scholarly journals Entanglement, Purity, and Information Entropies in Continuous Variable Systems

2005 ◽  
Vol 12 (02) ◽  
pp. 189-205 ◽  
Author(s):  
Gerardo Adesso ◽  
Alessio Serafini ◽  
Fabrizio Illuminati
Keyword(s):  
Tomographic Reconstruction ◽  
Continuous Variable ◽  
The State ◽  
Upper And Lower Bounds ◽  
Mixed States ◽  
Gaussian States ◽  
Pure States ◽  
Classical Correlations ◽  
Quantum And Classical Correlations

Quantum entanglement of pure states of a bipartite system is defined as the amount of local or marginal (i.e. referring to the subsystems) entropy. For mixed states this identification vanishes, since the global loss of information about the state makes it impossible to distinguish between quantum and classical correlations. Here we show how the joint knowledge of the global and marginal degrees of information of a quantum state, quantified by the purities or, in general, by information entropies, provides an accurate characterization of its entanglement. In particular, for Gaussian states of continuous variable systems, we classify the entanglement of two-mode states according to their degree of total and partial mixedness, comparing the different roles played by the purity and the generalized p-entropies in quantifying the mixedness and bounding the entanglement. We prove the existence of strict upper and lower bounds on the entanglement and the existence of extremally (maximally and minimally) entangled states at fixed global and marginal degrees of information. This results allow for a powerful, operative method to measure mixed-state entanglement without the full tomographic reconstruction of the state. Finally, we briefly discuss the ongoing extension of our analysis to the quantification of multipartite entanglement in highly symmetric Gaussian states of arbitrary 1 × N-mode partitions.

scholarly journals GAUSSIAN GEOMETRIC DISCORD

2011 ◽  
Vol 09 (07n08) ◽  
pp. 1773-1786 ◽  
Author(s):  
GERARDO ADESSO ◽  
DAVIDE GIROLAMI
Keyword(s):  
Quantum Discord ◽  
Gaussian Quadrature ◽  
Continuous Variable ◽  
Upper And Lower Bounds ◽  
Gaussian States ◽  
Physical Constraints ◽  
Geometric Measure ◽  
Classical Correlations ◽  
Qubit States ◽  
Thermal States

We extend the geometric measure of quantum discord, introduced and computed for two-qubit states, to quantify non-classical correlations in composite Gaussian states of continuous variable systems. We lay the formalism for the evaluation of a Gaussian geometric discord in two-mode Gaussian states, and present explicit formulas for the class of two-mode squeezed thermal states. In such a case, under physical constraints of bounded mean energy, geometric discord is shown to admit upper and lower bounds for a fixed value of the conventional (entropic) quantum discord. We finally discuss alternative geometric approaches to quantify Gaussian quadrature correlations.

scholarly journals THE BALANCE OF QUANTUM CORRELATIONS FOR A CLASS OF FEASIBLE TRIPARTITE CONTINUOUS VARIABLE STATES

2012 ◽  
Vol 27 (01n03) ◽  
pp. 1345024 ◽  
Author(s):  
STEFANO OLIVARES ◽  
MATTEO G. A. PARIS
Keyword(s):  
Conservation Laws ◽  
Quantum Discord ◽  
Quantum Correlations ◽  
Continuous Variable ◽  
Noisy Environment ◽  
Mixed States ◽  
Von Neumann ◽  
Entanglement Of Formation ◽  
Pure States ◽  
Gaussian Quantum Discord

We address the balance of quantum correlations for continuous variable (CV) states. In particular, we consider a class of feasible tripartite CV pure states and explicitly prove two Koashi–Winter-like conservation laws involving Gaussian entanglement of formation (EoF), Gaussian quantum discord and sub-system Von Neumann entropies. We also address the class of tripartite CV mixed states resulting from the propagation in a noisy environment, and discuss how the previous equalities evolve into inequalities.

scholarly journals Criteria for measures of quantum correlations

2012 ◽  
Vol 12 (9&10) ◽  
pp. 721-742
Author(s):  
Aharon Brodutch ◽  
Kavan Modi
Keyword(s):  
Quantum Discord ◽  
Quantum Correlations ◽  
Good Measure ◽  
General Mechanism ◽  
Mixed States ◽  
Entanglement Measures ◽  
Measurement Basis ◽  
Important Quality ◽  
Classical Correlations ◽  
Quantum And Classical Correlations

Entanglement does not describe all quantum correlations and several authors have shown the need to go beyond entanglement when dealing with mixed states. Various different measures have sprung up in the literature, for a variety of reasons, to describe bipartite and multipartite quantum correlations; some are known under the collective name {\it quantum discord}. Yet, in the same sprit as the criteria for entanglement measures, there is no general mechanism that determines whether a measure of quantum and classical correlations is a proper measure of correlations. This is partially due to the fact that the answer is a bit muddy. In this article we attempt tackle this muddy topic by writing down several criteria for a ``good" measure of correlations. We breakup our list into \emph{necessary}, \emph{reasonable}, and \emph{debatable} conditions. We then proceed to prove several of these conditions for generalized measures of quantum correlations. However, not all conditions are met by all measures; we show this via several examples. The reasonable conditions are related to continuity of correlations, which has not been previously discussed. Continuity is an important quality if one wants to probe quantum correlations in the laboratory. We show that most types of quantum discord are continuous but none are continuous with respect to the measurement basis used for optimization.

