scholarly journals Quantum coherence, discord and correlation measures based on Tsallis relative entropy

2020 ◽  
Vol 20 (7&8) ◽  
pp. 553-569
Author(s):  
Anna Vershynina
Keyword(s):  
Relative Entropy ◽  
Quantum Coherence ◽  
Distance Measure ◽  
Upper And Lower Bounds ◽  
General Definition ◽  
Strong Monotonicity ◽  
Correlation Measure ◽  
Coherence Measure ◽  
Pure States ◽  
Correlation Measures

Several ways have been proposed in the literature to define a coherence measure based on Tsallis relative entropy. One of them is defined as a distance between a state and a set of incoherent states with Tsallis relative entropy taken as a distance measure. Unfortunately, this measure does not satisfy the required strong monotonicity, but a modification of this coherence has been proposed that does. We introduce three new Tsallis coherence measures coming from a more general definition that also satisfy the strong monotonicity, and compare all five definitions between each other. Using three coherence measures that we discuss, one can also define a discord. Two of these have been used in the literature, and another one is new. We also discuss two correlation measures based on Tsallis relative entropy. We provide explicit expressions for all three discord and two correlation measure on pure states. Lastly, we provide tight upper and lower bounds on two discord and correlations measures on any quantum state, with the condition for equality.

scholarly journals Geometric measures of discordlike quantum correlations based on Tsallis relative entropy

2019 ◽  
Vol 17 (04) ◽  
pp. 1950034
Author(s):  
Weijing Li
Keyword(s):  
Relative Entropy ◽  
Quantum Correlation ◽  
Quantum Correlations ◽  
Partial Coherence ◽  
Good Measure ◽  
Coherence Measure ◽  
Geometric Measure ◽  
Special Cases ◽  
Analytic Expression ◽  
Correlation Measures

A kind of new geometric measure of quantum correlations is formulated. The proposed formulation is in terms of the quantum Tsallis relative entropy and can naturally be viewed as a one-parameter extension quantum discordlike measure that satisfies all requirements of a good measure of quantum correlations. It is of an elegant analytic expression and contains several existing good quantum correlation measures as special cases. The partial coherence measure is also investigated.

scholarly journals Proportional lumpability and proportional bisimilarity

2021 ◽  
Author(s):  
Andrea Marin ◽  
Carla Piazza ◽  
Sabina Rossi
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Upper And Lower Bounds ◽  
General Definition ◽  
Performance Indices ◽  
State Aggregation ◽  
Structural Regularity ◽  
Original Definition ◽  
Aggregation Technique ◽  
Definition Of

AbstractIn this paper, we deal with the lumpability approach to cope with the state space explosion problem inherent to the computation of the stationary performance indices of large stochastic models. The lumpability method is based on a state aggregation technique and applies to Markov chains exhibiting some structural regularity. Moreover, it allows one to efficiently compute the exact values of the stationary performance indices when the model is actually lumpable. The notion of quasi-lumpability is based on the idea that a Markov chain can be altered by relatively small perturbations of the transition rates in such a way that the new resulting Markov chain is lumpable. In this case, only upper and lower bounds on the performance indices can be derived. Here, we introduce a novel notion of quasi-lumpability, named proportional lumpability, which extends the original definition of lumpability but, differently from the general definition of quasi-lumpability, it allows one to derive exact stationary performance indices for the original process. We then introduce the notion of proportional bisimilarity for the terms of the performance process algebra PEPA. Proportional bisimilarity induces a proportional lumpability on the underlying continuous-time Markov chains. Finally, we prove some compositionality results and show the applicability of our theory through examples.

scholarly journals Symmetry-like Relation of Relative Entropy Measure of Quantum Coherence

2020 ◽  
Author(s):  
Cheng-yang Zhang ◽  
Zhi-hua Guo ◽  
H.X. Cao
Keyword(s):  
Relative Entropy ◽  
Quantum Coherence ◽  
Quantum Physics ◽  
Information Science ◽  
Upper Bounds ◽  
Von Neumann Entropy ◽  
Entropy Measure ◽  
Lower And Upper Bounds ◽  
Von Neumann ◽  
Natural Logarithm

Quantum coherence is an important physical resource in quantum information science, and also as one of the most fundamental and striking features in quantum physics. In this paper, we obtain a symmetry-like relation of relative entropy measure $C_r(\rho)$ of coherence for $n$-partite quantum states $\rho$, which gives lower and upper bounds for $C_r(\rho)$. Meanwhile, we discuss the conjecture about the validity of the inequality $C_r(\rho)\leq C_{\ell_1}(\rho)$ for any state $\rho$. We observe that every mixture $\eta$ of a state $\rho$ satisfying $C_r(\rho)\leq C_{\ell_1}(\rho)$ and any incoherent state $\sigma$ also satisfies the conjecture. We also note that if the von Neumann entropy is defined by the natural logarithm $\ln$ instead of $\log_2$, then the reduced relative entropy measure of coherence $\bar{C}_r(\rho)=-\rho_{\rm{diag}}\ln \rho_{\rm{diag}}+\rho\ln \rho$ satisfies the inequality ${\bar{C}}_r(\rho)\leq C_{\ell_1}(\rho)$ for any mixed state $\rho$.

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.

