Source coding with escort distributions and Rényi entropy bounds

We discuss the interest of escort distributions and Rényi entropy in the context of source coding. We first recall a source coding theorem by Campbell relating a generalized measure of length to the Rényi-Tsallis entropy. We show that the associated optimal codes can be obtained using considerations on escort-distributions. We propose a new family of measure of length involving escort-distributions and we show that these generalized lengths are also bounded below by the Rényi entropy. Furthermore, we obtain that the standard Shannon codes lengths are optimum for the new generalized lengths measures, whatever the entropic index. Finally, we show that there exists in this setting an interplay between standard and escort distributions.     

Equivalence among different formalisms in the Tsallis entropy framework

In a recent paper [T. Wada, A.M. Scarfone, Phys. Lett. A 335 (2005) 351] the authors discussed the equivalence among the various probability distribution functions of a system in equilibrium in the Tsallis entropy framework. In the present letter we extend these results to a system which is out of equilibrium and evolves to a stationary state according to a nonlinear Fokker-Planck equation (NFPE). By means of time-scale conversion, it is shown that there exists a “correspondence” among the self-similar solutions of the NFPEs associated with the different Tsallis formalisms. The time-scale conversion is related to the corresponding Lyapunov functions of the respective NFPEs.     

Tsallis distribution as a standard maximum entropy solution with ‘tail’ constraint

We show that Tsallis' distributions can be derived from the standard (Shannon) maximum entropy setting, by incorporating a constraint on the divergence between the distribution and another distribution imagined as its tail. In this setting, we find an underlying entropy which is the Rényi entropy. Furthermore, escort distributions and generalized means appear as a direct consequence of the construction. Finally, the “maximum entropy tail distribution” is identified as a Generalized Pareto Distribution.     

The world according to Rényi: thermodynamics of multifractal systems

We discuss basic statistical properties of systems with multifractal structure. This is possible by extending the notion of the usual Gibbs-Shannon entropy into more general framework—Rényi’s information entropy. We address the renormalization issue for Rényi’s entropy on (multi)fractal sets and consequently show how Rényi’s parameter is connected with multifractal singularity spectrum. The maximal entropy approach then provides a passage between Rényi’s information entropy and thermodynamics of multifractals. Important issues such as Rényi’s entropy versus Tsallis-Havrda-Charvat entropy and PDF reconstruction theorem are also studied. Finally, some further speculations on a possible relevance of our approach to cosmology are discussed.     

Asymmetric information and economics

We present an expression of the economic concept of asymmetric information with which it is possible to derive the dynamical laws of an economy. To illustrate the utility of this approach we show how the assumption of optimal information flow leads to a general class of investment strategies including the well-known Q theory of Tobin. Novel consequences of this formalism include a natural definition of market efficiency and an uncertainty principle relating capital stock and investment flow.     

Information and maximum power in a feedback controlled Brownian ratchet

Closed-loop or feedback controlled ratchets are Brownian motors that operate using information about the state of the system. For these ratchets, we compute the power output and we investigate its relation with the information used in the feedback control. We get analytical expressions for one-particle and few-particle flashing ratchets, and we find that the maximum power output has an upper bound proportional to the information. In addition, we show that the increase of the power output that results from changing the optimal open-loop ratchet to a closed-loop ratchet also has an upper bound that is linear in the information.     

Security risk during the transient in a key exchange protocol using random signals and feedback

Random signals and feedback may facilitate the exchange of shared keys in secure communications systems. In this Letter, the security risk during the initial turn-on is examined. Results indicate that causality plays a critical role. If signals are continuous, the eavesdropper can use extrapolation to breach the security. In digital signaling, however, two parties in communication are in control. They use sampling and set the quantization accuracy to limit the information available to eavesdropper. They can gradually increase the feedback coefficient. By using these countermeasures, they can prevent the eavesdropper from gathering useful information during the transient.     

Comment on: “A class of bound entangled states” [Phys. Lett. A 352 (2006) 321]

We generalize the condition keeping the special states bound entangled [S.-M. Fei, X. Li-Jost, B.-Z. Sun, Phys. Lett. A 352 (2006) 321] to the general states. Taking the original 4×4$4\times 4$ states as 2×8$2\times 8$ states we show that they still can be bound entangled.     

Blind encoding into qudits

We consider the problem of encoding classical information into unknown qudit states belonging to any basis, of a maximal set of mutually unbiased bases, by one party and then decoding by another party who has perfect knowledge of the basis. Working with qudits of prime dimensions, we point out a no-go theorem that forbids ‘shift’ operations on arbitrary unknown states. We then provide the necessary conditions for reliable encoding/decoding.     

Stability of incomplete entropy and incomplete expectation

Based on the principle of maximum entropy, the q-exponential distribution can be derived from several different nonextensive entropies including the incomplete entropy. It is widely used in nonextensive statistical mechanics. In the present paper, it is shown that the incomplete expectation value and incomplete entropy are stable under small deformation of the probability distribution function of the system.     

Is computation reversible?

Recent studies have suggested that computation is essentially reversible, provided no information is lost. This is a consequence of Landauer’s principle which only requires energy expenditure and entropy increase for information deletion. In this paper we propose to treat information as being intrinsic to points of non-analyticity, so that the movement of information is always associated with the dissipation of heat. This allows us to construct a theory consistent with causality, and the second law of thermodynamics. Since computation requires the movement of information bits through finite volume gates, energy is dissipated even when information is not destroyed, thus indicating that computation is fundamentally non-reversible.     

