量子噪声通道中的信息传送
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摘要
信息科学在改善人类的生活质量和推动社会文明发展中发挥着无可比拟和令人惊叹的作用。但是随着人们对信息需求的日益增加,现有的信息系统在传送信息的容量、速度、保密性和安全性等方面越来越不能满足人们的需求。于是一门新兴的信息科学——量子信息学便应运而生。它是量子力学和信息理论交叉产生的新兴学科,主要包括量子通信和量子计算等。量子信息的载体是量子态。量子态的制备、加工、传输以及存储的过程叫做量子信息过程。
如何有效地通过量子噪声通道传送信息是量子信息理论中的一个重要内容。本文应用量子信息理论研究了量子噪声通道中的信息传送。主要创新成果如下:
1、研究了通过振幅阻尼通道和广义振幅阻尼通道传送一个经典比特的问题。当允许通过通道进行两次传送时,研究了通道参数和输入态参数对出错率的影响。证明在一定的条件下,两次纠缠传送比两次直积态传送能增强接收者正确判断信息的能力。对于广义振幅阻尼通道而言,在一特定的条件下,纠缠态编码任何时候都比直积态编码能传递更多的信息。
2、运用量子通道经典容量的量子相对熵的表示公式,用数值方法研究了广义振幅阻尼通道的经典容量。结果表明经典容量依赖于表示耗散和周围环境温度的两个通道参数。特别地,在一定的条件下,广义振幅阻尼通道的经典容量可以用一个解析表达式表示。
3、利用量子互信息的部分对称性和凸性,研究了广义振幅阻尼通道的纠缠辅助经典容量。得到了在一定条件下,纠缠辅助经典容量的解析和数值结果。结果表明纠缠辅助经典容量与表示耗散和周围环境温度的两个通道参数有关,而且发现发送者和接收者之间的先验纠缠可以使广义振幅阻尼通道的经典容量增加近一倍。
4、研究了有记忆量子噪声通道经典信息的传送问题,首次得到具有相关噪声的泡利通道的互信息的一般公式。结果证明互信息依赖于通道收缩因子,输入态参数和通道记忆系数.通过对互信息公式的分析,发现:对于比特翻转通道和比特位相翻转通道等一些泡利通道,纠缠是一种有用的源,始终能增强其互信息。然而,对于位相翻转通道和位相阻尼通道等另一些泡利通道,纠缠态编码在任何情况下都不利于经典信息的有效传送。
Information science acts the great important part in improving the quality of people and promoting the development of the civilization of the society. However, with the people increasing desire for the capacity, speed, privacy and security of information transmission, and so on, the existing information system can not satisfy the requirement of people now. Therefore, a new subject——Quantum Information Theory—has been developed as the combination ofquantum mechanics and information theory. Quantum state is the carrier of quantum information. The process of preparation, operation, transmission andstorage of the quantum state are called quantum information process.
How to reliably transmit the information through the quantum noisy channel is one of the important questions in quantum information theory. In this thesis, the information transmission through the quantum noisy channel is investigated by the quantum information theory. Our mainly work is as follows:
1. The problem of sending a single classical bit through an amplitude-damping channel and a generalized amplitude damping channel is considered. When two transmissions through the channel are available as a resource, we find that two entangled transmissions can enhance the capability of receiver's judging information correctly under certain conditions compared with two product- state transmissions. In addition, for the generalized amplitude damping channel, we find a special case in which the two entangled transmissions can always make a classical bit more effectively disable the noise influence.
2. The classical capacity of a generalized amplitude-damping channel is investigated by using a numerical method. It is shown that the classical capacity depends on the channel parameters representing the ambient temperature and dissipation. In particular, under a special condition, the classical capacity of the generalized amplitude-damping channel can be written as an analytical expression.
3. The entanglement-assisted classical capacity of the generalized amplitude damping channel is investigated by using the properties of partial symmetry and concavity of mutual information. The numerical and analytical results of the entanglement-assisted classical capacity are obtained under some certain conditions. It is shown that, the entanglement-assisted classical capacity depends on the channel parameters representing the ambient temperature and dissipation, and the prior entanglement between sender and receiver can approximately double the classical capacity of the generalized amplitude damping channel.
4. The problem of the classical information transmission through the Pauli channel with memory is investigated. A general formula for the mutual information of the Pauli channels with memory modeled by correlated noise is derived. It is shown that the mutual information depends on the channel shrinking factor, the input state parameter and the channel memory coefficient. The analyses based on the general formula reveal that the entanglement is always a useful resource to enhance the mutual information of some Pauli channels, such as the bit flip channel and the bit-phase flip channel. Our analyses also show that the entanglement is not advantageous to the reliable transmission of classical information for some Pauli channels at any time, such as the phase flip channel and the phase damping channel.
