Upper bound for SL-invariant entanglement measures of mixed 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.

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ENTANGLEMENT DISTRIBUTION AND STATE DISCRIMINATION IN TRIPARTITE QUBIT SYSTEMS

International Journal of Quantum Information ◽

10.1142/s0219749908003487 ◽

2008 ◽

Vol 06 (02) ◽

pp. 237-253 ◽

Cited By ~ 3

Author(s):

J. BATLE ◽

M. CASAS

Keyword(s):

Monte Carlo ◽

Entangled States ◽

Perfect State ◽

Mixed States ◽

W States ◽

State Discrimination ◽

Entanglement Measures ◽

Entanglement Distribution ◽

Maximally Entangled States ◽

Qubit States

This work reviews and extends recent results concerning the distribution of entanglement, as well as nonlocality (in terms of inequality violations) in tripartite qubit systems. With recourse to a Monte Carlo generation of pure and mixed states of three-qubits, we explore several features related to the distribution of entanglement (expressed in the form of different measures of multiqubit entanglement based upon bipartitions). Also, special interest is paid to maximally entangled states (such as the GHZ for three-qubits) and W states. This study also sheds some light on the interesting relation existing between some entanglement measures and perfect state discrimination in LOCC measurements relevant to cryptographic protocols. We round off the results by studying the distribution of entanglement between Alice and Bob in a modified teleportation protocol toy model over three-qubit states.

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Relevance of rank for a mixed state quantum teleportation resource

Quantum Information and Computation ◽

10.26421/qic14.13-14-12 ◽

2014 ◽

Vol 14 (13&14) ◽

pp. 1227-1237

Author(s):

K.G. Paulson ◽

S.V.M. Satyanarayana

Keyword(s):

Lower Bound ◽

Mixed State ◽

Upper Bound ◽

Quantum Teleportation ◽

Low Rank ◽

Entangled States ◽

Mixed States ◽

Logarithmic Negativity ◽

Higher Rank ◽

Fidelity Measures

Mixed entangled states are generic resource for quantum teleportation. Optimal teleportation fidelity measures the success of quantum teleportation. The relevance of rank in the teleportation process is investigated by constructing three new maximally entangled mixed states (MEMS) of different ranks. Linear entropy, concurrence, optimal teleportation fidelity and Bell function are obtained for each of these states analytically. It is found that mixed states with higher rank are better resource for teleportation. In order to achieve a fixed value of optimal teleportation fidelity, we find that low rank states must have high concurrence. Further, for each of ranks 2, 3 and 4, we numerically generate 30000 maximally entangled mixed states. The analysis of these states reveals the existence of a rank dependent upper bound on optimal teleportation fidelity for a fixed purity. In order to achieve a fixed optimal teleportation fidelity, we find MEMS exhibit a rank dependent lower bound on concurrence. MEMS are classified in terms of their degree of nonlocality. The results are found to be same with logarithmic negativity used as a measure of entanglement.

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Criteria for measures of quantum correlations

Quantum Information and Computation ◽

10.26421/qic12.9-10-1 ◽

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.

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Connections between relative entropy of entanglement and geometric measure of entanglement

Quantum Information and Computation ◽

10.26421/qic4.4-2 ◽

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.

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MIXED STATE ENTANGLEMENT MEASURES FOR INTERMEDIATE SEPARABILITY

International Journal of Quantum Information ◽

10.1142/s0219749910006526 ◽

2010 ◽

Vol 08 (04) ◽

pp. 677-685 ◽

Cited By ~ 1

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.

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WITNESSED ENTANGLEMENT

International Journal of Quantum Information ◽

10.1142/s0219749906001803 ◽

2006 ◽

Vol 04 (02) ◽

pp. 331-340 ◽

Cited By ~ 28

Author(s):

FERNANDO G. S. L. BRANDÃO ◽

REINALDO O. VIANNA

Keyword(s):

Multipartite Entanglement ◽

Mixed States ◽

Entanglement Measures ◽

Different Types

We present a new measure of entanglement for mixed states. It can be approximately computable for every state and can be used to quantify all different types of multipartite entanglement. We show that it satisfies the usual properties of a good entanglement quantifier and derive relations between it and other entanglement measures.

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

International Journal of Modern Physics C ◽

10.1142/s0129183110015129 ◽

2010 ◽

Vol 21 (03) ◽

pp. 291-305 ◽

Cited By ~ 36

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.

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Matrices of fidelities for ensembles of quantum states and the Holevo quantity

Quantum Information and Computation ◽

10.26421/qic12.5-6-8 ◽

2012 ◽

Vol 12 (5&6) ◽

pp. 472-489

Author(s):

Mark Fannes ◽

Fernando de Melo ◽

Wojciech Roga ◽

Karol Zyczkowski

Keyword(s):

Upper Bound ◽

Quantum States ◽

Gram Matrix ◽

Mixed States ◽

Correlation Matrices

The entropy of the Gram matrix of a joint purification of an ensemble of $K$ mixed states yields an upper bound for the Holevo information $\chi$ of the ensemble. In this work we combine geometrical and probabilistic aspects of the ensemble in order to obtain meaningful bounds for $\chi$. This is done by constructing various correlation matrices involving fidelities between every pair of states from the ensemble. For $K=3$ quantum states we design a matrix of root fidelities that is positive and the entropy of which is conjectured to upper bound $\chi$. Slightly weaker bounds are established for arbitrary ensembles. Finally, we investigate correlation matrices involving multi-state correlations in relation to the Holevo quantity.

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