781 research outputs found
Optimality of entropic uncertainty relations
The entropic uncertainty relation proven by Maassen and Uffink for arbitrary
pairs of two observables is known to be non-optimal. Here, we call an
uncertainty relation optimal, if the lower bound can be attained for any value
of either of the corresponding uncertainties. In this work we establish optimal
uncertainty relations by characterising the optimal lower bound in scenarios
similar to the Maassen-Uffink type. We disprove a conjecture by Englert et al.
and generalise various previous results. However, we are still far from a
complete understanding and, based on numerical investigation and analytical
results in small dimension, we present a number of conjectures.Comment: 24 pages, 10 figure
Generalised asymptotic equivalence for extensive and non-extensive entropies
We extend the Hanel and Thurner asymptotic analysis to both extensive and
non-extensive entropies on the basis of a wide class of entropic forms. The
procedure is known to be capable to classify multiple entropy measures in terms
of their defining equivalence classes. Those are determined by a pair of
scaling exponents taking into account a large number of microstates as for the
thermodynamical limit. Yet, a generalisation to this formulation makes it
possible to establish an entropic connection between Markovian and
non-Markovian statistical systems through a set of fundamental entropies
, which have been studied in other contexts and exhibit, among their
attributes, two interesting aspects: They behave as additive for a large number
of degrees of freedom while they are substantially non-additive for a small
number of them. Furthermore, an ample amount of special entropy measures,
either additive or non-additive, are contained in such asymptotic
classification. Under this scheme we analyse the equivalence classes of
Tsallis, Sharma-Mittal and R\'enyi entropies and study their features in the
thermodynamic limit as well as the correspondences among them.Comment: 6 pages, 2 figure
Noise and Disturbance of Qubit Measurements: An Information-Theoretic Characterisation
Information-theoretic definitions for the noise associated with a quantum
measurement and the corresponding disturbance to the state of the system have
recently been introduced [F. Buscemi et al., Phys. Rev. Lett. 112, 050401
(2014)]. These definitions are invariant under relabelling of measurement
outcomes, and lend themselves readily to the formulation of state-independent
uncertainty relations both for the joint estimate of observables (noise-noise
relations) and the noise-disturbance tradeoff. Here we derive such relations
for incompatible qubit observables, which we prove to be tight in the case of
joint estimates, and present progress towards fully characterising the
noise-disturbance tradeoff. In doing so, we show that the set of obtainable
noise-noise values for such observables is convex, whereas the conjectured form
for the set of obtainable noise-disturbance values is not. Furthermore,
projective measurements are not optimal with respect to the joint-measurement
noise or noise-disturbance tradeoffs. Interestingly, it seems that four-outcome
measurements are needed in the former case, whereas three-outcome measurements
are optimal in the latter.Comment: Minor changes, corresponds to final published version. 14 pages, 5
figure
A new inequality for the von Neumann entropy
Strong subadditivity of von Neumann entropy, proved in 1973 by Lieb and
Ruskai, is a cornerstone of quantum coding theory. All other known inequalities
for entropies of quantum systems may be derived from it. Here we prove a new
inequality for the von Neumann entropy which we prove is independent of strong
subadditivity: it is an inequality which is true for any four party quantum
state, provided that it satisfies three linear relations (constraints) on the
entropies of certain reduced states.Comment: 8 pages, 1 eps figur
Universal geometric approach to uncertainty, entropy and information
It is shown that for any ensemble, whether classical or quantum, continuous
or discrete, there is only one measure of the "volume" of the ensemble that is
compatible with several basic geometric postulates. This volume measure is thus
a preferred and universal choice for characterising the inherent spread,
dispersion, localisation, etc, of the ensemble. Remarkably, this unique
"ensemble volume" is a simple function of the ensemble entropy, and hence
provides a new geometric characterisation of the latter quantity. Applications
include unified, volume-based derivations of the Holevo and Shannon bounds in
quantum and classical information theory; a precise geometric interpretation of
thermodynamic entropy for equilibrium ensembles; a geometric derivation of
semi-classical uncertainty relations; a new means for defining classical and
quantum localization for arbitrary evolution processes; a geometric
interpretation of relative entropy; and a new proposed definition for the
spot-size of an optical beam. Advantages of the ensemble volume over other
measures of localization (root-mean-square deviation, Renyi entropies, and
inverse participation ratio) are discussed.Comment: Latex, 38 pages + 2 figures; p(\alpha)->1/|T| in Eq. (72) [Eq. (A10)
of published version
Expectations for extreme-mass-ratio bursts from the Galactic Centre
When a compact object on a highly eccentric orbit about a much more massive
body passes through periapsis it emits a short gravitational wave signal known
as an extreme-mass-ratio burst (EMRB). We consider stellar mass objects
orbiting the massive black hole (MBH) found in the Galactic Centre. EMRBs
provide a novel means of extracting information about the MBH; an EMRB from the
Galactic MBH could be highly informative regarding the MBH's mass and spin if
the orbital periapsis is small enough. However, to be a useful astronomical
tool EMRBs must be both informative and sufficiently common to be detectable
with a space-based interferometer. We construct a simple model to predict the
event rate for Galactic EMRBs. We estimate there could be on average ~2 bursts
in a two year mission lifetime for LISA. Stellar mass black holes dominate the
event rate. Creating a sample of 100 mission realisations, we calculate what we
could learn about the MBH. On average, we expect to be able to determine the
MBH mass to ~1% and the spin to ~0.1 using EMRBs.Comment: 22 pages, 5 figures, 2 appendices. Minor changes to reflect published
versio
On the anonymity risk of time-varying user profiles.
Websites and applications use personalisation services to profile their users, collect their patterns and activities and eventually use this data to provide tailored suggestions. User preferences and social interactions are therefore aggregated and analysed. Every time a user publishes a new post or creates a link with another entity, either another user, or some online resource, new information is added to the user profile. Exposing private data does not only reveal information about single users’ preferences, increasing their privacy risk, but can expose more about their network that single actors intended. This mechanism is self-evident in social networks where users receive suggestions based on their friends’ activities. We propose an information-theoretic approach to measure the differential update of the anonymity risk of time-varying user profiles. This expresses how privacy is affected when new content is posted and how much third-party services get to know about the users when a new activity is shared. We use actual Facebook data to show how our model can be applied to a real-world scenario.Peer ReviewedPostprint (published version
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