Information-theoretic temporal Bell inequality and quantum computation

An information-theoretic temporal Bell inequality is formulated to contrast classical and quantum computations. Any classical algorithm satisfies the inequality, while quantum ones can violate it. Therefore, the violation of the inequality is an immediate consequence of the quantumness in the computation. Furthermore, this approach suggests a notion of temporal nonlocality in quantum computation.Comment: v2: 5 pages, refereces added, discussion slightly revised, main result unchanged. v3: typos correcte

Non-local correlations as an information theoretic resource

It is well known that measurements performed on spatially separated entangled quantum systems can give rise to correlations that are non-local, in the sense that a Bell inequality is violated. They cannot, however, be used for super-luminal signalling. It is also known that it is possible to write down sets of ``super-quantum'' correlations that are more non-local than is allowed by quantum mechanics, yet are still non-signalling. Viewed as an information theoretic resource, super-quantum correlations are very powerful at reducing the amount of communication needed for distributed computational tasks. An intriguing question is why quantum mechanics does not allow these more powerful correlations. We aim to shed light on the range of quantum possibilities by placing them within a wider context. With this in mind, we investigate the set of correlations that are constrained only by the no-signalling principle. These correlations form a polytope, which contains the quantum correlations as a (proper) subset. We determine the vertices of the no-signalling polytope in the case that two observers each choose from two possible measurements with d outcomes. We then consider how interconversions between different sorts of correlations may be achieved. Finally, we consider some multipartite examples.Comment: Revtex. 12 pages, 6 figure

The thermodynamics of prediction

A system responding to a stochastic driving signal can be interpreted as computing, by means of its dynamics, an implicit model of the environmental variables. The system's state retains information about past environmental fluctuations, and a fraction of this information is predictive of future ones. The remaining nonpredictive information reflects model complexity that does not improve predictive power, and thus represents the ineffectiveness of the model. We expose the fundamental equivalence between this model inefficiency and thermodynamic inefficiency, measured by dissipation. Our results hold arbitrarily far from thermodynamic equilibrium and are applicable to a wide range of systems, including biomolecular machines. They highlight a profound connection between the effective use of information and efficient thermodynamic operation: any system constructed to keep memory about its environment and to operate with maximal energetic efficiency has to be predictive.Comment: 5 pages, 1 figur

Quantum conditional operator and a criterion for separability

We analyze the properties of the conditional amplitude operator, the quantum analog of the conditional probability which has been introduced in [quant-ph/9512022]. The spectrum of the conditional operator characterizing a quantum bipartite system is invariant under local unitary transformations and reflects its inseparability. More specifically, it is shown that the conditional amplitude operator of a separable state cannot have an eigenvalue exceeding 1, which results in a necessary condition for separability. This leads us to consider a related separability criterion based on the positive map $\Gamma:\rho \to (Tr \rho) - \rho$, where $\rho$ is an Hermitian operator. Any separable state is mapped by the tensor product of this map and the identity into a non-negative operator, which provides a simple necessary condition for separability. In the special case where one subsystem is a quantum bit, $\Gamma$ reduces to time-reversal, so that this separability condition is equivalent to partial transposition. It is therefore also sufficient for $2\times 2$ and $2\times 3$ systems. Finally, a simple connection between this map and complex conjugation in the "magic" basis is displayed.Comment: 19 pages, RevTe

Assessing the Role of Spin Noise in the Precision Timing of Millisecond Pulsars

We investigate rotational spin noise (referred to as timing noise) in non-accreting pulsars: millisecond pulsars, canonical pulsars, and magnetars. Particular attention is placed on quantifying the strength and non-stationarity of timing noise in millisecond pulsars because the long-term stability of these objects is required to detect nanohertz gravitational radiation. We show that a single scaling law is sufficient to characterize timing noise in millisecond and canonical pulsars while the same scaling law underestimates the levels of timing noise in magnetars. The scaling law, along with a detailed study of the millisecond pulsar B1937+21, leads us to conclude that timing noise is latent in most millisecond pulsars and will be measurable in many objects when better arrival time estimates are obtained over long data spans. The sensitivity of a pulsar timing array to gravitational radiation is strongly affected by any timing noise. We conclude that detection of proposed gravitational wave backgrounds will require the analysis of more objects than previously suggested over data spans that depend on the spectra of both the gravitational wave background and of the timing noise. It is imperative to find additional millisecond pulsars in current and future surveys in order to reduce the effects of timing noise.Comment: 16 pages and 6 figures. ApJ, accepte

