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Journal of Modern Physics

Some Charicteristics and Applications for Quantum Information

Bi Qiao, Jinqing Fang, Harry E. Ruda

Published Jan 1, 2012• 4 citations
Publisher

Abstract

In this work some charicteristics and applications for quantum information is revealed. The various dynamical equations of quantum information density have been investigated, transmission characteristics of the dynamical mutual information have been studied, and the decoherence-free controlling procedure has been considered, which exposes that quantum information is holographic through the similarity structure of subdynamic kinetic equations for quantum information density.

DOI
10.4236/jmp.2012.39141

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In this work some charicteristics and applications for quantum information is revealed. The various dynamical equations of quantum information density have been investigated, transmission characteristics of the dynamical mutual information have been studied, and the decoherence-free controlling procedure has been considered, which exposes that quantum information is holographic through the similarity structure of subdynamic kinetic equations for quantum information density.
We establish several dynamical equations for quantum information density. It is demonstrated that quantum information density shares the same formalism of the Liouville equation, subdynamics kinetic equation and Fokker–Planck equation as the density operator and also possesses the superposition property. These allow one to use quantum information density directly to model quantum information. The kinetic equations for quantum information density reveal that the dynamical process of quantum information may be related to dissipative, Markovian, or diffusional information flows, together causing irreversibility. Finally, we discuss superposition of quantum information density, which allows us to construct a quantum information channel in the coherent state representation using harmonic oscillator based encoded quantum information, and obtain a formula for quantum dynamical mutual information.
Entanglement entropy is a proposal to quantify quantum entanglement of two disjoint regions in a pure system. It is a relatively new topic and is developing rapidly. The current main motives to study it are its applications to studying quantum gravity, thermalization and phase transitions in condensed matter systems. Mutual information is a related quantity and it can be used to measure the amount of information two disjoint regions share. The purpose of this thesis is to give an introduction to the general results of the field without considering specific systems. The two prominent approaches used are two-dimensional conformal field theory and the holographic entanglement entropy conjecture. The first approach is to calculate entanglement entropy using conformal field theory, the results of which are known to be exact although technically more difficult to derive and only available for 1+1 dimensional systems. Both static and dynamic systems will be discussed. The results are reproduced and generalized to higher dimensions using holography. As a more recent topic, the holographic approach is used to rederive the entanglement entropy of systems with a Fermi surface using AdS/Vaidya metric with Lifshitz scaling and hyperscaling violation. The final chapter of the thesis discusses mutual information in various static and dynamic cases considered in the previous chapters. For the first time, the evolution of mutual information is calculated in AdS/Vaidya metric with Lifshitz scaling and hyperscaling violation in the critical theta=d-1 case.
In this work,we present a nonequilibrium ensemble formalism based on the subdynamic kine-tic equation.The constructing procedure is to take a similarity transformation to Gibbsian ensemble formalism.This allows one to obtain subdynamic ensemble formalism for projected density operator.This formalism can be considered as a generalization of the equilibrium quantum canonical ensemble formula in the sense as 1) if the similarity operator is unitary,then the new formula is just an effective representation of the old canonical ensemble formula,2) if the similarity operator is non-unitary,then the new formula is an extension of the old formula,which represents a kind of non-equilibrium quantum canonical ensembles formula and reflects irreversibility of the system,and 3) if the similarity operator can be deduced by some approximations,such as Markovian/non-Markovian approximations,then the new formula can expose some irreversibility characteristics of the system which can not be gained from the equilibrium quantum ensemble formulas.As an application,we study a quantum strongly correlated network described by an Anderson model.A Green function based on this formalism is also introduced.We find the complex entropy of the system to be related to appearance of the local magnetic vectors in the nodes of the network,where the image part of the entropy represents an evolution rate of the entropy.
Quantum entropy is a fundamental concept for quantum information recently developed in various directions. We will review the mathematical aspects of quantum entropy (entropies) and discuss some applications to quantum communication, statistical physics. All topics taken here are somehow related to the quantum entropy that the present authors have been studied. Many other fields recently developed in quantum information theory, such as quantum algorithm, quantum teleportation, quantum cryptography, etc., are totally discussed in the book (reference number 60).
The work is devoted to study of quantum mutual information and coherent information -- the two important characteristics of quantum communication channel. Appropriate definitions of these quantities in the infinite-dimensional case are given and their properties are studied in detail. Basic identities relating quantum mutual information and coherent information of a pair of complementary channels are proved. An unexpected continuity property of quantum mutual information and coherent information following from the above identities is observed. An upper bound for the coherent information is obtained.
A formulation for performing quantum computing in a projected subspace is presented, based on the subdynamical kinetic equation (SKE) for an open quantum system. The eigenvectors of the kinetic equation are shown to remain invariant before and after interaction with the environment. However, the eigenvalues in the projected subspace exhibit a type of phase shift to the evolutionary states. This phase shift does not destroy the decoherence-free (DF) property of the subspace because the associated fidelity is 1.1.1. This permits a universal formalism to be presented---the eigenprojectors of the free part of the Hamiltonian for the system and bath may be used to construct a DF projected subspace based on the SKE. To eliminate possible phase or unitary errors induced by the change in the eigenvalues, a cancellation technique is proposed, using the adjustment of the coupling time, and applied to a two-qubit computing system. A general criteria for constructing a DF-projected subspace from the SKE is discussed. Finally, a proposal for using triangulation to realize a decoherence-free subsystem based on SKE is presented. The concrete formulation for a two-qubit model is given exactly. Our approach is general and appears to be applicable to any type of decoherence.