenomenon, as it implies that both the classical and quantum zero-error capacities of these channels can be superactivated simultaneously, while being a strictly stronger property of capacities. Superactivation of the quantum zero-error capacity was not previously known.
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. Superactivation of the quantum zero-error capacity was not previously known.
This paper introduces a fundamentally new method of finding the conditions for the superactivation of the zero-error capacity of quantum channels.The zero-error capacity of the quantum channel describes the amount of information which can be transmitted perfectly through a noisy quantum channel.The superactivation of the zero-error capacity of quantum channels makes it possible to use two quantum channels, each with zero zero-error capacity, with a positive joint zero-error capacity.Currently we have no theoretical background for describing all possible combinations of superactive quantum channels, hence there should be many other possible combinations.We give an algorithmic solution to the problem.To analyze the superactivation of the zero-error capacity, we introduce a new geometrical representation, called the quantum superball.Our method can be the first efficient algorithmic solution to discover the still unknown combinations to determine the superactivation of the zero-error capacity of quantum channels, without the extremely high computational costs.