arXiv subjects
Publications and source records attributed to J. Batle.
Solutions of the Schr\"odinger equation by spanning the wave function is a complete basis is a common practice is many-body interacting systems. We shall study the case of a two-dimensional quantum system composed by two interacting spin-less electrons and see that the correctness of the matrix approach depends inexplicably on the type of interaction existing between particles.
We present a formula for an infinite number of universal quantum logic gates, which are $4$ by $4$ unitary solutions to the Yang-Baxter (Y-B) equation. We obtain this family from a certain representation of the cyclic group of order $n$. We then show that this {\it discrete} family, parametrized by integers $n$, is in fact, a small sub-class of a larger {\it continuous} family, parametrized by real numbers $\theta$, of universal quantum gates. We discuss the corresponding Yang-Baxterization and related symmetries in the concomitant Hamiltonian.
The number of steps required in order to maximize a Bell inequality for arbitrary number of qubits is shown to grow exponentially with either the number of steps and the number of parties involved. The proof that the optimization of such correlation measure is a NP-problem is based on an operational perspective involving a Turing machine, which follows a general algorithm. The implications for the computability of the so called {\it nonlocality} for any number of qubits is similar to recent results involving entanglement or similar quantum correlation-based measures.
The use of the so-called entropic inequalities is revisited in the light of new quantum correlation measures, specially nonlocality. We introduce the concept of {\it classicality} as the non-violation of these classical inequalities by quantum states of several multiqubit systems and compare it with the non-violation of Bell inequalities, that is, {\it locality}. We explore --numerically and analytically-- the relationship between several other quantum measures and discover the deep connection existing between them. The results are surprising due to the fact that these measures are very different in their nature and application. The cases for $n=2,3,4$ qubits and a generalization to systems with arbitrary number of qubits are studied here when discriminated according to their degree of mixture.
There has been great interest in finding sets of $m$ mutually unbiased bases which are compatible with a given space $\mathbb{C}^d$, specially in physics due to their interesting applications in quantum information theory. Several general results have been obtained so far, but surprising results may occur for definite $(m,d)$-values. One such case that has remained an open question (the simplest case) is the one regarding the existence of $m=4$ mutually orthogonal bases for $d=6$. In the present work we introduce a new approach to the problem by translating it into an optimization procedure for a given pair $(m,d)$.
Systems of identical particles with equal charge are studied under a special type of confinement. These classical particles are free to move inside some convex region S and on the boundary of it $\Omega$ (the $S^{d-1}-$ sphere, in our case). We shall show how particles arrange themselves under the sole action of the Coulomb repulsion in many dimensions in the usual Euclidean space, therefore generalizing the so called Thomson problem to many dimensions. Also, we explore how the problem varies when non-Euclidean geometries are considered. We shall see that optimal configurations in all cases possess a high degree of symmetry, regardless of the concomitant dimension or geometry.
Any set of pure states living in an given Hilbert space possesses a natural and unique metric --the Haar measure-- on the group $U(N)$ of unitary matrices. However, there is no specific measure induced on the set of eigenvalues $\Delta$ of any density matrix $\rho$. Therefore, a general approach to the global properties of mixed states depends on the specific metric defined on $\Delta$. In the present work we shall employ a simple measure on $\Delta$ that has the advantage of possessing a clear geometric visualization whenever discussing how arbitrary states are distributed according to some measure of mixedness. The degree of mixture will be that of the participation ratio $R=1/Tr(\rho^2)$ and the concomitant maximum eigenvalue $\lambda_m$. The cases studied will be the qubit-qubit system and the qubit-qutrit system, whereas some discussion will be made on higher-dimensional bipartite cases in both the $R$-domain and the $\lambda_m$-domain.
We investigate quantum states that posses both maximum entanglement and maximum discord between the pertinent parties. Since entanglement (discord) is defined only for bipartite (two qubit) systems, we shall introduce an appropriate sum over of all bi-partitions as the associated measure. The ensuing definition --not new for entanglement-- is thus extended here to quantum discord. Also, additional dimensions within the parties are considered ({\it qudits}). We also discuss nonlocality (in the form of maximum violation of a Bell inequality) for all multiqubit systems. The emergence of more nonlocal states than local ones, all of them possessing maximum entanglement, will be linked, surprisingly enough, to whether quantum mechanics is defined over the fields of real or complex numbers.
We investigate thermal properties of quantum correlations in the thermodynamic limit with reference to the XY-model
We discuss some properties of the quantum discord based on the geometric distance advanced by Dakic, Vedral, and Brukner [Phys. Rev. Lett. {\bf 105}, 190502 (2010)], with emphasis on Werner- and MEM-states. We ascertain just how good the measure is in representing quantum discord. We explore the dependence of quantum discord on the degree of mixedness of the bipartite states, and also its connection with non-locality as measured by the maximum violation of a Bell inequality within the CHSH scenario.
