Entropies of general quantum states

The entropy and the mutual entropy for states and channels are totally studied in general quantum systems described byC*-algebras. In particular, we discuss the relations among theS-entropy and the mutual entropy introduced by Ohya and the Connes-Narnhofer-Thirring entropy.     

OPEN QUANTUM SYSTEMS AND COMPLETE POSITIVITY

The dynamics of an open quantum system is usually assumed to be generated by time-evolution equations of Lindblad form. It follows that the time evolutors are completely positive. Far from being just a useful technicality, complete positivity is one of the many aspects of quantum entanglement and is necessary to avoid physical inconsistencies. This issue is addressed by examining the open dynamics of two level systems in situations typical of chemical physics and quantum optics.     

On Entangled Information and Quantum Capacity

The pure quantum entanglement is generalized to the case of mixed compound states to include the classical and quantum encodings as particular cases. The true quantum entanglements are characterized as transpose-CP but not CP maps. The entangled information is introduced as the relative entropy of the mutual and the input state and total information of the entangled states leads to two different types of entropy for a given quantum state: the von Neumann entropy, which is achieved as the supremum of the information over all c-entanglements, and the true quantum entropy, which is achieved at the standard entanglement. The q-capacity, defined as the supremum over all entanglements, doubles the c-capacity in the case of the simple algebra. The conditional q-entropy is positive, and q-information of a quantum channel is additive.     

Quantum coherence of two-qubit over quantum channels with memory

Using the axiomatic definition of the quantum coherence measure, such as the \(l_{1}\) norm and the relative entropy, we study the phenomena of two-qubit system quantum coherence through quantum channels where successive uses of the channels are memory. Different types of noisy channels with memory, such as amplitude damping, phase damping, and depolarizing channels effect on quantum coherence have been discussed in detail. The results show that quantum channels with memory can efficiently protect coherence from noisy channels. Particularly, as channels with perfect memory, quantum coherence is unaffected by the phase damping as well as depolarizing channels. Besides, we also investigate the cohering and decohering power of quantum channels with memory.     

基于邻域粗糙互信息熵的非单调性属性约简

属性约简是粗糙集理论一项重要的应用,目前已广泛运用于机器学习和数据挖掘等领域,邻域粗糙集是粗糙集理论中处理连续型数据的一种重要方法.针对目前邻域粗糙集模型中属性约简存在的缺陷,构造一种基于邻域粗糙集的邻域粗糙熵模型,并基于此给出邻域粗糙联合熵、邻域粗糙条件熵和邻域粗糙互信息熵等概念.邻域粗糙互信息熵是评估属性集相关性的一种重要的方法,具有非单调性变化的特性,对此,提出一种基于邻域粗糙互信息熵的非单调性属性约简算法.实验分析表明,所提出算法不仅比目前已有的单调性属性约简算法具有更优越的属性约简结果,而且具有更高的约简效率.     

隐私保护的信息熵模型及其度量方法

隐私的量化是隐私保护技术的重要支撑,信息熵作为信息的量化手段,自然可以用于解决隐私度量问题. 基于Shannon信息论的通信框架,提出了几种隐私保护信息熵模型,以解决隐私保护系统的相关度量问题,主要包括：隐私保护基本信息熵模型、含敌手攻击的隐私保护信息熵模型、带主观感受的信息熵模型和多隐私信源的隐私保护信息熵模型.在这些模型中,将信息拥有者假设为发送方,隐私谋取者假设为接收方,隐私的泄露渠道假设为通信信道;基于这样的假设,分别引入信息熵、平均互信息量、条件熵及条件互信息等来分别描述隐私保护系统信息源的隐私度量、隐私泄露度量、含背景知识的隐私度量及泄露度量;以此为基础,进一步提出了隐私保护方法的强度和敌手攻击能力的量化测评,为隐私泄露的量化风险评估提供了一种支撑;最后,针对位置隐私保护的应用场景,给出了具体的信息熵模型及隐私保护机制和攻击能力的度量及分析.所提出的模型和隐私量化方法,可以为隐私保护技术和隐私泄露风险分析与评估提供可行的理论基础.     

The monogamy relation and quantum phase transition in one-dimensional anisotropic XXZ model

In the paper, we have researched the monogamy relation and the quantum phase transition (QPT) in the anisotropic spin XXZ model by exploiting the quantum renormalization group method. The results show that there exits QPT after several iterations of renormalization in the present system. And we can find out that the monogamy inequality of entanglement of formation (EOF) and entropy quantum discord develop two saturated values which associate with spin-liquid and Néel phases after several iterations of the renormalization. Furthermore, we can also find out that the renormalization of EOF and entropy quantum discord violate the monogamy property while the renormalized geometric quantum discord obeys it no matter whether the QPT iterations are carried out. As a byproduct, the nonanalytic phenomenon and scaling behavior of the spin system are analyzed in detail.     

