scholarly journals Entanglement of Three-Qubit Random Pure States

Entropy ◽  
2018 ◽  
Vol 20 (10) ◽  
pp. 745 ◽  
Author(s):  
Marco Enríquez ◽  
Francisco Delgado ◽  
Karol Życzkowski
Keyword(s):  
Haar Measure ◽  
Probability Distributions ◽  
Classical Communication ◽  
Polynomial Invariants ◽  
Analytical Work ◽  
Pure States ◽  
Random State

We study entanglement properties of generic three-qubit pure states. First, we obtain the distributions of both the coefficients and the only phase in the five-term decomposition of Acín et al. for an ensemble of random pure states generated by the Haar measure on U ( 8 ) . Furthermore, we analyze the probability distributions of two sets of polynomial invariants. One of these sets allows us to classify three-qubit pure states into four classes. Entanglement in each class is characterized using the minimal Rényi-Ingarden-Urbanik entropy. Besides, the fidelity of a three-qubit random state with the closest state in each entanglement class is investigated. We also present a characterization of these classes in terms of the corresponding entanglement polytope. The entanglement classes related to stochastic local operations and classical communication (SLOCC) are analyzed as well from this geometric perspective. The numerical findings suggest some conjectures relating some of those invariants with entanglement properties to be ground in future analytical work.

scholarly journals Relations between bipartite entanglement measures

2018 ◽  
Vol 18 (1&2) ◽  
pp. 85-113 ◽  
Author(s):  
Katharina Schwaiger ◽  
Barbara Kraus
Keyword(s):  
Pure State ◽  
Single Copy ◽  
Point Of View ◽  
Multipartite Entanglement ◽  
Bipartite Entanglement ◽  
Classical Communication ◽  
Main Emphasis ◽  
Entanglement Measures ◽  
Pure States

We investigate the entanglement of bipartite systems from an operational point of view. Main emphasis is put on bipartite pure states in the single copy regime. First, we present an operational characterization of bipartite pure state entanglement, viewing the state as a multipartite state. Then, we investigate the properties and relations of two classes of operational bipartite and multipartite entanglement measures, the so-called source and the accessible entanglement. The former measures how easy it is to generate a given state via local operations and classical communication (LOCC) from some other state, whereas the latter measures the potentiality of a state to be convertible to other states via LOCC. We investigate which parameter regime is physically available, i.e. for which values of these measures does there exist a bipartite pure state. Moreover, we determine, given some state, which parameter regime can be accessed by it and from which parameter regime it can be accessed. We show that this regime can be determined analytically using the Positivstellensatz. We compute the boundaries of these sets and the boundaries of the corresponding source and accessible sets. Furthermore, we relate these results to other entanglement measures and compare their behaviors.

scholarly journals A Brief Review of Generalized Entropies

Entropy ◽  
2018 ◽  
Vol 20 (11) ◽  
pp. 813 ◽  
Author(s):  
José Amigó ◽  
Sámuel Balogh ◽  
Sergio Hernández
Keyword(s):  
Diffusion Processes ◽  
Complex Dynamics ◽  
Probability Distributions ◽  
Point Of View ◽  
Scaling Exponent ◽  
Scaling Exponents ◽  
Physical Systems ◽  
New Phenomena ◽  
Generalized Entropies

Entropy appears in many contexts (thermodynamics, statistical mechanics, information theory, measure-preserving dynamical systems, topological dynamics, etc.) as a measure of different properties (energy that cannot produce work, disorder, uncertainty, randomness, complexity, etc.). In this review, we focus on the so-called generalized entropies, which from a mathematical point of view are nonnegative functions defined on probability distributions that satisfy the first three Shannon–Khinchin axioms: continuity, maximality and expansibility. While these three axioms are expected to be satisfied by all macroscopic physical systems, the fourth axiom (separability or strong additivity) is in general violated by non-ergodic systems with long range forces, this having been the main reason for exploring weaker axiomatic settings. Currently, non-additive generalized entropies are being used also to study new phenomena in complex dynamics (multifractality), quantum systems (entanglement), soft sciences, and more. Besides going through the axiomatic framework, we review the characterization of generalized entropies via two scaling exponents introduced by Hanel and Thurner. In turn, the first of these exponents is related to the diffusion scaling exponent of diffusion processes, as we also discuss. Applications are addressed as the description of the main generalized entropies advances.

