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 Local transformations of multiple multipartite states

2021 ◽  
Vol 11 (2) ◽  
Author(s):  
Antoine Neven ◽  
David Kenworthy Gunn ◽  
Martin Hebenstreit ◽  
Barbara Kraus
Keyword(s):  
Single Copy ◽  
Partial Ordering ◽  
Multipartite Entanglement ◽  
Entanglement Measure ◽  
Classical Communication ◽  
Wide Range ◽  
Pure States ◽  
Multipartite States ◽  
The Individual ◽  
Probabilistic Protocols

Understanding multipartite entanglement is vital, as it underpins a wide range of phenomena across physics. The study of transformations of states via Local Operations assisted by Classical Communication (LOCC) allows one to quantitatively analyse entanglement, as it induces a partial order in the Hilbert space. However, it has been shown that, for systems with fixed local dimensions, this order is generically trivial, which prevents relating multipartite states to each other with respect to any entanglement measure. In order to obtain a non-trivial partial ordering, we study a physically motivated extension of LOCC: multi-state LOCC. Here, one considers simultaneous LOCC transformations acting on a finite number of entangled pure states. We study both multipartite and bipartite multi-state transformations. In the multipartite case, we demonstrate that one can change the stochastic LOCC (SLOCC) class of the individual initial states by only applying Local Unitaries (LUs). We show that, by transferring entanglement from one state to the other, one can perform state conversions not possible in the single copy case; provide examples of multipartite entanglement catalysis; and demonstrate improved probabilistic protocols. In the bipartite case, we identify numerous non-trivial LU transformations and show that the source entanglement is not additive. These results demonstrate that multi-state LOCC has a much richer landscape than single-state LOCC.

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 Correlation loss and multipartite entanglement across a black hole horizon (

2009 ◽  
Vol 9 (7&8) ◽  
pp. 657-665
Author(s):  
G. Adesso ◽  
I. Fuentes-Schuller
Keyword(s):  
Black Hole ◽  
Small Mass ◽  
Point Of View ◽  
Multipartite Entanglement ◽  
Squeezed State ◽  
Schwarzschild Black Hole ◽  
Bipartite Entanglement ◽  
Hawking Effect ◽  
Entanglement Distribution ◽  
Quantum And Classical Correlations

We investigate the Hawking effect on entangled fields. By considering a scalar field which is in a two-mode squeezed state from the point of view of freely falling (Kruskal) observers crossing the horizon of a Schwarzschild black hole, we study the degradation of quantum and classical correlations in the state from the perspective of physical (Schwarzschild) observers confined outside the horizon. Due to monogamy constraints on the entanglement distribution, we show that the lost bipartite entanglement is recovered as multipartite entanglement among modes inside and outside the horizon. In the limit of a small-mass black hole, no bipartite entanglement is detected outside the horizon, while the genuine multipartite entanglement interlinking the inner and outer regions grows infinitely.

scholarly journals CHARACTERIZING AND MEASURING MULTIPARTITE ENTANGLEMENT

2007 ◽  
Vol 05 (01n02) ◽  
pp. 97-103 ◽  
Author(s):  
P. FACCHI ◽  
G. FLORIO ◽  
S. PASCAZIO
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Multipartite Entanglement ◽  
Bipartite Entanglement ◽  
Pure States

A method is proposed to characterize and quantify multipartite entanglement in terms of the probability density function of bipartite entanglement over all possible balanced bipartitions of an ensemble of qubits. The method is tested on a class of random pure states.

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 ENTANGLEMENT FRUSTRATION IN MULTIMODE GAUSSIAN STATES

2012 ◽  
Vol 09 (02) ◽  
pp. 1260022 ◽  
Author(s):  
COSMO LUPO ◽  
STEFANO MANCINI ◽  
PAOLO FACCHI ◽  
GIUSEPPE FLORIO ◽  
SAVERIO PASCAZIO
Keyword(s):  
Pure State ◽  
High Energy ◽  
Quantum Systems ◽  
Entangled States ◽  
Multipartite Entanglement ◽  
Total System ◽  
Continuous Variables ◽  
Bipartite Entanglement ◽  
Gaussian States ◽  
Composite Quantum System

Bipartite entanglement between two parties of a composite quantum system can be quantified in terms of the purity of one party and there always exists a pure state of the total system that maximizes it (and minimizes purity). When many different bipartitions are considered, the requirement that purity be minimal for all bipartitions gives rise to the phenomenon of entanglement frustration. This feature, observed in quantum systems with both discrete and continuous variables, can be studied by means of a suitable cost function whose minimizers are the maximally multipartite-entangled states (MMES). In this paper we extend the analysis of multipartite entanglement frustration of Gaussian states in multimode bosonic systems. We derive bounds on the frustration, under the constraint of finite mean energy, in the low- and high-energy limits.

