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Journal of the Physical Society of Japan

A Model of Quantum Mechanical Communication Channel with Quantum and Thermal Noises

Sachiko Kitajima, Masashi Ban, Fumiaki Shibata

Published May 15, 2005• 1 citation
Publisher

Abstract

Dynamics of quantum information channel is studied with the use of a boson detector model. Explicit results are given for quantum dense coding process on the basis of an equivalence relation between a classical capacity and von Neumann mutual information. Reduced density matrices for an entangled spin degree of freedom of an incoming particle are exactly solved to give complete time evolution under influence of fluctuations due to a detector. Quantum mechanical channel matrix and the classical capacity are determined as functions of time and temperature. Decoherence properties of the entangled states under quantum and thermal noises are fully discussed.

DOI
10.1143/jpsj.74.1436

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Dynamics of quantum information channel is studied with the use of a boson detector model. Explicit results are given for quantum dense coding process on the basis of an equivalence relation between a classical capacity and von Neumann mutual information. Reduced density matrices for an entangled spin degree of freedom of an incoming particle are exactly solved to give complete time evolution under influence of fluctuations due to a detector. Quantum mechanical channel matrix and the classical capacity are determined as functions of time and temperature. Decoherence properties of the entangled states under quantum and thermal noises are fully discussed.
Quantum mechanical decoherence process is studied with the use of a boson detector model. Generalizing the previous method, thermal effect is taken into account: Coherence of an incident particle is destroyed in interaction with its surroundings at finite temperature. Vanishing of off-diagonal elements of a reduced density matrix is rigorously shown when the detector (surroundings) is composed of a large number of constituents and/or its temperature is very high. The temperature effect reveals itself through a renormalization of interaction strength between an incident particle and the detector. Time evolution of the von Neumann entropy is explicitly determined.
Theory of quantum mechanical communication channel under the influence of environmental fluctuations is formulated on the basis of non-equilibrium statistical mechanics. The formalism is explicitly applied to the quantum dense coding process where a qubit signal is sent into the channel in which the temperature and/or quantum fluctuations disturb an initial coherency of the signal. A quantum channel matrix, the Shannon's mutual information and quantum mechanical mutual information are analytically obtained and numerically evaluated. It is shown that the temperature effect is quite large in the signal transmission processes.
We discuss the capacity of quantum channels for information transmission and storage. Quantum channels have dual uses: they can be used to transmit known quantum states which code for classical information, and they can be used in a purely quantum manner, for transmitting or storing quantum entanglement. We propose here a definition of the von Neumann capacity of quantum channels, which is a quantum-mechanical extension of the Shannon capacity and reverts to it in the classical limit. As such, the von Neumann capacity assumes the role of a classical or quantum capacity depending on the usage of the channel. In analogy to the classical construction, this capacity is defined as the maximum von Neumann mutual entropy processed by the channel, a measure which reduces to the capacity for classical information transmission through quantum channels (the ``Kholevo capacity'') when known quantum states are sent. The quantum mutual entropy fulfills all basic requirements for a measure of information, and observes quantum data-processing inequalities. We also derive a quantum Fano inequality relating the quantum loss of the channel to the fidelity of the quantum code. The quantities introduced are calculated explicitly for the quantum depolarizing channel. The von Neumann capacity is interpreted within the context of superdense coding, and an extended Hamming bound is derived that is consistent with that capacity.
Decoherence of entanglement of qubits is investigated which is caused by the phenomenological quantum channel (the Bloch channel) equivalent to the Bloch equations. It is shown how the decoherence of entanglement depends on the longitudinal and transverse relaxation times and the equilibrium value of the qubit. The quantum dense coding system under the influence of the Bloch channel is also investigated. The Shannon mutual information obtained by the Bell measurement and the Holevo capacity is calculated. Furthermore, the microscopic system-reservoir model which yields the Bloch equations is considered. The result shows that the temperature of the thermal reservoir significantly affects the decoherence of entanglement.
Abstract Quantum entanglement is a special nonlocal association between quantum systems. It is a fundamental feature of the difference between quantum and classical. At the same time, it acts as a very important resource in quantum informatics. Therefore, entanglement plays a noble role in quantum information and quantum computing. Quantum dense coding, as an application of quantum entanglement in practice, is a quantum communication protocol characterized by security and secrecy, which plays a very important role in quantum informatics and cryptography. This paper is mainly in the Non-Markovian environment of Heisenberg Spin chain system, using the quantum dense coding scheme proposed by Bennett et al. to study The dynamics of quantum dense coding channel capacity evolution with time and the entanglement dynamics of the system are discussed in detail.Based on the theoretical knowledge of open quantum systems and the maximum entangled state ∣ψAB⟩=12(∣11⟩+∣00⟩)\left| {{{\mathbf{\psi }}_{{\mathbf{AB}}}}} \right\rangle =\frac{1}{{\sqrt 2 }}\left( {\left| {11} \right\rangle +\left| {00} \right\rangle } \right)∣ψAB​⟩=2​1​(∣11⟩+∣00⟩) as the initial state, the density matrix of the system evolution over time is obtained by quantum state diffusion method,and the evolution of channel capacity and entanglement over time of Heisenberg spin chains is simulated accurately. The results show that: the capacity and entanglement of dense coded channel oscillate with time and reach different stable values. Environmental correlation coefficientγ\gammaγand coupling strengthη\etaηand a play an important role in dense coding and quantum entanglement of quantum systems.Proper selection of parameters can keep the system good entanglement characteristics and increase the channel capacityχ\chiχof the system, thus ensuring the effective transmission and processing of quantum information in practice. It lays a solid foundation for quantum computing