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Modern Physics Letters B

ENTANGLEMENT-ASSISTED CLASSICAL CAPACITIES OF SOME SINGLE QUBIT QUANTUM NOISY CHANNELS

Xian-Ting Liang, Hong-Yi Fan

Published May 20, 2002• 6 citations
Publisher

Abstract

In this paper, we calculate the entanglement-assisted classical capacities of the depolarizing channel, the phase damping channel, the phase flip channel, the bit flip channel, the bit-phase flip channel, the two-Pauli channel and the amplitude channel, and discuss the analytical results obtained. The Stokes papametrization representation of a qubit and the characteristic of unitary covariance of some quantum noisy channels are used in the calculations.

DOI
10.1142/s0217984902003890

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In this paper, we calculate the entanglement-assisted classical capacities of the depolarizing channel, the phase damping channel, the phase flip channel, the bit flip channel, the bit-phase flip channel, the two-Pauli channel and the amplitude channel, and discuss the analytical results obtained. The Stokes papametrization representation of a qubit and the characteristic of unitary covariance of some quantum noisy channels are used in the calculations.
In this paper, we calculate the entanglement-assisted classical information capacity of the amplitude damping channel and compare it with the particular mutual information which is considered as the entanglement-assisted classical information capacity of this channel in Ref. 6. It is shown that the difference between them is very small. In addition, we point out that using partial symmetry and concavity of mutual information derived from the dense coding scheme, one can simplify the calculation of entanglement-assisted classical information capacities for non-unitary-covariant quantum noisy channels.
We study four well-known capacities of a two-parameter family of qubit Pauli channels. These are the channels that are covariant under the SO(2)\text{SO}(2)SO(2) group and contain the depolarizing channel as a special case. We find exact expressions for the classical capacity and entanglement-assisted capacities, and analytically determine the regions where the quantum capacity of the channel vanishes. We then use a flag extension to find upper bound for the quantum capacity and private capacity of these channels in the entire region of parameter space and also obtain the lower bound for the quantum capacity by calculating the single-shot quantum capacity numerically. In conjunction with previous results on depolarizing channels, our result is one step forward for determining the capacities of the full Pauli channel.
We derive explicit, absolutely convergent series formulas for the quantum, unassisted classical, and entanglement-assisted classical capacities of the qubit amplitude damping channel, eliminating residual optimizations and implicit roots. We also obtain analytical upper and lower bounds on two-way assisted quantum and private communication capacities by evaluating a balanced-squashing bound and optimizing the reverse coherent information. Our results characterize the optimal input populations, establish convergence and truncation properties, and provide analytical benchmarks for quantum and classical communication over dissipative channels.
The amount of coherent quantum information that can be reliably transmitted down the memory Pauli channels with Markovian correlated noise is investigated. Two methods for evaluating the quantum capacity of the memory Pauli channels are proposed to try to trace the memory effect on the transmissions of quantum information. We show that the evaluation of quantum capacity can be reduced to the calculation of the initial memory state of each successive transmission. Furthermore, we derive quantum capacities of the memory phase flip channel, bit flip channel and bit-phase flip channel. Also, a lower bound of the quantum capacity of the memory depolarizing channel is obtained. An increase of the degree of memory of the channels has a positive effect on the increase of their quantum capacities.