THERMAL DISCORD AND NEGATIVITY IN A TWO-SPIN-QUTRIT SYSTEM UNDER DIFFERENT MAGNETIC FIELDS

2013 ◽  
Vol 11 (08) ◽  
pp. 1350070 ◽  
Author(s):  
XIAO-JING LI ◽  
HUI-HUI JI ◽  
XI-WEN HOU
Keyword(s):  
Magnetic Fields ◽  
Quantum Discord ◽  
Strong Field ◽  
Quantum Correlations ◽  
Mixed States ◽  
Classical Correlation ◽  
Lower Temperature ◽  
Qubit States ◽  
Quantum And Classical Correlations

The characterization of quantum discord (QD) has been well understood only for two-qubit states and is little known for mixed states beyond qubits. In this work, thermal quantum discord is studied for a qutrit system in different magnetic fields, where classical correlation and entanglement negativity are calculated for comparison. It is shown that the discord is more robust against temperature than the negativity. For a suitable region of magnetic field and its direction, the discord is non-zero while the negativity is zero. When the system is at a lower temperature, these three quantities, however, display a similar behavior for the varied field and direction, and their discontinuities come from crossovers between different ground states in the system. Moreover, the inequality between the quantum and classical correlations depends upon the system parameters as well as the temperature. In particular, both correlations are equal at a suitable field, direction, and temperature. Remarkably, such an equality remains for a strong field in the antiparallel direction, while both correlations in two-qubit systems are identical for any antiparallel field and temperature. These are useful for quantum information and understanding quantum correlations in qutrit mixed states.

scholarly journals Characterization of Combinatorially Independent Permutation Separability Criteria

2005 ◽  
Vol 12 (04) ◽  
pp. 331-345 ◽  
Author(s):  
Paweł Wocjan ◽  
Michał Horodecki
Keyword(s):  
Density Matrix ◽  
The State ◽  
Necessary Condition ◽  
Trace Norm ◽  
Mixed States ◽  
Graphical Notation ◽  
Operational Conditions ◽  
Separability Criteria ◽  
Partial Transpose

The so-called permutation separability criteria are simple operational conditions that are necessary for separability of mixed states of multipartite systems: (1) permute the indices of the density matrix and (2) check if the trace norm of at least one of the resulting operators is greater than one. If it is greater than one then the state is necessarily entangled. A shortcoming of the permutation separability criteria is that many permutations give rise to equivalent separability criteria. Therefore, we introduce a necessary condition for two permutations to yield independent criteria called combinatorial independence. This condition basically means that the map corresponding to one permutation cannot be obtained by concatenating the map corresponding to the second permutation with a norm-preserving map. We characterize completely combinatorially independent criteria, and determine simple permutations that represent all independent criteria. The representatives can be visualized by means of a simple graphical notation. They are composed of three basic operations: partial transpose, and two types of so-called reshufflings. In particular, for a four-partite system all criteria except one are composed of partial transpose and only one type of reshuffling; the exceptional one requires the second type of reshuffling. Furthermore, we show how to obtain efficiently a simple representative for every permutation. This method allows to check easily if two permutations are combinatorially equivalent or not.

scholarly journals Pairwise quantum and classical correlations in multi-qubits states via linear relative entropy

2014 ◽  
Vol 12 (06) ◽  
pp. 1450035 ◽  
Author(s):  
M. Daoud ◽  
R. Ahl Laamara ◽  
H. El Hadfi
Keyword(s):  
Relative Entropy ◽  
Qubit State ◽  
Mixed States ◽  
Special Cases ◽  
Pairwise Correlations ◽  
Classical Correlations ◽  
Additivity Relation ◽  
Quantum And Classical Correlations ◽  
Explicit Expressions

The pairwise correlations in a multi-qubit state are quantified through a linear variant of relative entropy. In particular, we derive the explicit expressions of total, quantum and classical bipartite correlations. Two different bi-partioning schemes are considered. We discuss the derivation of closest product, quantum–classical and quantum–classical product states. We also investigate the additivity relation between the various pairwise correlations existing in pure and mixed states. As illustration, some special cases are examined.

scholarly journals Monogamy of entanglement without inequalities

Quantum ◽  
2018 ◽  
Vol 2 ◽  
pp. 81 ◽  
Author(s):  
Gilad Gour ◽  
Yu Guo
Keyword(s):  
Quantum State ◽  
The State ◽  
Mixed States ◽  
Bipartite State ◽  
Fine Grained ◽  
Entanglement Of Formation ◽  
Rank 2 ◽  
Pure States ◽  
Entanglement Of Assistance ◽  
Monogamy Relations

We provide a fine-grained definition for monogamous measure of entanglement that does not invoke any particular monogamy relation. Our definition is given in terms an equality, as oppose to inequality, that we call the "disentangling condition". We relate our definition to the more traditional one, by showing that it generates standard monogamy relations. We then show that all quantum Markov states satisfy the disentangling condition for any entanglement monotone. In addition, we demonstrate that entanglement monotones that are given in terms of a convex roof extension are monogamous if they are monogamous on pure states, and show that for any quantum state that satisfies the disentangling condition, its entanglement of formation equals the entanglement of assistance. We characterize all bipartite mixed states with this property, and use it to show that the G-concurrence is monogamous. In the case of two qubits, we show that the equality between entanglement of formation and assistance holds if and only if the state is a rank 2 bipartite state that can be expressed as the marginal of a pure 3-qubit state in the W class.

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