Nonlocality and quantum entanglement constitute two special aspects of the quantum correlations existing in quantum systems, which are of paramount importance in quantum-information theory. Traditionally, they have been regarded as identical (equivalent, in fact, for pure two qubit states, that is, {\it Gisin's Theorem}), yet they constitute different resources. Describing nonlocality by means of the violation of several Bell inequalities, we obtain by direct optimization those states of two qubits that maximally violate a Bell inequality, in terms of their degree of mixture as measured by either their participation ratio $R=1/Tr(\rho^2)$ or their maximum eigenvalue $\lambda_{max}$. This optimum value is obtained as well, which coincides with previous results. Comparison with entanglement is performed too. An example of an application is given in the XY model. In this novel approximation, we also concentrate on the nonlocality for linear combinations of pure states of two qubits, providing a closed form for their maximal nonlocality measure. The case of Bell diagonal mixed states of two qubits is also extensively studied. Special attention concerning the connection between nonlocality and entanglement for mixed states of two qubits is paid to the so called maximally entangled mixed states. Additional aspects for the case of two qubits are also described in detail. Since we deal with qubit systems, we will perform an analogous study for three qubits, employing similar tools. Relation between distillability and nonlocality is explored quantitatively for the whole space of states of three qubits. We finally extend our analysis to four qubit systems, where nonlocality for generalized Greenberger-Horne-Zeilinger states of arbitrary number of parties is computed.
Nonlocality and quantum entanglement constitute two special features of quantum systems of paramount importance in quantum information theory (QIT). Essentially regarded as identical or equivalent for many years, they constitute different concepts. Describing nonlocality by means of the maximal violation of two Bell inequalities, we study both entanglement and nonlocality for two and three spins in the XY model. Our results shed a new light into the description of nonlocality and the possible information-theoretic task limitations of entanglement in an infinite quantum system.
A comparison is made of various searching procedures, based upon different entanglement measures or entanglement indicators, for highly entangled multi-qubits states. In particular, our present results are compared with those recently reported by Brown et al. [J. Phys. A: Math. Gen. 38 (2005) 1119]. The statistical distribution of entanglement values for the aforementioned multi-qubit systems is also explored.
We ascertain, following ideas of Arnesen, Bose, and Vedral concerning thermal entanglement [Phys. Rev. Lett. {\bf 87} (2001) 017901] and using the statistical tool called {\it entropic non-triviality} [Lamberti, Martin, Plastino, and Rosso, Physica A {\bf 334} (2004) 119], that there is a one to one correspondence between (i) the mixing coefficient $x$ of a Werner state, on the one hand, and (ii) the temperature $T$ of the one-dimensional Heisenberg two-spin chain with a magnetic field $B$ along the $z-$axis, on the other one. This is true for each value of $B$ below a certain critical value $B_c$. The pertinent mapping depends on the particular $B-$value one selects within such a range.
We consider the change of entanglement of formation $ΔE$ produced by a unitary transformation acting on a general (pure or mixed) state $ρ$ describing a system of two qubits. We study numerically the probabilities of obtaining different values of $ΔE$, assuming that the initial state is randomly distributed in the space of all states according to the product measure introduced by Zyczkowski {\it et al.} [Phys. Rev. A {\bf 58} (1998) 883].
The maximally entangled mixed states of Munro, James, White, and Kwiat [Phys. Rev. A {\bf 64} (2001) 030302] are shown to exhibit interesting features vis a vis conditional entropic measures. The same happens with the Ishizaka and Hiroshima states [Phys. Rev. A {\bf 62} 022310 (2000)], whose entanglement-degree can not be increased by acting on them with logic gates. Special types of entangled states that do not violate classical entropic inequalities are seen to exist in the space of two qubits. Special meaning can be assigned to the Munro {\it et al.} special participation ratio of 1.8.
It has been recently pointed out by Caves, Fuchs, and Rungta that real quantum mechanics (that is, quantum mechanics defined over real vector spaces provides an interesting foil theory whose study may shed some light on just which particular aspects of quantum entanglement are unique to standard quantum theory, and which ones are more generic over other physical theories endowed with this phenomenon. Following this work, we discuss some entanglement properties of two-rebits systems, making a comparison with the basic properties of two-qubits systems, i.e., the ones described by standard complex quantum mechanics. We also discuss the use of quaternionic quantum mechanics as applied to the phenomenon of entanglement.
Quantum gates, that play a fundamental role in quantum computation and other quantum information processes, are unitary evolution operators $\hat U$ that act on a composite system changing its entanglement. In the present contribution we study some aspects of these entanglement changes. By recourse of a Monte Carlo procedure, we compute the so called "entangling power" for several paradigmatic quantum gates and discuss results concerning the action of the CNOT gate. We pay special attention to the distribution of entanglement among the several parties involved.