Information-theoretic differential geometry of quantum phase transitions

The manifold of coupling constants parametrizing a quantum Hamiltonian is equipped with a natural Riemannian metric with an operational distinguishability content. We argue that the singularities of this metric are in correspondence with the quantum phase transitions featured by the corresponding system. This approach provides a universal conceptual framework to study quantum critical phenomena which is differential geometric and information theoretic at the same time.     

Information-theoretic basis of spectral estimates of entropy minimax

The relationship between the entropy-minimax method and other methods of spectral analysis, such as the maximum-entropy, eigenvector, and maximum-likelihood methods, is discussed. The method is shown to be optimal in the Kullback-Leibler information metric for a wide class of problems.Nizhny Novgorod State Technical University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 36, No. 11, pp. 1002–1010, November, 1993.     

Information-theoretic stability and evolution criteria in irreversible thermodynamics

Generalized information gains are used to derive stability conditions, for steady states, periodic orbits and invariant sets, and general evolution criteria, both global and local with respect to the time variable, in irreversible thermodynamics. Meixner's passivity condition and the Glansdorff-Prigogine stability and evolution criteria are found to be special cases thereof. The information gain quantities include Kullback's three kinds of divergences, the first two of which are dual to each other and yield criteria which are symmetric in the average densities of the system's extensive variables and the conjugate parameters, but which are nonsymmetric in the irreversible fluxes and forces, while the third one does not involve the entropy function of the system. Furthermore, Renyi's information gain of order and Csiszar'sf-divergence are treated. The latter is used to construct a most general information gain quantity as a Liapunov function and evolution criterion, which, however, for local stability and evolution conditions is still equivalent to the use of the second-order variation of the entropy.Supported by the Deutsche Forschungsgemeinschaft.     

Information-theoretic bell inequalities and the relative measure of probability

The information-theoretic Bell inequalities of Braunstein and Caves are studied in relation to the concept of the relative probability measure which one allows to overcome the limitations of the considered theorem.I esteem it an honor to dedicate this paper to Sir Karl Popper on the occasion of his 90th birthday.     

Information-theoretic approach to single-particle and two-particle interference in multipath interferometers

We propose entropic measures for the strength of single-particle and two-particle interference in interferometric experiments where each particle of a pair traverses a multipath interferometer. Optimal single-particle interference excludes any two-particle interference, and vice versa. We report an inequality that states the compromises allowed by quantum mechanics in intermediate situations, and identify a class of two-particle states for which the upper bound is reached. Our approach is applicable to symmetric two-partite systems of any finite dimension.     

Information-theoretic approach to the problem of detection of genuine multipartite classical correlations

We examine the problem of detection of genuine multipartite classical correlations. We first show that genuine multipartite classical correlations cannot be reliably detected by analyzing the multipartite covariances, and then we introduce an information-theoretic approach to the problem of detection of such correlations.     

Information-theoretic stochastic resonance in noise-floor limited systems: the case for adding noise

We show that in systems whose output must compete with a noise source, stochastic resonance (maximization of output signal-noise separation as a nonmonotonic function of input noise strength) exists even when measured in terms of fundamental statistical measures and optimal detector performance. This is in contrast to the commonly considered scenario where, without the competing noise, the system (e.g., a driven, overdamped particle moving in a double well potential) is essentially invertible and optimal detector performance monotonically deteriorates with increasing input noise strength.     