TWO-MODE GAUSSIAN DENSITY MATRICES AND SQUEEZING OF PHOTONS

1992 ◽  
Vol 06 (10) ◽  
pp. 1657-1709 ◽  
Author(s):  
ROBERT R. TUCCI
Keyword(s):  
Fundamental Equation ◽  
Sufficient Conditions ◽  
Upper And Lower Bounds ◽  
Second Law ◽  
Gaussian State ◽  
Density Matrices ◽  
Duhem Equation ◽  
Gaussian Density ◽  
Pure States ◽  
Necessary And Sufficient

In this paper, we generalize to 2-mode states the 1-mode state results obtained in a previous paper. We study 2-mode Gaussian density matrices (i.e., density matrices of the form: exponential of a quadratic polynomial in the creation and annihilation operators for the two modes). We find a linear transformation which maps the two annihilation operators, one for each mode, into two new annihilation operators that are uncorrelated and unsqueezed. This allows us to express the density matrix as a product of two 1mode density matrices. We find general conditions under which 2-mode Gaussian density matrices become pure states. Possible pure states include the 2-mode squeezed pure states commonly mentioned in the literature, plus other pure states never mentioned before. We discuss the entropy and thermodynamic laws (Second Law, Fundamental Equation, and Gibbs-Duhem Equation) for the 2-mode states being considered. We study the change in entropy that is produced when a 2-mode Gaussian state is subjected to a measurement of the complex amplitude of one of its two modes. We derive upper and lower bounds for the final (i.e., after the measurement) entropy of the unmeasured mode, and we give necessary and sufficient conditions for the achievement of these bounds. The existence of the bounds is shown to be a consequence of the concavity property of the entropy function.

Connections between relative entropy of entanglement and geometric measure of entanglement

2004 ◽  
Vol 4 (4) ◽  
pp. 252-272
Author(s):  
T.-C. Wei ◽  
M. Ericsson ◽  
P.M. Goldbart ◽  
W.J. Munro
Keyword(s):  
Lower Bound ◽  
Relative Entropy ◽  
Upper Bounds ◽  
Mixed States ◽  
Geometric Measure ◽  
Entanglement Of Formation ◽  
Entanglement Measures ◽  
Pure States ◽  
Relative Entropy Of Entanglement ◽  
Geometric Measure Of Entanglement

As two of the most important entanglement measures---the entanglement of formation and the entanglement of distillation---have so far been limited to bipartite settings, the study of other entanglement measures for multipartite systems appears necessary. Here, connections between two other entanglement measures---the relative entropy of entanglement and the geometric measure of entanglement---are investigated. It is found that for arbitrary pure states the latter gives rise to a lower bound on the former. For certain pure states, some bipartite and some multipartite, this lower bound is saturated, and thus their relative entropy of entanglement can be found analytically in terms of their known geometric measure of entanglement. For certain mixed states, upper bounds on the relative entropy of entanglement are also established. Numerical evidence strongly suggests that these upper bounds are tight, i.e., they are actually the relative entropy of entanglement.

scholarly journals Geometric lower bound for a quantum coherence measure

2015 ◽  
Vol 91 (4) ◽  
Author(s):  
Diego Paiva Pires ◽  
Lucas C. Céleri ◽  
Diogo O. Soares-Pinto
Keyword(s):  
Lower Bound ◽  
Quantum Coherence ◽  
Coherence Measure

scholarly journals Recommendation of Attributes for Heart Disease Prediction using Correlation Measure

2019 ◽  
Vol 8 (2S3) ◽  
pp. 870-875
Keyword(s):  
Heart Disease ◽  
Mortality Rate ◽  
Early Stage ◽  
Heart Diseases ◽  
Correct Diagnosis ◽  
Disease Prediction ◽  
Correlation Measure ◽  
Heart Problem ◽  
Correlation Measures ◽  
General Prediction

Heart diseases are the major cause for human mortality rate. Correct diagnosis and treatment at an early stage will save people from heart disease and will decrease mortality rate due to heart problem. Since ten years various data mining techniques have been used to facilitate the prediction of heart diseases .In general prediction algorithms for trained with huge, known dataset to arrive at a classifier which then predicts the diseases for unknown data with the help of classifying attributes. These attributes also called as features. In this work relevant features are determined for heart disease prediction with known dataset using correlation measures. The results are presented.

A Novel Hybrid Correlation Measure for Query Expansion-Based Information Retrieval

2020 ◽  
pp. 1-19 ◽  
Author(s):  
Ilyes Khennak ◽  
Habiba Drias
Keyword(s):  
Information Retrieval ◽  
Information Needs ◽  
Query Expansion ◽  
Relevant Information ◽  
Medical Database ◽  
Correlation Measure ◽  
Real Dataset ◽  
Correlation Measures ◽  
Degree Of Similarity ◽  
Prior State

Query expansion (QE) is one of the most effective techniques to enhance the retrieval performance and to retrieve more relevant information. It attempts to build more useful queries by enriching the original queries with additional expansion terms that best characterize the users' information needs. In this chapter, the authors propose a new correlation measure for query expansion to evaluate the degree of similarity between the expansion term candidates and the original query terms. The proposed correlation measure is a hybrid of two correlation measures. The first one is considered as an external correlation and it is based on the term co-occurrence, and the second one is considered as an internal correlation and it is based on the term proximity. Extensive experiments have been performed on MEDLINE, a real dataset from a large online medical database. The results show the effectiveness of the proposed approach compared to prior state-of-the-art approaches.

Sign in / Sign up

Export Citation Format

Share Document