Fundamentals of Quantum Mutual Entropy and Capacity

The rigorous formulation of the quantum mutual entropy is reviewed. The capacities for various channels, such as quantum channel, classical-quantum channel and classical-quantum-classical channel are discussed exhaustively and some misuses of these capacities are indicated.     

A Survey on quantum computing technology

The power of quantum computing technologies is based on the fundamentals of quantum mechanics, such as quantum superposition, quantum entanglement, or the no-cloning theorem. Since these phenomena have no classical analogue, similar results cannot be achieved within the framework of traditional computing. The experimental insights of quantum computing technologies have already been demonstrated, and several studies are in progress. Here we review the most recent results of quantum computation technology and address the open problems of the field.     

On Classification of Quantum Channels

Quantum mutual entropy and quantum capacity are rigorously defined by Ohya, and they are quite useful in the study of quantum communication processes. Mathematical models of optical communication processes are described by a quantum channel and optical states, and quantum capacity is one of the most important criteria to measure the efficiency of information transmission. In actual optical communication, a laser beam is used for a signal, and it is denoted mathematically by a coherent state. Further, optical communication using a squeezed state, which is expected to be more efficient than that using a coherent state is proposed. In this paper, we define several quantum channels, that is, a squeezed channel and a coherent channel and so on. We compare them by calculating quantum capacity.     

Quantum inspired method of feature fusion based on von Neumann entropy

One mission of feature fusion is to obtain a complete yet concise presentation of all existing feature data by detecting and fusing the duplicate feature data. In contrast to the already developed feature fusion methods which have shown their limitations, this paper applies the theories of quantum information to feature fusion. Further, a novel and effective step-wise quantum inspired feature fusion method, which detects the duplicate feature data based on maximum von Neumann mutual information and fuses the duplicate feature data using the operations on quantum state, is developed. This same idea is also used for feature dimensionality reduction, and the corresponding models are investigated. For comparison, another quantum inspired feature fusion method based on average quantum phase is presented here. The experimental results show that the quantum inspired feature fusion method based on von Neumann entropy gives better results on completeness and conciseness than the method based on average quantum phase.     

Dynamic analyses of information encoding in neural ensembles

Neural spike train decoding algorithms and techniques to compute Shannon mutual information are important methods for analyzing how neural systems represent biological signals. Decoding algorithms are also one of several strategies being used to design controls for brain-machine interfaces. Developing optimal strategies to design decoding algorithms and compute mutual information are therefore important problems in computational neuroscience. We present a general recursive filter decoding algorithm based on a point process model of individual neuron spiking activity and a linear stochastic state-space model of the biological signal. We derive from the algorithm new instantaneous estimates of the entropy, entropy rate, and the mutual information between the signal and the ensemble spiking activity. We assess the accuracy of the algorithm by computing, along with the decoding error, the true coverage probability of the approximate 0.95 confidence regions for the individual signal estimates. We illustrate the new algorithm by reanalyzing the position and ensemble neural spiking activity of CA1 hippocampal neurons from two rats foraging in an open circular environment. We compare the performance of this algorithm with a linear filter constructed by the widely used reverse correlation method. The median decoding error for Animal 1 (2) during 10 minutes of open foraging was 5.9 (5.5) cm, the median entropy was 6.9 (7.0) bits, the median information was 9.4 (9.4) bits, and the true coverage probability for 0.95 confidence regions was 0.67 (0.75) using 34 (32) neurons. These findings improve significantly on our previous results and suggest an integrated approach to dynamically reading neural codes, measuring their properties, and quantifying the accuracy with which encoded information is extracted.     

An efficient quantum neuro-fuzzy classifier based on fuzzy entropy and compensatory operation

In this paper, a quantum neuro-fuzzy classifier (QNFC) for classification applications is proposed. The proposed QNFC model is a five-layer structure, which combines the compensatory-based fuzzy reasoning method with the traditional Takagi–Sugeno–Kang (TSK) fuzzy model. The compensatory-based fuzzy reasoning method uses adaptive fuzzy operations of neuro-fuzzy systems that can make the fuzzy logic system more adaptive and effective. Layer 2 of the QNFC model contains quantum membership functions, which are multilevel activation functions. Each quantum membership function is composed of the sum of sigmoid functions shifted by quantum intervals. A self-constructing learning algorithm, which consists of the self-clustering algorithm (SCA), quantum fuzzy entropy and the backpropagation algorithm, is also proposed. The proposed SCA method is a fast, one-pass algorithm that dynamically estimates the number of clusters in an input data space. Quantum fuzzy entropy is employed to evaluate the information on pattern distribution in the pattern space. With this information, we can determine the number of quantum levels. The backpropagation algorithm is used to tune the adjustable parameters. The simulation results have shown that (1) the QNFC model converges quickly; (2) the QNFC model has a higher correct classification rate than other models.     