Some new approaches to probability distributions

1980 ◽  
Vol 12 (04) ◽  
pp. 903-921 ◽  
Author(s):  
S. Kotz ◽  
D. N. Shanbhag
Keyword(s):  
Probability Distributions ◽  
Random Variables ◽  
Representation Theorems ◽  
Practical Applications ◽  
New Approaches ◽  
Characterization Of Distributions ◽  
Stability Theorems ◽  
Conditional Expectations

We develop some approaches to the characterization of distributions of real-valued random variables, useful in practical applications, in terms of conditional expectations and hazard measures. We prove several representation theorems generalizing earlier results, and establish stability theorems for two general characteristics introduced in this paper.

On Stochastic Characterization of Temporal Storm Patterns

1984 ◽  
Vol 16 (8-9) ◽  
pp. 147-153 ◽  
Author(s):  
Van-Thanh-Van Nguyen
Keyword(s):  
Markov Chain ◽  
Probability Distributions ◽  
First Order ◽  
Stochastic Characterization ◽  
Order Markov Chain ◽  
Hourly Rainfall ◽  
General Stochastic Model ◽  
Storm Rainfall ◽  
Rainfall Occurrences

The present study, a continuation of a previous work by the author, suggests a new theoretical approach to the characterization of the temporal pattern of storms. A storm is defined as a continuous run of non-zero one-hour rainfall depths. A general stochastic model is developed to determine the probability distributions of cumulative storm rainfall amounts at successive time intervals after the storm began. The previous model for characterizing storm temporal patterns was based on the assumption that hourly rainfall depths were independent and identically exponentially distributed random variables, while sequences of wet hours were modeled by a first-order stationary Markov chain. Hence, the model did only introduce dependence of wet hour occurences into the rainfall process through the first-order Markov chain. The present paper proposes a more general model that can take into account both the persistence in hourly rainfall occurrences and the dependence between successive hourly rainfall depths. Results of an illustrative example show that by accounting for the correlation structure of consecutive rainfall depths the present model gives a better fit to the observations than the previous one.

Entropy of a Convolution Operator

2004 ◽  
Vol 11 (01) ◽  
pp. 79-85 ◽  
Author(s):  
Aleksander Urbański
Keyword(s):  
Abelian Group ◽  
Probability Measure ◽  
Haar Measure ◽  
Compact Abelian Group ◽  
Convolution Operator ◽  
Absolutely Continuous ◽  
Doubly Stochastic ◽  
Doubly Stochastic Operator ◽  
Zero Entropy

The concept of the entropy of a doubly stochastic operator was introduced in 1999 by Ghys, Langevin, and Walczak. The idea was developed further by Kamiński and de Sam Lazaro, who also conjectured that the entropy of a convolution operator determined by a probability measure on a compact abelian group is equal to zero. We prove that this is true when the group is connected and the convolution operator is determined by a measure absolutely continuous with respect to the normalized Haar measure. Our result provides also a characterization of the set of doubly stochastic operators with non-zero entropy.

On an elementary characterization of the increasing convex ordering, by an application

1994 ◽  
Vol 31 (3) ◽  
pp. 834-840 ◽  
Author(s):  
Armand M. Makowski
Keyword(s):  
Short Proof ◽  
Probability Distributions ◽  
Short Note ◽  
Comparison Result ◽  
Convex Ordering ◽  
Simple Characterization ◽  
Increasing Convex Ordering ◽  
Elementary Characterization

In this short note, we present a simple characterization of the increasing convex ordering on the set of probability distributions on ℝ. We show its usefulness by providing a very short proof of a comparison result for M/GI/1 queues due to Daley and Rolski, and obtained by completely different means.

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