Madelung Fluid Model for The Most Likely Wave Function of a Single Free Particle in Two Dimensional Space with a Given Average Energy

We consider spatially two dimensional Madelung fluid whose irrotational motion reduces into the Schr\"odinger equation for a single free particle. In this respect, we regard the former as a direct generalization of the latter, allowing a rotational quantum flow. We then ask for the most likely wave function possessing a given average energy by maximizing the Shannon information entropy over the quantum probability density. We show that there exists a class of solutions in which the wave function is self-trapped, rotationally symmetric, spatially localized with finite support, and spinning around its center, yet stationary. The stationarity comes from the balance between the attractive quantum force field of a trapping quantum potential generated by quantum probability density and the repulsive centrifugal force of a rotating velocity vector field. We further show that there is a limiting case where the wave function is non-spinning and yet still stationary. This special state turns out to be the lowest stationary state of the ordinary Schr\"odinger equation for a particle in a cylindrical tube classical potential.Comment: 19 page

Entropy Encoding, Hilbert Space and Karhunen-Loeve Transforms

By introducing Hilbert space and operators, we show how probabilities, approximations and entropy encoding from signal and image processing allow precise formulas and quantitative estimates. Our main results yield orthogonal bases which optimize distinct measures of data encoding.Comment: 25 pages, 1 figur

The Physics of Maxwell's demon and information

Maxwell's demon was born in 1867 and still thrives in modern physics. He plays important roles in clarifying the connections between two theories: thermodynamics and information. Here, we present the history of the demon and a variety of interesting consequences of the second law of thermodynamics, mainly in quantum mechanics, but also in the theory of gravity. We also highlight some of the recent work that explores the role of information, illuminated by Maxwell's demon, in the arena of quantum information theory.Comment: 24 pages, 13 figures. v2: some refs added, figs improve

Conceptual Inadequacy of the Shannon Information in Quantum Measurements

In a classical measurement the Shannon information is a natural measure of our ignorance about properties of a system. There, observation removes that ignorance in revealing properties of the system which can be considered to preexist prior to and independent of observation. Because of the completely different root of a quantum measurement as compared to a classical measurement conceptual difficulties arise when we try to define the information gain in a quantum measurement using the notion of Shannon information. The reason is that, in contrast to classical measurement, quantum measurement, with very few exceptions, cannot be claimed to reveal a property of the individual quantum system existing before the measurement is performed.Comment: 11 pages, 5 figures, important Ref. [6] is now cited in all appropriate place

Gravitational wave astronomy of single sources with a pulsar timing array

Abbreviated: We investigate the potential of detecting the gravitational wave from individual binary black hole systems using pulsar timing arrays (PTAs) and calculate the accuracy for determining the GW properties. This is done in a consistent analysis, which at the same time accounts for the measurement of the pulsar distances via the timing parallax. We find that, at low redshift, a PTA is able to detect the nano-Hertz GW from super massive black hole binary systems with masses of \sim10^8 - 10^{10}\,M_{\sun} less than $\sim10^5$\,years before the final merger, and those with less than $\sim10^3 - 10^4$ years before merger may allow us to detect the evolution of binaries. We derive an analytical expression to describe the accuracy of a pulsar distance measurement via timing parallax. We consider five years of bi-weekly observations at a precision of 15\,ns for close-by ($\sim 0.5 - 1$\,kpc) pulsars. Timing twenty pulsars would allow us to detect a GW source with an amplitude larger than $5\times 10^{-17}$. We calculate the corresponding GW and binary orbital parameters and their measurement precision. The accuracy of measuring the binary orbital inclination angle, the sky position, and the GW frequency are calculated as functions of the GW amplitude. We note that the "pulsar term", which is commonly regarded as noise, is essential for obtaining an accurate measurement for the GW source location. We also show that utilizing the information encoded in the GW signal passing the Earth also increases the accuracy of pulsar distance measurements. If the gravitational wave is strong enough, one can achieve sub-parsec distance measurements for nearby pulsars with distance less than $\sim 0.5 - 1$\,kpc.Comment: 16 pages, 5 figure,, accepted by MNRA