scholarly journals Distance between Bound Entangled States from Unextendible Product Bases and Separable States

2020 ◽  
Vol 2 (1) ◽  
pp. 49-56 ◽  
Author(s):  
Marcin Wieśniak ◽  
Palash Pandya ◽  
Omer Sakarya ◽  
Bianka Woloncewicz
Keyword(s):  
The State ◽  
Reference State ◽  
Entangled States ◽  
Separable State ◽  
Entanglement Witness ◽  
Separable States ◽  
Entanglement Witnesses

We discuss the use of the Gilbert algorithm to tailor entanglement witnesses for unextendible product basis bound entangled states (UPB BE states). The method relies on the fact that an optimal entanglement witness is given by a plane perpendicular to a line between the reference state, entanglement of which is to be witnessed, and its closest separable state (CSS). The Gilbert algorithm finds an approximation of CSS. In this article, we investigate if this approximation can be good enough to yield a valid entanglement witness. We compare witnesses found with Gilbert algorithm and those given by Bandyopadhyay–Ghosh–Roychowdhury (BGR) construction. This comparison allows us to learn about the amount of entanglement and we find a relationship between it and a feature of the construction of UPBBE states, namely the size of their central tile. We show that in most studied cases, witnesses found with the Gilbert algorithm in this work are more optimal than ones obtained by Bandyopadhyay, Ghosh, and Roychowdhury. This result implies the increased tolerance to experimental imperfections in a realization of the state.

scholarly journals DUALITIES AND POSITIVITY IN THE STUDY OF QUANTUM ENTANGLEMENT

2010 ◽  
Vol 08 (05) ◽  
pp. 721-754 ◽  
Author(s):  
ŁUKASZ SKOWRONEK
Keyword(s):  
Related Problem ◽  
Separable State ◽  
Real Numbers ◽  
Entanglement Witness ◽  
Separable States ◽  
Entanglement Witnesses ◽  
Real Polynomials ◽  
Schmidt Rank ◽  
Families Of Operators ◽  
Cone Duality

We present a survey on mathematical topics relating to separable states and entanglement witnesses. The convex cone duality between separable states and entanglement witnesses is discussed and later generalized to other families of operators, leading to their characterization via multiplicative properties. The condition for an operator to be an entanglement witness is rephrased as a problem of positivity of a family of real polynomials. By solving the latter in a specific case of a three-parameter family of operators, we obtain explicit description of entanglement witnesses belonging to that family. A related problem of block positivity over real numbers is discussed. We also consider a broad family of block positivity tests and prove that they can never be sufficient, which should be useful in case of future efforts in that direction. Finally, we introduce the concept of length of a separable state and present new results concerning relationships between the length and Schmidt rank. In particular, we prove that separable states of length lower or equal to 3 have Schmidt ranks equal to their lengths. We also give an example of a state which has length 4 and Schmidt rank 3.

scholarly journals Characterization and properties of weakly optimal entanglement witnesses

2015 ◽  
Vol 15 (13&14) ◽  
pp. 1109-1121
Author(s):  
Bang-Hai Wang ◽  
Hai-Ru Xu ◽  
Steve Campbell ◽  
Simone Severini
Keyword(s):  
Hilbert Space ◽  
Identity Matrix ◽  
Negative Number ◽  
Entanglement Witness ◽  
Geometric Properties ◽  
Expectation Value ◽  
Separable States ◽  
Entanglement Witnesses ◽  
Product Vector

We present an analysis of the properties and characteristics of weakly optimal entanglement witnesses, that is witnesses whose expectation value vanishes on at least one product vector. Any weakly optimal entanglement witness can be written as the form of $W^{wopt}=\sigma-c_{\sigma}^{max} I$, where $c_{\sigma}^{max}$ is a non-negative number and $I$ is the identity matrix. We show the relation between the weakly optimal witness $W^{wopt}$ and the eigenvalues of the separable states $\sigma$. Further we give an application of weakly optimal witnesses for constructing entanglement witnesses in a larger Hilbert space by extending the result of [P. Badzi\c{a}g {\it et al}, Phys. Rev. A {\bf 88}, 010301(R) (2013)], and we examine their geometric properties.

scholarly journals MIXING AND DECOHERENCE TO NEAREST SEPARABLE STATES

2009 ◽  
Vol 07 (04) ◽  
pp. 829-846
Author(s):  
AVIJIT LAHIRI ◽  
GAUTAM GHOSH ◽  
SANKHASUBHRA NAG
Keyword(s):  
Quantum Correlation ◽  
Entangled States ◽  
Entanglement Measure ◽  
Measuring Apparatus ◽  
Separable State ◽  
Mixed States ◽  
Classical Correlation ◽  
Separable States ◽  
System A ◽  
Pure States

We consider a class of entangled states of a quantum system (S) and a second system (A) where pure states of the former are correlated with mixed states of the latter, and work out the entanglement measure with reference to the nearest separable state. Such "pure-mixed" entanglement is expected when the system S interacts with a macroscopic measuring apparatus in a quantum measurement, where the quantum correlation is destroyed in the process of environment-induced decoherence whereafter only the classical correlation between S and A remains, the latter being large compared to the former. We present numerical evidence that the entangled S–A state drifts towards the nearest separable state through decoherence, with an additional tendency of equimixing among relevant groups of apparatus states.

