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If a quantum logic16, 18, 21 admits hidden variables in a certain sense, then the resulting structure is called a “generalized measurable space”. Such a structure is more general than a traditional measurable space and is distinct from the conventional Hilbert space quantum mechanics. A generalized measurable space consists of a set Ω together with a collection of subsets C which form a α-orthocomplete orthomodular poset2 under set theoretic inclusion and complementation. We may think of (Ω, C) as giving a quantum set theory. Besides being of interest in quantum mechanics, there are possible applications of this theory to investigations in “elementary lengths”, pattern recognition, and computer science.1, 3, 5, 9, 12, 15, 22, 23, 24

Basic concepts and results of physical information theory are presented. The entropy defect and Shannon’s measure of information are introduced and the entropy defect principle is formulated for both quasiclassical and consistently quantum description of a physical system. Results related to ideal physical information channels are discussed. The entropy defect and the amount of information coincide in the quasiclassical case, but the latter quantity is, in general, smaller than the former in quantum case due to the quantum-mechanical irreversibility of measurement. The physical meaning of both quantities is analyzed in connection with Gibbs paradox and the maximum work obtainable from a non-equilibrium system. Indirect (generalized) vs. direct (von Neumann’s) quantum measurements are considered. It is shown that in any separable infinite-dimensional Hilbert space direct and indirect quantum measurements yield equal maximum information.

A sandwich technique is presented. Two white light speckle patterns corresponding to the two different loading states of the structure are recorded. Analysis can be carried on in the Fourier transformation optical system as well as the point diffraction optical system. The coherent conditions of the speckle sandwich patterns are derived. The mismatch of the two corresponding speckle patterns can be made use of to introduce the linear photo-carrier with which the sensitivity and accuracy of the white light speckle are improved, meanwhile the measurement scale is enlarged.

Despite its long history, which stretches back in excess of one hundred years, to many the term entropy still appears esoteric. In the early days of classical thermodynamics, perhaps it was; even though its original meaning was grounded in a bedrock of physical facts. But nowadays it is becoming increasingly popular in one field after another. To some extent, these more recent adaptations are related in a purely formal way to a simple algebraic formula which is the cloak under which entropy grows more familiar to an increasing number of social scientists.1

The idea of infinite structures, even of continua, in information theory is not a new one [K056]. This note is devoted to infinite one-time pads [BL81] and an infinite projective geometric [BL79] threshold scheme. Perhaps an infinite structure can be better understood than its finite analog if it is amenable to investigation by methods from calculus or harmonic analysis. It is conceivable that an existing error control code, pool/split/restitute process [AS82], or cryptosystem can be better understood by examining an infinite version.

The classical entropy of a coherent state of an N-spin system, which is N/(N + 1), was conjectured by Lieb to be the minimum. To prove it was our original motive. But we can only prove that this entropy is a local minimum. An arbitrary N-spin state is represented by N points on the surface of the unit sphere. For a coherent state, these N points condense into a single point. This is the basis for the proof. It is also conjectured here that the maximum entropy is attained when the N points form a possible regular polyhedron.

A generalization of linear-response theory is attempted, within the framework of information theory [Y. Alhassid and R. D. Levine, Phys. Rev. A 18, 89 (1978); D. Otero et al., Phys. Rev. A 26, 1209 (1982)], with the idea of developing a formalism capable of dealing with small deviations from a previously well-described situation. The ensuing approach is applied to diverse physical situations that illustrate its flexibility.

A noncommutative version of the Cramer theorem is used to show that if two quantum systems are prepared independently, and if their center of mass is found to be in a coherent state, then each of the component systems is also in a coherent state, centered around the position in phase space predicted by the classical theory. Thermal coherent states are also shown to possess properties similar to classical ones.

Publisher Summary This chapter discusses various generalized entropies and generalized divergence measures. The focus is on the generalizations of Shannon's entropy, directed divergence, J-divergence, and information radius. The applications of these generalized information measures to statistical pattern recognition are discussed in the chapter. These studies have been conducted on the generalized information measures written in unified expressions. Information theory studies all theoretical problems connected with the transmission of information over communication channels. This includes the study of uncertainty (information) measure and various practical and economical methods of coding information for transmission. A key feature of Shannon information theory is that the term information can often be given a mathematical meaning as a numerically measurable quantity, on the basis of a probabilistic model, in such a way that the solutions of many important problems of information storage and transmission can be formulated in terms of this measure of the amount of information. This important measure has a very concrete operational interpretation. It roughly equals the minimum number of binary digits needed, on the average, to encode the message in question.

This paper presents several fundamental concepts necessary for the successful analysis of dynamic scenes containing occluding objects by discussing various systems which have been developed to perform this analysis. The dynamic scenes are represented by time ordered sequences of images. Data must be extracted from each of these images and then integrated into coherent information about the sequence as a whole.