An application of the information theory to filtering problems

The purpose of this paper is to study the filtering problems from the viewpoint of the information theory. For a linear system it is proved that the necessary and sufficient condition for maximizing the mutual information between a state and the estimate is to minimize the entropy of the estimation error. Then we derive the Kalman-Bucy filter for both the discrete-time and the continuous-time systems by an application of the information theory. Furthermore, the approach is extended to the nonlinear dynamical systems with noisy observations and then the information structures of the optimal filter for a continuous-time nonlinear system are made clear, which has been presented as the interesting open problems by Bucy.     

Comparison of quantum discord and relative entropy in some bipartite quantum systems

The study of quantum correlations in high-dimensional bipartite systems is crucial for the development of quantum computing. We propose relative entropy as a distance measure of correlations may be measured by means of the distance from the quantum state to the closest classical–classical state. In particular, we establish relations between relative entropy and quantum discord quantifiers obtained by means of orthogonal projection measurements. We show that for symmetrical X-states density matrices the quantum discord is equal to relative entropy. At the end of paper, various examples of X-states such as two-qubit and qubit–qutrit have been demonstrated.     

State transfer with quantum side information

We first consider quantum communication protocols between a sender Alice and a receiver Bob, which transfer Alice’s quantum information to Bob by means of non-local resources, such as classical communication, quantum communication, and entanglement. In these protocols, we assume that Alice and Bob may have quantum side information, not transferred. In this work, these protocols are called the state transfer with quantum side information. We determine the optimal costs for non-local resources in the protocols and study what the effects of the use of quantum side information are. Our results can give new operational meanings to the quantum mutual information and the quantum conditional mutual information, which directly provide us with an operational interpretation of the chain rule for the quantum mutual information.     

互信息的序决策信息系统属性约简研究

优势关系粗糙集理论是粗糙集理论有意义的推广,决策信息系统知识约简是粗糙集理论的核心内容之一.通过在协调序决策信息系统中引入条件熵、互信息概念,给出了基于条件熵、互信息的协调序决策信息系统属性约简算法,并通过学生评价决策信息系统验证了该算法的有效性,使协调序决策信息系统的属性约简得到了扩展.在不协调序决策信息系统中引入限定条件熵、限定互信息概念,并给出基于限定互信息的不协调序决策信息系统属性约简算法,为不协调序决策信息系统的属性约简的应用提供了可行的解决方法.     

Some Properties of Conservative Port Contact Systems

The dynamics of open irreversible thermodynamic systems, that is systems including both the balance equation of the energy and the entropy, has been formulated as contact vector fields with generating functions depending on some external (control) variable and called conservative port contact systems. In this paper we relate the dynamical properties of these systems (equilibrium points, asymptotic stability) to properties of the generating functions (the contact Hamiltonian functions). We show that the equilibrium points of the system satisfy certain conditions involving the contact Hamiltonian function. We also consider Lyapunov's first theorem to emphasize a stability criterion for the equilibrium points in terms of this contact Hamiltonian function and relate it to some thermodynamical properties. These results are then related to the physical phenomena that are taking place in the system.     

Does “cooling by heating” protect quantum correlations?

The connection between nonequilibrium quantum correlations, such as entanglement and quantum discord, and cooling by heating is investigated for a system composed by two atoms interacting with a single electromagnetic mode of a lossy cavity. This Hamiltonian model is experimentally feasible in the quantum optics domain and presents the occurrence of both nonequilibrium quantum correlations and cooling by heating for a range of parameters. Since in the cooling by heating phenomenon the effective temperature of the system decreases even increasing the environments temperature, it could be expected that quantum correlations could be enhanced. Interestingly, for some parameters we show that, contrary to this expectation, in the case studied here the lowering of the system effective temperature leads to no enhancement in the quantum correlations. In addition, we found that at both zero and finite temperature, depending on the parameters used, quantum correlations can be enhanced even when increasing the damping rates, a somewhat counterintuitive result.     

Topological phenomena in quantum walks: elementary introduction to the physics of topological phases

Discrete quantum walks are dynamical protocols for controlling a single quantum particle. Despite of its simplicity, quantum walks display rich topological phenomena and provide one of the simplest systems to study and understand topological phases. In this article, we review the physics of discrete quantum walks in one and two dimensions in light of topological phenomena and provide elementary explanations of topological phases and their physical consequence, namely the existence of boundary states. We demonstrate that quantum walks are versatile systems that simulate many topological phases whose classifications are known for static Hamiltonians. Furthermore, topological phenomena appearing in quantum walks go beyond what has been known in static systems; there are phenomena unique to quantum walks, being an example of periodically driven systems, that do not exist in static systems. Thus the quantum walks not only provide a powerful tool as a quantum simulator for static topological phases but also give unique opportunity to study topological phenomena in driven systems.