scholarly journals Exposedness of Choi-Type Entanglement Witnesses and Applications to Lengths of Separable States

2013 ◽  
Vol 20 (04) ◽  
pp. 1350012 ◽  
Author(s):  
Kil-Chan Ha ◽  
Seung-Hyeok Kye
Keyword(s):  
Large Class ◽  
Separable State ◽  
Matrix Algebras ◽  
Separable States ◽  
Linear Maps ◽  
Entanglement Witnesses ◽  
Positive Linear Maps

We present a large class of indecomposable exposed positive linear maps between 3 × 3 matrix algebras. We also construct two-qutrit separable states with lengths ten in the interior of their dual faces. With these examples, we show that the length of a separable state may decrease strictly when we mix it with another separable state.

A matrix realignment method for recognizing entanglement

2003 ◽  
Vol 3 (3) ◽  
pp. 193-202
Author(s):  
K. Chen ◽  
L.-A. Wu
Keyword(s):  
Kronecker Product ◽  
Singular Values ◽  
Quantum Systems ◽  
Entangled States ◽  
Approximation Technique ◽  
Separable State ◽  
Simple Method ◽  
Bipartite Quantum Systems ◽  
The Matrix ◽  
Degree Of Entanglement

Motivated by the Kronecker product approximation technique, we have developed a very simple method to assess the inseparability of bipartite quantum systems, which is based on a realigned matrix constructed from the density matrix. For any separable state, the sum of the singular values of the matrix should be less than or equal to $1$. This condition provides a very simple, computable necessary criterion for separability, and shows powerful ability to identify most bound entangled states discussed in the literature. As a byproduct of the criterion, we give an estimate for the degree of entanglement of the quantum state.

scholarly journals Probabilistic Resumable Quantum Teleportation of a Two-Qubit Entangled State

Entropy ◽  
2019 ◽  
Vol 21 (4) ◽  
pp. 352 ◽  
Author(s):  
Zhan-Yun Wang ◽  
Yi-Tao Gou ◽  
Jin-Xing Hou ◽  
Li-Ke Cao ◽  
Xiao-Hui Wang
Keyword(s):  
Quantum Channel ◽  
Quantum Teleportation ◽  
The State ◽  
Entangled States ◽  
Entangled State ◽  
Quantum Channels ◽  
Unknown State ◽  
Optimal Probability

We explicitly present a generalized quantum teleportation of a two-qubit entangled state protocol, which uses two pairs of partially entangled particles as quantum channel. We verify that the optimal probability of successful teleportation is determined by the smallest superposition coefficient of these partially entangled particles. However, the two-qubit entangled state to be teleported will be destroyed if teleportation fails. To solve this problem, we show a more sophisticated probabilistic resumable quantum teleportation scheme of a two-qubit entangled state, where the state to be teleported can be recovered by the sender when teleportation fails. Thus the information of the unknown state is retained during the process. Accordingly, we can repeat the teleportion process as many times as one has available quantum channels. Therefore, the quantum channels with weak entanglement can also be used to teleport unknown two-qubit entangled states successfully with a high number of repetitions, and for channels with strong entanglement only a small number of repetitions are required to guarantee successful teleportation.

Indecomposability of entanglement witnesses

2015 ◽  
pp. 478-488
Author(s):  
Xiao-Fei Qi ◽  
Jin-Chuan Hou
Keyword(s):  
Entangled States ◽  
Finite Dimensional ◽  
Entanglement Witnesses ◽  
Ppt Criterion

We present a way to construct indecomposable entanglement witnesses from any permutations pi with pi^2 not equal to id for any finite dimensional bipartite systems. Some new bounded entangled states are also found, which can be detected by such witnesses while cannot be distinguished by PPT criterion, realignment criterion, etc.

scholarly journals An elementary introduction to the geometry of quantum states with pictures

2019 ◽  
Vol 32 (02) ◽  
pp. 2030001 ◽  
Author(s):  
J. Avron ◽  
O. Kenneth
Keyword(s):  
Convex Body ◽  
Cross Sections ◽  
Convex Bodies ◽  
Quantum States ◽  
Entangled States ◽  
High Dimensions ◽  
Geometric Properties ◽  
Precise Sense ◽  
Entanglement Witnesses ◽  
Elementary Methods

This is a review of the geometry of quantum states using elementary methods and pictures. Quantum states are represented by a convex body, often in high dimensions. In the case of [Formula: see text] qubits, the dimension is exponentially large in [Formula: see text]. The space of states can be visualized, to some extent, by its simple cross sections: Regular simplexes, balls and hyper-octahedra. a When the dimension gets large, there is a precise sense in which the space of states resembles, almost in every direction, a ball. The ball turns out to be a ball of rather low purity states. We also address some of the corresponding, but harder, geometric properties of separable and entangled states and entanglement witnesses. “All convex bodies behave a bit like Euclidean balls.” Keith Ball

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