Today the data acquisition systems used during drilling operations tend to collect large amounts of data which people in charge do not have time to fully interpret. This is unfortunate because miraculous post-analysis will often prove that there are a lot of "gold nuggets" in these cubic metres of data records, reports, tapes or disks. One way to facilitate real-time analysis, is to improve the accuracy and reliability of the sensors. A more efficient way is to increase the confidence in the data by screening all the insignificant measurements. PC-ANA and INKA use expert systems technology to sort out only the pertinent data in the current context. This needs an extremely precise real-time analysis of the rig status to store, to use or to visualise only relevant and coherent information.

The operationally defined invariant information introduced by Brukner and Zeilinger is related to the problem of estimation of quantum states. It quantifies how the estimated states differ in average from the true states in the sense of Hilbert-Schmidt norm. This information evaluates the quality of the measurement and data treatment adopted. Its ultimate limitation is given by the trace of inverse of Fisher information matrix.

The ability to protect quantum information from the effect of noise is one of the major goals of quantum information processing. In this article, we study limitations on the asymptotic stability of quantum information stored in passive N-qubit systems. We consider the effect of small imperfections in the implementation of the protecting Hamiltonian in the form of perturbations or weak coupling to a ground state environment. We prove that, regardless of the protecting Hamiltonian, there exists a perturbed evolution that necessitates a final error correcting step when the state of the memory is read. Such an error correction step is shown to require a finite error threshold, the lack thereof being exemplified by the 3D compass model. We go on to present explicit weak Hamiltonian perturbations which destroy the logical information stored in the 2D toric code in a time O(log(N))., 17 pages and appendices

We study the roles of quantum correlations, entanglement, discord, and dissonance needed for performing unambiguous quantum state discrimination assisted by an auxiliary system.

Networks of open quantum systems with feedback have become an active area of research for applications such as quantum control, quantum communication and coherent information processing. A canonical formalism for the interconnection of open quantum systems using quantum stochastic differential equations (QSDEs) has been developed by Gough, James and co-workers and has been used to develop practical modeling approaches for complex quantum optical, microwave and optomechanical circuits/networks. In this paper we fill a significant gap in existing methodology by showing how trapped modes resulting from feedback via coupled channels with finite propagation delays can be identified systematically in a given passive linear network. Our method is based on the Blaschke-Potapov multiplicative factorization theorem for inner matrix-valued functions, which has been applied in the past to analog electronic networks. Our results provide a basis for extending the Quantum Hardware Description Language (QHDL) framework for automated quantum network model construction (Tezak \textit{et al.} in Philos. Trans. R. Soc. A, Math. Phys. Eng. Sci. 370(1979):5270-5290, to efficiently treat scenarios in which each interconnection of components has an associated signal propagation time delay.

We prove that a general upper bound on the maximal mutual information of quantum channels is saturated in the case of Pauli channels with an arbitrary degree of memory. For a subset of such channels we explicitly identify the optimal signal states. We show analytically that for such a class of channels entangled states are indeed optimal above a given memory threshold.

We construct new polar coding schemes for the transmission of quantum or private classical information over arbitrary quantum channels. In the former case, our coding scheme achieves the symmetric coherent information and in the latter the symmetric private information. Both schemes are built from a polar coding construction capable of transmitting classical information over a quantum channel [Wilde and Guha, IEEE Transactions on Information Theory, in press]. Appropriately merging two such classical-quantum schemes, one for transmitting "amplitude" information and the other for transmitting "phase," leads to the new private and quantum coding schemes, similar to the construction for Pauli and erasure channels in [Renes, Dupuis, and Renner, Physical Review Letters 109, 050504 (2012)]. The encoding is entirely similar to the classical case, and thus efficient. The decoding can also be performed by successive cancellation, as in the classical case, but no efficient successive cancellation scheme is yet known for arbitrary quantum channels. An efficient code construction is unfortunately still unknown. Generally, our two coding schemes require entanglement or secret-key assistance, respectively, but we extend two known conditions under which the needed assistance rate vanishes. Finally, although our results are formulated for qubit channels, we show how the scheme can be extended to multiple qubits. This then demonstrates a near-explicit coding method for realizing one of the most striking phenomena in quantum information theory: the superactivation effect, whereby two quantum channels which individually have zero quantum capacity can have a non-zero quantum capacity when used together., Comment: 14 pages; subsumes and extends the results of arXiv:1201.2906 and arXiv:1203.5794

The classical information entropy defined by Wehrl in terms of the Husimi Q-function is discussed and generalized over the concepts of the Wehrl phase distribution, and the Wehrl intermode-correlation parameters. The classical entropic functions are applied to describe the quantum properties of single and/or two-mode optical fields.

