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Journal of the Physical Society of Japan

Dynamics of Quantum Fluctuations in a Communication Channel –Quantum Dense Coding–

Sachiko Kitajima

Published May 15, 2004• 5 citations
Publisher

Abstract

The dynamics of a quantum dense coding process is formulated and exactly solved on the basis of a boson detector model previously used to study decoherence processes. When classical information encoded in entangled states is transmitted through a quantum mechanical communication channel, an environment exerts quantum noise causing decoherence and disentanglement. Shannon's mutual information and error probability function are obtained explicitly as functions of time.

DOI
10.1143/jpsj.73.1093

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The dynamics of a quantum dense coding process is formulated and exactly solved on the basis of a boson detector model previously used to study decoherence processes. When classical information encoded in entangled states is transmitted through a quantum mechanical communication channel, an environment exerts quantum noise causing decoherence and disentanglement. Shannon's mutual information and error probability function are obtained explicitly as functions of time.
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.
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.
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.
Quantum communication is at the heart of the quantum information theory. We study two types of quantum communication protocols over noisy transmission channels, i.e. super dense coding and cryptography protocols. In the first part of this thesis, for various scenarios, it is discussed how the super dense coding capacity is influenced by noisy quantum channels. The case of memoryless channels as well as those channels with memory which are modelled by uncorrelated and correlated noise, respectively, are considered. Explicitly Pauli channels over arbitrary dimensions are treated and the super dense coding capacity for some resource states is derived. For the qubit depolarizing channel, when noise is uncorrelated, the super dense coding capacity with respect to the input state is also optimized. This illustrates a threshold value of the noise parameter below which the super dense coding capacity is optimized by a maximally entangled initial state, while above the threshold value the super dense coding capacity is optimized by a product state. For Pauli channels, with correlated noise, the case of non unitary encoding is studied and the super dense coding capacity is derived. The first part of this thesis is concluded with an example for multipartite super dense coding. In the second part of the thesis, the problem of optimal eavesdropping on noisy states in quantum key distribution is investigated. The case of the six state protocol, when the signal states are mixed with white noise, is considered. This situation may arise either when Alice deliberately adds noise to the signal states before they leave her lab, or, in a realistic scenario when the eavesdropper (referred to as Eve) is not assumed to replace the noisy quantum channel by a noiseless one. For individual attacks, we find Eve's optimal mutual information with Alice as a function of the quantum bit error rate. Finally, the results illustrate that adding noise on the quantum level can make quantum key distribution more robust against eavesdropping.
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
An upper limit is given to the amount of quantum information that can be transmitted reliably down a noisy, decoherent quantum channel using the high-probability states of quantum sources. A class of quantum error-correcting codes is presented that allows the information transmitted to attain this limit. The result is a quantum analog of Shannon's bound and code for the noisy classical channel [C. E. Shannon and W. Weaver, The Mathematical Theory of Communication (University of Illinois Press, Chicago, 1948)].