Realization of Greenberg-Horne-Zeilinger （GHZ） and W Entangled States with Multiple Superconducting Quantum-Interference Device Qubits in Cavity QED

An alternative scheme is proposed for generating the Greenberg-Horne-Zeilinger （GHZ） and W types of the entangled states with multiple superconducting quantum-interference device （SQUID） qubits in a single-mode microwave cavity field. In this scheme, there is no transfer of quantum information between the SQUIDs and the cavity, the cavity is always in the vacuum and thus the requirement on the quality of cavity is greatly loosened. In addition, during the process of the generation of the W entangled state, the present method does not involve a real excitation of intermediate levels. Thus, decoherence due to energy relaxation of intermediate levels is minimized.     

Anisotropic and large low-field magnetic entropy change in a La4/3Sr5/3Mn2O7 single crystal

We have studied the magnetocaloric effect (MCE) in a bilayered La_4/3Sr_5/3Mn₂O₇ single crystal with applied field along both ab-plane and c-direction. Due to the quasi-two-dimensional structure, the crystal exhibits a strong anisotropy in the MCE. The difference of magnetic entropy change between two crystallographic directions depends on external magnetic fields and has a maximum of 2 J/kg K. A large low-field magnetic entropy change, reaching 3.2 J/kg K for a magnetic field change of 15 kOe, is observed when the applied field is along ab-plane. This large low-field magnetic entropy change is attributed to the rapid change of magnetization in response to external magnetic fields in the easy magnetizing plane.     

Information flow between composite stock index and individual stocks

We investigate the strength and the direction of information transfer in the US stock market between the composite stock price index of stock market and prices of individual stocks using the transfer entropy. Through the directionality of the information transfer, we find that individual stocks are influenced by the index of the market.     

Landauer's principle and non-equilibrium statistical ensembles

Landauer's principle is fundamental for the physics of information. It establishes the least amount of energy that needs to be dissipated in order to erase a bit of information. Using the Beck-Cohen representation of statistical ensemble distributions, we explore an extension of Landauer's principle to systems out of equilibrium.     

Existence conditions and spreading properties of extreme entropy D-dimensional distributions

The extremization of the information-theoretic measures (Fisher information, Shannon entropy, Tsallis entropy), which complementary describe the spreading of the physical states of natural systems, gives rise to fundamental equations of motion and/or conservation laws. At times, the associated extreme entropy distributions are known for some given constraints, usually moments or radial expectation values. In this work, first we give the existence conditions of the maxent probability distributions in a D-dimensional scenario where two moments (not necessarily of consecutive order) are known. Then we find general relations which involve four elements (the extremized entropy, the other two information-theoretic measures and the variance of the extremum density) in scenarios with different dimensionalities and moment constraints.     

The distribution of quantum fidelities

When applied to different input states, an imperfect quantum operation yields output states with varying fidelities, defined as the absolute square of their overlap with the desired states. We present an expression for the distribution of fidelities for a class of operations applied to a general qubit state, and we present general expressions for the variance and input-space averaged fidelities of arbitrary linear operations on state vectors in finite-dimensional Hilbert spaces.     

Encryption test of pseudo-aleatory messages embedded on chaotic laser signals: An information theory approach

In this Letter, we make use of two information-theory based indicators to measure the goodness of two encryption schemes commonly used within the context of chaotic communications. In particular, we have shown that the computation of the normalized Shannon entropy and the MPR-Statistical Complexity measure [M.T. Martín, A. Plastino, O.A. Rosso, Phys. Lett. A 311 (2003) 126, P.W. Lamberti, M.T. Martín, A. Plastino, O.A. Rosso, Physica A 334 (2004) 119] for different chaotic laser signals can lead to statistically significant criteria to assess the quality of several encryption techniques. The proposed measures allow, in some cases, to detect the presence of a message embedded within a chaotic carrier. They also reveal that the Chaos Modulation scheme is more reliable from the statistical point of view, when compared with the Chaos Shift Keying.     

On entropy, financial markets and minority games

The paper builds upon an earlier statistical analysis of financial time series with Shannon information entropy, published in [L. Molgedey, W. Ebeling, Local order, entropy and predictability of financial time series, European Physical Journal B—Condensed Matter and Complex Systems 15/4 (2000) 733-737]. A novel generic procedure is proposed for making multistep-ahead predictions of time series by building a statistical model of entropy. The approach is first demonstrated on the chaotic Mackey-Glass time series and later applied to Japanese Yen/US dollar intraday currency data. The paper also reinterprets Minority Games [E. Moro, The minority game: An introductory guide, Advances in Condensed Matter and Statistical Physics (2004)] within the context of physical entropy, and uses models derived from minority game theory as a tool for measuring the entropy of a model in response to time series. This entropy conditional upon a model is subsequently used in place of information-theoretic entropy in the proposed multistep prediction algorithm.     

Secret Key Distillation for Continuous Variable Quantum Key Distribution against Gaussian Classical Eve

The continuous variable quantum key distribution is expected to provide high secret key rate without single photon source and detector, while the lack of the effective key distillation method makes it unpractical under the high loss condition. Here we present a single-bit-reverse-reconciliation protocol against Oaussian classical Eve, which can distill the secret key through practical imperfect error correction with high efficiency. The simulation results show that this protocol can distill secret keys even when the transmission fibre is longer than 150 km, which may make the continuous variable scheme to outvie the single photon one.