5 Quantum channels and operations

Marciel Medeiros de Oliveira, Francisco de Assis, Micael Andrade Dias

In this paper, we present a condition for the zeroerror capacity of quantum channels.To achieve this result we first prove that the eigenvectors (or eigenstates) common to the Kraus operators representing the quantum channel are fixed points of the channel.From this fact and assuming that these Kraus operators have at least two eigenstates in common and also considering that every quantum channel has at least one fixed point, it is proved that the zero-error capacity of the quantum channel is positive.Moreover, this zero-error capacity condition is a lower bound for the zero-error capacity of the quantum channel.This zero-error capacity condition of quantum channels has a peculiar feature that it is easy to verify when one knows the Kraus operators representing the quantum channel.

Joseph M. Renes

Marciel Medeiros de Oliveira, Francisco de Assis, Micael Andrade Dias

In this paper, we present a condition for the zero-error capacity of quantum channels. To achieve this result we first prove that the eigenvectors (or eigenstates) common to the Kraus operators representing the quantum channel are fixed points of the channel. From this fact and assuming that these Kraus operators have at least two eigenstates in common and also considering that every quantum channel has at least one fixed point, it is proved that the zero-error capacity of the quantum channel is positive. Moreover, this zero-error capacity condition is a lower bound for the zero-error capacity of the quantum channel. This zero-error capacity condition of quantum channels has a peculiar feature that it is easy to verify when one knows the Kraus operators representing the quantum channel.

Christopher A. Fuchs

Motohisa Fukuda

Abstract. In this study, we generate quantum channels with random Kraus operators to typically obtain almost twirling quantum channels and quantum expanders. To prove the concentration phenomena, we use matrix Bernstein’s inequality. In this way, our random models do not utilize Haar-distributed unitary matrices or Gaussian matrices. Rather, as in the preceding research, we use unitary t-designs to generate mixed tensor-product unitary channels acting on Cdt. Although our bounds in Schatten p-norm are valid only for 1 ≤p≤ 2, we show that they are typically quantum ϵ-twirling channels with the tail bound proportional to 1/poly(dt), while such bounds were previously constants. The number of required Kraus operators was also improved by powers of logdandt, to be proportional todtlogd/ϵ2. Such random quantum channels are also typically quantum expanders, but the number of Kraus operators must grow proportionally totlogdin our case. Finally, a new non-unital model of super-operators generated by bounded and isotropic random Kraus operators was introduced, which can be typically rectified to yield almost randomizing quantum channels and quantum expanders.