We consider the noisy thermal amplifier channel, where signal modes are amplified together with environmental thermal modes. We focus on the secret-key capacity of this channel, which is the maximum amount of secret bits that two remote parties can generate by means of the most general adaptive protocol, assisted by unlimited and two-way classical communication. For this channel only upper and lower bounds are known, and in this work we improve the lower bound. We consider a protocol based on squeezed states and homodyne detections, in both direct and reverse reconciliation. In particular, we assume that trusted thermal noise is mixed on beam splitters controlled by the parties in a way to assist their homodyne detections. The new improved lower bounds to the secret-key capacity are obtained by optimizing the key rates over the variance of the trusted noise injected, and the transmissivity of the parties' beam splitters. Our results confirm that there is a separation between the coherent information of the thermal amplifier channel and its secret key capacity.

Information theory, as developed by Shannon (1948), provides a theoretical framework for describing the transfer of information through a communication channel. A coherent imaging system is a particular form of communication channel which is characterized by a multiplicative noise process termed 'speckle'. Information theory is used to assess the effect of speckle on the information content of coherent images. Particular emphasis is given to relating information theoretic concepts to intuitive descriptions of information based on physical models.

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We present some new results which relate information to chaotic dynamics. In our approach the quantity of information is measured by the Algorithmic Information Content (Kolmogorov complexity) or by a sort of computable version of it (Computable Information Content) in which the information is measured by the use of a suitable universal data compression algorithm. We apply these notions to the study of dynamical systems by considering the asymptotic behavior of the quantity of information necessary to describe their orbits. When a system is ergodic, this method provides an indicator which equals the Kolmogorov-Sinai entropy almost everywhere. Moreover, if the entropy is 0, our method gives new indicators which measure the unpredictability of the system and allows to classify various kind of weak chaos. Actually this is the main motivation of this work. The behaviour of a zero entropy dynamical system is far to be completely predictable exept that in particular cases. In fact there are 0 entropy systems which exibit a sort of {\it weak chaos} where the information necessary to describe the orbit behavior increases with time more than logarithmically (periodic case) even if less than linearly (positive entropy case). Also, we believe that the above method is useful for the classification of zero entropy time series. To support this point of view, we show some theoretical and experimenthal results in specific cases., Comment: 33 pages, 10 figures. Submitted to Continuous Dynamical Systems A

The paper is devoted to consideration of a number of recently published results, possessing, in main, a conceptual feature, and related with attempts of generalization of the notion of "entropy". It has been shown the attempts to be scientific delusions. A correct variant of a generalized description for the entropy within the framework is proposed.

Summary form only given. We analyze a general quantum channel, and describe how entanglement changes due to interaction with the environment. We obtain information, entanglement and coherent information. Based on these we relate the two separate uses of the quantum channel and obtain a relationship between the Holevo bound and the amount of entanglement transmitted through a quantum channel. The channel might also represent actions of an eavesdropper so that these results can be used to put bounds on the eavesdropping efficiency. We apply these results to a pair of entangled optical modes when one of them is transmitted through a decoherent channel.

It has been suggested that long-range lateral connections in the cortex play a contextual role in that they modulate the gain of the response to primary receptive field input. In the first part of this paper I show that a network with a set of such pre-wired connections has a short-term dynamics that enhances and stabilizes coherent information defined across multiple, non-overlapping receptive fields. In the second part, I suggest a simple Hebbian rule that can develop the required pattern of synaptic strengths and describe two simulations where the networks discover information that is defined only by its coherence across receptive fields.

Using information theory and Glauber's definition of (first-order) coherence the statistical operator of a partially coherent radiation field is derived. Von Laue's formula for the entropy is proved.

It is shown how coherent states may be constructed from a condition of minimum uncertainty.

We reflect on the information paradigm in quantum and gravitational physics and on how it may assist us in approaching quantum gravity. We begin by arguing, using a reconstruction of its formalism, that quantum theory can be regarded as a universal framework governing an observer's acquisition of information from physical systems taken as information carriers. We continue by observing that the structure of spacetime is encoded in the communication relations among observers and more generally the information flow in spacetime. Combining these insights with an information-theoretic Machian view, we argue that the quantum architecture of spacetime can operationally be viewed as a locally finite network of degrees of freedom exchanging information. An advantage -- and simultaneous limitation -- of an informational perspective is its quasi-universality, i.e. quasi-independence of the precise physical incarnation of the underlying degrees of freedom. This suggests to exploit these informational insights to develop a largely microphysics independent top-down approach to quantum gravity to complement extant bottom-up approaches by closing the scale gap between the unknown Planck scale physics and the familiar physics of quantum (field) theory and general relativity systematically from two sides. While some ideas have been pronounced before in similar guise and others are speculative, the way they are strung together and justified is new and supports approaches attempting to derive emergent spacetime structures from correlations of quantum degrees of freedom.