119 research outputs found
Cryptanalysis of public-key cryptosystems that use subcodes of algebraic geometry codes
We give a polynomial time attack on the McEliece public key cryptosystem
based on subcodes of algebraic geometry (AG) codes. The proposed attack reposes
on the distinguishability of such codes from random codes using the Schur
product. Wieschebrink treated the genus zero case a few years ago but his
approach cannot be extent straightforwardly to other genera. We address this
problem by introducing and using a new notion, which we call the t-closure of a
code
Gauging functional brain activity: from distinguishability to accessibility
Standard neuroimaging techniques provide non-invasive access not only to
human brain anatomy but also to its physiology. The activity recorded with
these techniques is generally called functional imaging, but what is observed
per se is an instance of dynamics, from which functional brain activity should
be extracted. Distinguishing between bare dynamics and genuine function is a
highly non-trivial task, but a crucially important one when comparing
experimental observations and interpreting their significance. Here we
illustrate how the ability of neuroimaging to extract genuine functional brain
activity is bounded by the structure of functional representations. To do so,
we first provide a simple definition of functional brain activity from a
system-level brain imaging perspective. We then review how the properties of
the space on which brain activity is represented allow defining relations
ranging from distinguishability to accessibility of observed imaging data. We
show how these properties result from the structure defined on dynamical data
and dynamics-to-function projections, and consider some implications that the
way and extent to which these are defined have for the interpretation of
experimental data from standard system-level brain recording techniques.Comment: 7 pages, 0 figure
Multipartite quantum correlations: symplectic and algebraic geometry approach
We review a geometric approach to classification and examination of quantum
correlations in composite systems. Since quantum information tasks are usually
achieved by manipulating spin and alike systems or, in general, systems with a
finite number of energy levels, classification problems are usually treated in
frames of linear algebra. We proposed to shift the attention to a geometric
description. Treating consistently quantum states as points of a projective
space rather than as vectors in a Hilbert space we were able to apply powerful
methods of differential, symplectic and algebraic geometry to attack the
problem of equivalence of states with respect to the strength of correlations,
or, in other words, to classify them from this point of view. Such
classifications are interpreted as identification of states with `the same
correlations properties' i.e. ones that can be used for the same information
purposes, or, from yet another point of view, states that can be mutually
transformed one to another by specific, experimentally accessible operations.
It is clear that the latter characterization answers the fundamental question
`what can be transformed into what \textit{via} available means?'. Exactly such
an interpretations, i.e, in terms of mutual transformability can be clearly
formulated in terms of actions of specific groups on the space of states and is
the starting point for the proposed methods.Comment: 29 pages, 9 figures, 2 tables, final form submitted to the journa
The structure of Renyi entropic inequalities
We investigate the universal inequalities relating the alpha-Renyi entropies
of the marginals of a multi-partite quantum state. This is in analogy to the
same question for the Shannon and von Neumann entropy (alpha=1) which are known
to satisfy several non-trivial inequalities such as strong subadditivity.
Somewhat surprisingly, we find for 0<alpha<1, that the only inequality is
non-negativity: In other words, any collection of non-negative numbers assigned
to the nonempty subsets of n parties can be arbitrarily well approximated by
the alpha-entropies of the 2^n-1 marginals of a quantum state.
For alpha>1 we show analogously that there are no non-trivial homogeneous (in
particular no linear) inequalities. On the other hand, it is known that there
are further, non-linear and indeed non-homogeneous, inequalities delimiting the
alpha-entropies of a general quantum state.
Finally, we also treat the case of Renyi entropies restricted to classical
states (i.e. probability distributions), which in addition to non-negativity
are also subject to monotonicity. For alpha different from 0 and 1 we show that
this is the only other homogeneous relation.Comment: 15 pages. v2: minor technical changes in Theorems 10 and 1
Sub-shot-noise quantum metrology with entangled identical particles
The usual notion of separability has to be reconsidered when applied to
states describing identical particles. A definition of separability not related
to any a priori Hilbert space tensor product structure is needed: this can be
given in terms of commuting subalgebras of observables. Accordingly, the
results concerning the use of the quantum Fisher information in quantum
metrology are generalized and physically reinterpreted.Comment: 17 pages, LaTe
Sublinear scaling in non-Markovian open quantum systems simulations
While several numerical techniques are available for predicting the dynamics
of non-Markovian open quantum systems, most struggle with simulations for very
long memory and propagation times, e.g., due to superlinear scaling with the
number of time steps . Here, we introduce a numerically exact algorithm to
calculate process tensors -- compact representations of environmental
influences -- which provides a scaling advantage over previous algorithms by
leveraging self-similarity of the tensor networks that represent Gaussian
environments. Based on a divide-and-conquer strategy, our approach requires
only singular value decompositions for environments with
infinite memory. Where the memory can be truncated after time steps, a
scaling is found, which is independent of . This
improved scaling is enabled by identifying process tensors with repeatable
blocks. To demonstrate the power and utility of our approach we provide three
examples. (1) We calculate the fluorescence spectra of a quantum dot under both
strong driving and strong dot-phonon couplings, a task requiring simulations
over millions of time steps, which we are able to perform in minutes. (2) We
efficiently find process tensors describing superradiance of multiple emitters.
(3) We explore the limits of our algorithm by considering coherence decay with
a very strongly coupled environment. The algorithm we present here not only
significantly extends the scope of numerically exact techniques to open quantum
systems with long memory times, but also has fundamental implications for
simulation complexity
On affine maps on non-compact convex sets and some characterizations of finite-dimensional solid ellipsoids
Convex geometry has recently attracted great attention as a framework to
formulate general probabilistic theories. In this framework, convex sets and
affine maps represent the state spaces of physical systems and the possible
dynamics, respectively. In the first part of this paper, we present a result on
separation of simplices and balls (up to affine equivalence) among all compact
convex sets in two- and three-dimensional Euclidean spaces, which focuses on
the set of extreme points and the action of affine transformations on it.
Regarding the above-mentioned axiomatization of quantum physics, our result
corresponds to the case of simplest (2-level) quantum system. We also discuss a
possible extension to higher dimensions. In the second part, towards
generalizations of the framework of general probabilistic theories and several
existing results including ones in the first part from the case of compact and
finite-dimensional physical systems as in most of the literatures to more
general cases, we study some fundamental properties of convex sets and affine
maps that are relevant to the above subject.Comment: 25 pages, a part of this work is to be presented at QIP 2011,
Singapore, January 10-14, 2011; (v2) References updated (v3) Introduction and
references updated (v4) Re-organization of the paper (results not added
Aggregating fuzzy subgroups and T-vague groups
Fuzzy subgroups and T-vague groups are interesting fuzzy algebraic structures that have been widely studied. While fuzzy subgroups fuzzify the concept of crisp subgroup, T-vague groups can be identified with quotient groups of a group by a normal fuzzy subgroup and there is a close relation between both structures and T-indistinguishability operators (fuzzy equivalence relations).
In this paper the functions that aggregate fuzzy subgroups and T-vague groups will be studied. The functions aggregating T-indistinguishability operators have been characterized [9] and the main result of this paper is that the functions aggregating T-indistinguishability operators coincide with the ones that aggregate fuzzy subgroups and T-vague groups. In particular, quasi-arithmetic means and some OWA operators aggregate them if the t-norm is continuous Archimedean.Peer ReviewedPostprint (author's final draft
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