11 research outputs found
The Berezinskii Kosterlitz Thouless phase transition is of second-order in the microcanonical ensemble
A paradigmatic example of a phase transition taking place in the absence of
symmetry-breaking is provided by the Berezinkii-Kosterlitz-Thouless (BKT)
transition in the two-dimensional XY model. In the framework of canonical
ensemble, this phase transition is defined as an infinite-order one. To the
contrary, by tackling the transitional behavior of the two dimensional XY model
in the microcanonical ensemble, we show that the BKT phase transition is of
second order. This provides a new example of statistical ensemble inequivalence
that could apply to a broad class of systems undergoing BKT phase transitions
Entanglement and quantum correlation measures for quantum multipartite mixed states
Entanglement, and quantum correlation, are precious resources for quantum technologies
implementation based on quantum information science, such as quantum communication, quantum
computing, and quantum interferometry. Nevertheless, to our best knowledge, a directly or
numerically computable measure for the entanglement of multipartite mixed states is still lacking. In
this work, (i) we derive a measure of the degree of quantum correlation for mixed multipartite states.
The latter possesses a closed-form expression valid in the general case unlike, to our best knowledge,
all other known measures of quantum correlation. (ii) We further propose an entanglement measure,
derived from this quantum correlation measure using a novel regularization procedure for the
density matrix. Therefore, a comparison of the proposed measures, of quantum correlation and
entanglement, allows one to distinguish between quantum correlation detached from entanglement
and the one induced by entanglement and, hence, to identify separable but non-classical states. We
have tested our quantum correlation and entanglement measures, on states well-known in literature:
a general Bell diagonal state and the Werner states, which are easily tractable with our regularization
procedure, and we have verifed the accordance between our measures and the expected results for
these states. Finally, we validate the two measures in two cases of multipartite states. The frst is a
generalization to three qubits of the Werner state, the second is a one-parameter three qubits mixed
state interpolating between a bi-separable state and a genuine multipartite state, passing through a
fully separable state
Entanglement and Quantum Correlation Measures from a Minimum Distance Principle
Entanglement, and quantum correlation, are precious resources for quantum
technologies implementation based on quantum information science, such as, for
instance, quantum communication, quantum computing, and quantum interferometry.
Nevertheless, to our best knowledge, a directly computable measure for the
entanglement of multipartite mixed-states is still lacking. In this work, {\it
i)} we derive from a minimum distance principle, an explicit measure able to
quantify the degree of quantum correlation for pure or mixed multipartite
states; {\it ii)} through a regularization process of the density matrix, we
derive an entanglement measure from such quantum correlation measure; {\it
iii)} we prove that our entanglement measure is \textit{faithful} in the sense
that it vanishes only on the set of separable states. Then, a comparison of the
proposed measures, of quantum correlation and entanglement, allows one to
distinguish between quantum correlation detached from entanglement and the one
induced by entanglement, hence to define the set of separable but non-classical
states.
Since all the relevant quantities in our approach, descend from the geometry
structure of the projective Hilbert space, the proposed method is of general
application.
Finally, we apply the derived measures as an example to a general Bell
diagonal state and to the Werner states, for which our regularization procedure
is easily tractable.Comment: 7 Pages, 3 Figure
Geometrical aspects in the analysis of microcanonical phase-transitions
In the present work, we discuss how the functional form of thermodynamic
observables can be deduced from the geometric properties of subsets of phase
space. The geometric quantities taken into account are mainly extrinsic
curvatures of the energy level sets of the Hamiltonian of a system under
investigation. In particular, it turns out that peculiar behaviours of
thermodynamic observables at a phase transition point are rooted in more
fundamental changes of the geometry of the energy level sets in phase space.
More specifically, we discuss how microcanonical and geometrical descriptions
of phase-transitions are shaped in the special case of models with
either nearest-neighbours and mean-field interactions
Entanglement distance for an arbitrary state of M qubits
We propose a measure of entanglement that can be computed for any pure state
of an -qubit system. The entanglement measure has the form of a distance
that we derive from an adapted application of the Fubini-Study metric. This
measure is invariant under local unitary transformations and defined as trace
of a suitable metric that we derive, the entanglement metric .
Furthermore, the analysis of the eigenvalues of gives information
about the robustness of entanglement.Comment: 6 pages, 5 figure
Geometric study of classical phase transitions and quantum entanglement
The present work is divided into two parts. First, we discuss how the functional form of thermodynamic observables can be deduced from the geometric properties of subsets of phase space. The geometric quantities taken into account are mainly extrinsic curvatures of the energy level sets of the Hamiltonian of a system under investigation. In particular, it turns out that peculiar behaviors of thermodynamic observables at a phase transition point are rooted in more fundamental changes in the geometry of the energy level sets in phase space. More specifically, we discuss how microcanonical and geometrical descriptions of phase transitions are shaped in the peculiar Kosterlitz–Thouless phase transition and in the special case of the ϕ4 model. In the second part, we will focus on the
quantification of quantum entanglement, a topic of great current interest. We will derive entanglement and quantum correlation measures, from a geometrical procedure, which are valid for multipartite hybrid states. We also provide a physical and operational meaning of the proposed entanglement measures for pure states. Furthermore, we show that the proposed measures can either be analytically or numerically computed. Finally, we test the validity of the proposed measure through a variety of examples
Topological origin of protein folding transition
In this paper, a geometrical and thermodynamical analysis of the global properties of the potential energy landscape of a minimalistic model of a polypeptide is presented. The global geometry of the potential energy landscape is supposed to contain relevant information about the properties of a given sequence of amino acids, that is, to discriminate between a random heteropolymer and a protein. By considering the SH3 and PYP protein-sequences and their randomized versions it turns out that in addition to the standard signatures of the folding transition-discriminating between protein sequences of amino acids and random heteropolymer sequences-also peculiar geometric signatures of the equipotential hypersurfaces in configuration space can discriminate between proteins and random heteropolymers. Interestingly, these geometric signatures are the "shadows" of deeper topological changes that take place in correspondence with the protein folding transition. The protein folding transition takes place in systems with a small number of degrees of freedom (very far from the Avogadro number) and in the absence of a symmetry-breaking phenomenon. Nevertheless, seen from the deepest level of topology changes of equipotential submanifolds of phase space, the protein folding transition fully qualifies as a phase transition
Geometrical and topological study of the Kosterlitz–Thouless phase transition in the XY model in two dimensions
International audiencePhase transitions do not necessarily correspond to a symmetry-breaking phenomenon. This is the case of the Kosterlitz–Thouless (KT) phase transition in a two-dimensional classical XY model, a typical example of a transition stemming from a deeper phenomenon than a symmetry-breaking. Actually, the KT transition is a paradigmatic example of the successful application of topological concepts to the study of phase transition phenomena in the absence of an order parameter. Topology conceptually enters through the meaning of defects in real space. In the present work, the same kind of KT phase transition in a two-dimensional classical XY model is tackled by resorting again to a topological viewpoint, however focussed on the energy level sets in phase space rather than on topological defects in real space. Also from this point of view, the origin of the KT transition can be attributed to a topological phenomenon. In fact, the transition is detected through peculiar geometrical changes of the energy level sets which, after a theorem in differential topology, are direct probes of topological changes of these level sets