如何有效地通过量子噪声通道传送信息是量子信息理论中的一个重要内容。本文应用量子信息理论研究了量子噪声通道中的信息传送。主要创新成果如下:
1、研究了通过振幅阻尼通道和广义振幅阻尼通道传送一个经典比特的问题。当允许通过通道进行两次传送时,研究了通道参数和输入态参数对出错率的影响。证明在一定的条件下,两次纠缠传送比两次直积态传送能增强接收者正确判断信息的能力。对于广义振幅阻尼通道而言,在一特定的条件下,纠缠态编码任何时候都比直积态编码能传递更多的信息。
2、运用量子通道经典容量的量子相对熵的表示公式,用数值方法研究了广义振幅阻尼通道的经典容量。结果表明经典容量依赖于表示耗散和周围环境温度的两个通道参数。特别地,在一定的条件下,广义振幅阻尼通道的经典容量可以用一个解析表达式表示。
3、利用量子互信息的部分对称性和凸性,研究了广义振幅阻尼通道的纠缠辅助经典容量。得到了在一定条件下,纠缠辅助经典容量的解析和数值结果。结果表明纠缠辅助经典容量与表示耗散和周围环境温度的两个通道参数有关,而且发现发送者和接收者之间的先验纠缠可以使广义振幅阻尼通道的经典容量增加近一倍。
4、研究了有记忆量子噪声通道经典信息的传送问题,首次得到具有相关噪声的泡利通道的互信息的一般公式。结果证明互信息依赖于通道收缩因子,输入态参数和通道记忆系数.通过对互信息公式的分析,发现:对于比特翻转通道和比特位相翻转通道等一些泡利通道,纠缠是一种有用的源,始终能增强其互信息。然而,对于位相翻转通道和位相阻尼通道等另一些泡利通道,纠缠态编码在任何情况下都不利于经典信息的有效传送。
Information science acts the great important part in improving the quality of people and promoting the development of the civilization of the society. However, with the people increasing desire for the capacity, speed, privacy and security of information transmission, and so on, the existing information system can not satisfy the requirement of people now. Therefore, a new subject——Quantum Information Theory—has been developed as the combination ofquantum mechanics and information theory. Quantum state is the carrier of quantum information. The process of preparation, operation, transmission andstorage of the quantum state are called quantum information process.
How to reliably transmit the information through the quantum noisy channel is one of the important questions in quantum information theory. In this thesis, the information transmission through the quantum noisy channel is investigated by the quantum information theory. Our mainly work is as follows:
1. The problem of sending a single classical bit through an amplitude-damping channel and a generalized amplitude damping channel is considered. When two transmissions through the channel are available as a resource, we find that two entangled transmissions can enhance the capability of receiver's judging information correctly under certain conditions compared with two product- state transmissions. In addition, for the generalized amplitude damping channel, we find a special case in which the two entangled transmissions can always make a classical bit more effectively disable the noise influence.
2. The classical capacity of a generalized amplitude-damping channel is investigated by using a numerical method. It is shown that the classical capacity depends on the channel parameters representing the ambient temperature and dissipation. In particular, under a special condition, the classical capacity of the generalized amplitude-damping channel can be written as an analytical expression.
3. The entanglement-assisted classical capacity of the generalized amplitude damping channel is investigated by using the properties of partial symmetry and concavity of mutual information. The numerical and analytical results of the entanglement-assisted classical capacity are obtained under some certain conditions. It is shown that, the entanglement-assisted classical capacity depends on the channel parameters representing the ambient temperature and dissipation, and the prior entanglement between sender and receiver can approximately double the classical capacity of the generalized amplitude damping channel.
4. The problem of the classical information transmission through the Pauli channel with memory is investigated. A general formula for the mutual information of the Pauli channels with memory modeled by correlated noise is derived. It is shown that the mutual information depends on the channel shrinking factor, the input state parameter and the channel memory coefficient. The analyses based on the general formula reveal that the entanglement is always a useful resource to enhance the mutual information of some Pauli channels, such as the bit flip channel and the bit-phase flip channel. Our analyses also show that the entanglement is not advantageous to the reliable transmission of classical information for some Pauli channels at any time, such as the phase flip channel and the phase damping channel.
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