scholarly journals Some Remarks on the Entanglement Number

Quanta ◽  
2020 ◽  
Vol 9 (1) ◽  
pp. 22-36
Author(s):  
George Androulakis ◽  
Ryan McGaha
Keyword(s):  
Point Of View ◽  
Entanglement Measure ◽  
Mixed States ◽  
Interesting Point ◽  
Finite Dimensional ◽  
Entanglement Measures ◽  
Pure States ◽  
Tree Representations ◽  
Natural Way ◽  
Finite Dimensional Hilbert Space

Gudder, in a recent paper, defined a candidate entanglement measure which is called the entanglement number. The entanglement number is first defined on pure states and then it extends to mixed states by the convex roof construction. In Gudder's article it was left as an open problem to show that Optimal Pure State Ensembles (OPSE) exist for the convex roof extension of the entanglement number from pure to mixed states. We answer Gudder's question in the affirmative, and therefore we obtain that the entanglement number vanishes only on the separable states. More generally we show that OPSE exist for the convex roof extension of any function that is norm continuous on the pure states of a finite dimensional Hilbert space. Further we prove that the entanglement number is an LOCC monotone, (and thus an entanglement measure), by using a criterion that was developed by Vidal in 2000. We present a simplified proof of Vidal's result where moreover we use an interesting point of view of tree representations for LOCC communications. Lastly, we generalize Gudder's entanglement number by producing a monotonic family of entanglement measures which converge in a natural way to the entropy of entanglement.Quanta 2020; 9: 22–36.

Coherence measures and optimal conversion for coherent states

2015 ◽  
Vol 15 (15&16) ◽  
pp. 1307-1316 ◽  
Author(s):  
Shuanping Du ◽  
Zhaofang Bai ◽  
Xiaofei Qi
Keyword(s):  
Relative Entropy ◽  
Pure State ◽  
Coherent States ◽  
Single Copy ◽  
Entropy Measure ◽  
General Strategy ◽  
Important Class ◽  
L1 Norm ◽  
Pure States

We discuss a general strategy to construct coherence measures. One can build an important class of coherence measures which cover the relative entropy measure for pure states, the l1-norm measure for pure states and the α-entropy measure. The optimal conversion of coherent states under incoherent operations is presented which sheds some light on the coherence of a single copy of a pure state.

Multipartite entanglement and hyperdeterminants

2002 ◽  
Vol 2 (Special) ◽  
pp. 540-555
Author(s):  
A. Miyake ◽  
M. Wadati
Keyword(s):  
Entangled States ◽  
Multipartite Entanglement ◽  
Maximal Dimension ◽  
Ordered Structure ◽  
Mixed States ◽  
Separable States ◽  
Entanglement Measures ◽  
Pure States ◽  
Maximally Entangled States ◽  
Bipartite Case

We classify multipartite entanglement in a unified manner, focusing on a duality between the set of separable states and that of entangled states. Hyperdeterminants, derived from the duality, are natural generalizations of entanglement measures, the concurrence, 3-tangle for 2, 3 qubits respectively. Our approach reveals how inequivalent multipartite entangled classes of pure states constitute a partially ordered structure under local actions, significantly different from a totally ordered one in the bipartite case. Moreover, the generic entangled class of the maximal dimension, given by the nonzero hyperdeterminant, does not include the maximally entangled states in Bell's inequalities in general (e.g., in the \(n \!\geq\! 4\) qubits), contrary to the widely known bipartite or 3-qubit cases. It suggests that not only are they never locally interconvertible with the majority of multipartite entangled states, but they would have no grounds for the canonical \(n\)-partite entangled states. Our classification is also useful for that of mixed states.

scholarly journals MULTIPARTITE ENTANGLEMENT UNDER STOCHASTIC LOCAL OPERATIONS AND CLASSICAL COMMUNICATION

2004 ◽  
Vol 02 (01) ◽  
pp. 65-77 ◽  
Author(s):  
AKIMASA MIYAKE
Keyword(s):  
Quantum System ◽  
Multipartite Entanglement ◽  
Mixed States ◽  
Classical Communication ◽  
Quantum Operations ◽  
Essential Properties ◽  
Pure States ◽  
Local Filtering

Stochastic local operations and classical communication (SLOCC), also called local filtering operations, are a convenient, useful set of quantum operations in grasping essential properties of entanglement. We give a quick overview of the characteristics of multipartite entanglement in terms of SLOCC, illustrating the 2-qubit and the rest (2×2×n) quantum system. This not only includes celebrated results of 3-qubit pure states, but also has implications to 2-qubit mixed states.

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