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 Construction of genuinely entangled subspacesand the associated bounds on entanglement measures for mixed states

2021 ◽  
Author(s):  
Konstantin Antipin
Keyword(s):  
Information Processing ◽  
Quantum Information ◽  
Lower Bounds ◽  
Quantum Information Processing ◽  
Multipartite Entanglement ◽  
Quantum Channels ◽  
Valuable Resource ◽  
Mixed States ◽  
Entanglement Measures ◽  
Pure States

Abstract Genuine entanglement is the strongest form of multipartite entanglement. Genuinely entangled pure states contain entanglement in every bipartition and as such can be regarded as a valuable resource in the protocols of quantum information processing. A recent direction of research is the construction of genuinely entangled subspaces — the class of subspaces consisting entirely of genuinely entangled pure states. In this paper we present methods of construction of such subspaces including those of maximal possible dimension. The approach is based on the composition of bipartite entangled subspaces and quantum channels of certain types. The examples include maximal subspaces for systems of three qubits, four qubits, three qutrits. We also provide lower bounds on two entanglement measures for mixed states, the concurrence and the convex-roof extended negativity, which are directly connected with the projection on genuinely entangled subspaces.

scholarly journals Generalized geometric measure of entanglement for multiparty mixed states

2016 ◽  
Vol 94 (2) ◽  
Author(s):  
Tamoghna Das ◽  
Sudipto Singha Roy ◽  
Shrobona Bagchi ◽  
Avijit Misra ◽  
Aditi Sen(De) ◽  
...  
Keyword(s):  
Mixed States ◽  
Geometric Measure ◽  
Geometric Measure Of Entanglement

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 Upper bounds for relative entropy of entanglement based on active learning

2020 ◽  
Vol 5 (4) ◽  
pp. 045019
Author(s):  
Shi-Yao Hou ◽  
Chenfeng Cao ◽  
D L Zhou ◽  
Bei Zeng
Keyword(s):  
Active Learning ◽  
Relative Entropy ◽  
Upper Bounds ◽  
Relative Entropy Of Entanglement

scholarly journals MIXED STATE ENTANGLEMENT MEASURES FOR INTERMEDIATE SEPARABILITY

2010 ◽  
Vol 08 (04) ◽  
pp. 677-685 ◽  
Author(s):  
TSUBASA ICHIKAWA ◽  
MARCUS HUBER ◽  
PHILIPP KRAMMER ◽  
BEATRIX C. HIESMAYR
Keyword(s):  
Lower Bound ◽  
Quantum State ◽  
Mixed State ◽  
Building Blocks ◽  
Mixed States ◽  
Entanglement Measures

To determine whether a given multipartite quantum state is separable with respect to some partition, we construct a family of entanglement measures {Rm(ρ)}. This is done utilizing generalized concurrences as building blocks which are defined by flipping of M constituents and indicate states that are separable with regard to bipartitions when vanishing. Furthermore, we provide an analytically computable lower bound for {Rm(ρ)} via a simple ordering relation of the convex roof extension. Using the derived lower bound, we illustrate the effect of the isotropic noise on a family of four-qubit mixed states for each intermediate separability.

A COMPARATIVE STUDY OF NEGATIVITY AND CONCURRENCE BASED ON SPIN COHERENT STATES

2010 ◽  
Vol 21 (03) ◽  
pp. 291-305 ◽  
Author(s):  
K. BERRADA ◽  
M. El BAZ ◽  
H. ELEUCH ◽  
Y. HASSOUNI
Keyword(s):  
Comparative Study ◽  
Coherent States ◽  
Mixed States ◽  
Qubit System ◽  
Entanglement Measures ◽  
Pure States ◽  
Spin Coherent States

In this paper, we investigate two different entanglement measures, the negativity and concurrence, in the case of pure and mixed states of two-qubit system basing on the spin coherent states. For two-qubit pure states, the negativity is the same as the concurrence. For mixed states, using a simplified expression of concurrence in Wootters' measure of entanglement, we write the bounds of Verstraete et al.1as a function of some new parameters and we compare the both measures for a class of mixed states.

scholarly journals MEASUREMENT-INDUCED NONLOCALITY OVER TWO-SIDED PROJECTIVE MEASUREMENTS

2013 ◽  
Vol 27 (16) ◽  
pp. 1350067 ◽  
Author(s):  
YU GUO
Keyword(s):  
Lower Bound ◽  
Quantum Correlation ◽  
Quantum Discord ◽  
Mixed States ◽  
Infinite Dimensional ◽  
Pure States ◽  
Projective Measurements

Measurement-induced nonlocality (MIN), introduced by Luo and Fu [Phys. Rev. Lett.106, 120401 (2011)], is a kind of quantum correlation which is different from entanglement and quantum discord (QD). MIN is defined over one-sided projective measurements. In this paper, we introduce a MIN over two-sided projective measurements. The nullity of this two-sided MIN is characterized, a formula for calculating two-sided MIN for pure states is proposed, and a lower bound of (two-sided) MIN for maximally entangled mixed states is given. In addition, we find that (two-sided) MIN is not continuous. Both finite- and infinite-dimensional cases are considered.

ENTANGLEMENT OF FORMATION FOR A CLASS OF SPECIAL QUANTUM STATES

2007 ◽  
Vol 05 (03) ◽  
pp. 343-352 ◽  
Author(s):  
HUI ZHAO ◽  
ZHI-XI WANG
Keyword(s):  
Lower Bound ◽  
Large Class ◽  
Special Kind ◽  
Quantum States ◽  
The Other ◽  
High Dimensional ◽  
Entangled States ◽  
Mixed States ◽  
Density Matrices ◽  
Entanglement Of Formation

The entanglement of formation for a class of high-dimensional quantum mixed states is investigated. A special kind of D-computable states is defined and the lower bound of entanglement of formation for a large class of density matrices whose decompositions lie in these D-computable quantum states is obtained. Moreover we present a kind of construction for this special state which is defined by a class of special matrices with two non-zero different eigenvalues and the other eigenvalues are zero. Making use of the D-computable we construct a class of bound entangled states.

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