Realistic limits to computation I. Physical limits

The ultimate limits of computation have been determined in the hypothesis that computation is a physical process occurring in a medium immersed in a thermal reservoir at assigned (room) temperature and thus obeying the underlying physical laws. Whichever is the information carrier, the computational figure of merit is inherently reduced by the need of transforming the microscopic computation outcome into a macroscopic event. The resulting loss of performance has been estimated in the hypothesis that the microscopic state is sensed with an apparatus undergoing repeated measurements. PACS 81.07.-b; 81.07.Nb; 85.35.-p     

Physical limits of inference

We show that physical devices that perform observation, prediction, or recollection share an underlying mathematical structure. We call devices with that structure “inference devices”. We present a set of existence and impossibility results concerning inference devices. These results hold independent of the precise physical laws governing our universe. In a limited sense, the impossibility results establish that Laplace was wrong to claim that even in a classical, non-chaotic universe the future can be unerringly predicted, given sufficient knowledge of the present. Alternatively, these impossibility results can be viewed as a non-quantum-mechanical “uncertainty principle”.The mathematics of inference devices has close connections to the mathematics of Turing Machines (TMs). In particular, the impossibility results for inference devices are similar to the Halting theorem for TMs. Furthermore, one can define an analog of Universal TMs (UTMs) for inference devices. We call those analogs “strong inference devices”. We use strong inference devices to define the “inference complexity” of an inference task, which is the analog of the Kolmogorov complexity of computing a string. A task-independent bound is derived on how much the inference complexity of an inference task can differ for two different inference devices. This is analogous to the “encoding” bound governing how much the Kolmogorov complexity of a string can differ between two UTMs used to compute that string. However no universe can contain more than one strong inference device. So whereas the Kolmogorov complexity of a string is arbitrary up to specification of the UTM, there is no such arbitrariness in the inference complexity of an inference task.We informally discuss the philosophical implications of these results, e.g., for whether the universe “is” a computer. We also derive some graph-theoretic properties governing any set of multiple inference devices. We also present an extension of the framework to address physical devices used for control. We end with an extension of the framework to address probabilistic inference.     

Fundamental limits of optical superresolution

Increasing the resolving power of optical systems beyond the limits imposed by diffraction, or superresolution, has considerable theoretical as well as practical interest. Several schemes have been proposed to achieve superresolution with reasonable success, but there are no criteria that enable one to determine what improvement can ultimately be attained for a certain level of resolution. We have determined fundamental limits imposed on the performance of any superresolution strategy. A brief analysis indicates that current optical-superresolution techniques can still have their performance considerably augmented.     

Comments on limits of space-times

A coordinate-dependent investigation of limits of space-times is given. Different kinds of limits are found and analyzed. A criterion selecting limits which appear to be nothing but technical tricks is proposed. Finally, a method for generating new limits from a given family of metrics is examined.This work was carried out under the auspices of the National Group for Mathematical Physics of C.N.R.     

Physical limits to communication

The limits posed by physics to the quantity of information that can be transmitted with a certain amount of power are investigated. The same ultimate limits are found for transmission of information encoded using matter and massless fields.     

Cosmological limits on computation

A true universal Turing machine can be constructed only if it is possible to actually process and store an infinite number of bits between now and the end of the universe. Conditions on the universe are derived that must hold if such processing and storage is to be possible. In particular, it is shown that it is possible only if the universe is closed and only if its futurec-boundary consists of a single point.     

Physical limits of heat-bath algorithmic cooling

Simultaneous near-certain preparation of qubits (quantum bits) in their ground states is a key hurdle in quantum computing proposals as varied as liquid-state NMR and ion traps. "Closed-system" cooling mechanisms are of limited applicability due to the need for a continual supply of ancillas for fault tolerance, and to the high initial temperatures of some systems. "Open-system" mechanisms are therefore required. We describe a new, efficient initialization procedure for such open systems. With this procedure, an n-qubit device that is originally maximally mixed, but is in contact with a heat bath of bias epsilon>2(-n), can be almost perfectly initialized. This performance is optimal due to a newly discovered threshold effect: for bias epsilon<2(-n) no cooling procedure can, even in principle (running indefinitely without any decoherence), significantly initialize even a single qubit.     

Newtonian limits of warp drive spacetimes

We find a class of warp drive spacetimes possessing Newtonian limits, which we then determine. The same method is used to compute Newtonian limits of the Schwarzschild solution and spatially flat Friedmann-Robertson-Walker cosmological models. This work was partially supported by FCT/POCTI/FEDER.