Separability of diagonal symmetric states: a quadratic conic optimization problem

We study the separability problem in mixtures of Dicke states i.e., the separability of the so-called Diagonal Symmetric (DS) states. First, we show that separability in the case of DS inCd⊗Cd(symmetric qudits) can be reformulated as a quadratic conic optimization problem. This connection allows us to exchange concepts and ideas between quantum information and this field of mathematics. For instance, copositive matrices can be understood as indecomposable entanglement witnesses for DS states. As a consequence, we show that positivity of the partial transposition (PPT) is sufficient and necessary for separability of DS states ford≤4. Furthermore, ford≥5, we provide analytic examples of PPT-entangled states. Second, we develop new sufficient separability conditions beyond the PPT criterion for bipartite DS states. Finally, we focus onN-partite DS qubits, where PPT is known to be necessary and sufficient for separability. In this case, we present a family of almost DS states that are PPT with respect to each partition but nevertheless entangled.

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Indecomposability of entanglement witnesses

Quantum Information and Computation ◽

10.26421/qic15.5-6-7 ◽

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.

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Free versus bound entanglement, a NP-hard problem tackled by machine learning

Scientific Reports ◽

10.1038/s41598-021-98523-6 ◽

2021 ◽

Vol 11 (1) ◽

Author(s):

Beatrix C. Hiesmayr

Keyword(s):

Machine Learning ◽

Large Family ◽

Machine Learning Algorithms ◽

Entangled States ◽

Hard Problem ◽

Np Hard ◽

Entanglement Witnesses ◽

Np Hard Problem ◽

Bound Entanglement ◽

Ppt Criterion

AbstractEntanglement detection in high dimensional systems is a NP-hard problem since it is lacking an efficient way. Given a bipartite quantum state of interest free entanglement can be detected efficiently by the PPT-criterion (Peres-Horodecki criterion), in contrast to detecting bound entanglement, i.e. a curious form of entanglement that can also not be distilled into maximally (free) entangled states. Only a few bound entangled states have been found, typically by constructing dedicated entanglement witnesses, so naturally the question arises how large is the volume of those states. We define a large family of magically symmetric states of bipartite qutrits for which we find $$82\%$$ 82 % to be free entangled, $$2\%$$ 2 % to be certainly separable and as much as $$10\%$$ 10 % to be bound entangled, which shows that this kind of entanglement is not rare. Via various machine learning algorithms we can confirm that the remaining $$6\%$$ 6 % of states are more likely to belonging to the set of separable states than bound entangled states. Most important we find via dimension reduction algorithms that there is a strong two-dimensional (linear) sub-structure in the set of bound entangled states. This revealed structure opens a novel path to find and characterize bound entanglement towards solving the long-standing problem of what the existence of bound entanglement is implying.

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PPT classifiable families of bipartite states

Quantum Information and Computation ◽

10.26421/qic10.5-6-10 ◽

2010 ◽

Vol 10 (5&6) ◽

pp. 535-538

Author(s):

F.E.S. Steinhoff ◽

M.C. de Oliveira

Keyword(s):

Numerical Analysis ◽

Density Matrix ◽

Block Structure ◽

Arbitrary Dimension ◽

Entangled States ◽

Sufficient Criterion ◽

Necessary And Sufficient ◽

Ppt Criterion ◽

Bipartite States ◽

Global Density

We construct a family of bipartite states of arbitrary dimension whose eigenvalues of the partially transposed matrix can be inferred directly from the block structure of the global density matrix. We identify from this several subfamilies in which the PPT criterion is both necessary and sufficient. A sufficient criterion of separability is obtained, which is fundamental for the discussion. We show how several examples of states known to be classifiable by the PPT criterion indeed belong to this general set. Possible uses of these states in numerical analysis of entanglement and in the search of PPT bound entangled states are briefly discussed.

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Separability criteria based on the Bloch representation of density matrices

Quantum Information and Computation ◽

10.26421/qic7.7-5 ◽

2007 ◽

Vol 7 (7) ◽

pp. 624-638

Author(s):

J. de Vicente

Keyword(s):

Sufficient Conditions ◽

Quantum Systems ◽

Necessary Condition ◽

Sufficient Condition ◽

Pure Product ◽

Separability Problem ◽

Arbitrary Dimensions ◽

Necessary And Sufficient ◽

Bloch Representation

We study the separability of bipartite quantum systems in arbitrary dimensions using the Bloch representation of their density matrix. This approach enables us to find an alternative characterization of the separability problem, from which we derive a necessary condition and sufficient conditions for separability. For a certain class of states the necessary condition and a sufficient condition turn out to be equivalent, therefore yielding a necessary and sufficient condition. The proofs of the sufficient conditions are constructive, thus providing decompositions in pure product states for the states that satisfy them. We provide examples that show the ability of these conditions to detect entanglement. In particular, the necessary condition is proved to be strong enough to detect bound entangled states.

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A note on the degree conjecture for separability of multipartite quantum states

International Journal of Quantum Information ◽

10.1142/s0219749920500483 ◽

2021 ◽

pp. 2050048

Author(s):

Zhen Wang ◽

Ming-Jing Zhao ◽

Zhi-Xi Wang

Keyword(s):

Quantum States ◽

Point Graph ◽

Mixed States ◽

Degree Condition ◽

Graph Laplacians ◽

Nearest Point ◽

Necessary And Sufficient ◽

Ppt Criterion ◽

Math Phys ◽

Degree Criterion

The degree conjecture for bipartite quantum states which are normalized graph Laplacians was first put forward by Braunstein et al. [Phys. Rev. A 73 (2006) 012320]. The degree criterion, which is equivalent to PPT criterion, is simpler and more efficient to detect the separability of quantum states associated with graphs. Hassan et al. settled the degree conjecture for the separability of multipartite quantum states in [J. Math. Phys. 49 (2008) 0121105]. It is proved that the conjecture is true for pure multipartite quantum states. However, the degree condition is only necessary for separability of a class of quantum mixed states. It does not apply to all mixed states. In this paper, we show that the degree conjecture holds for the mixed quantum states of nearest point graph. As a byproduct, the degree criterion is necessary and sufficient for multipartite separability of [Formula: see text]-qubit quantum states associated with graphs.

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Robust Approximate Optimality Conditions for Uncertain Nonsmooth Optimization with Infinite Number of Constraints

Mathematics ◽

10.3390/math7010012 ◽

2018 ◽

Vol 7 (1) ◽

pp. 12 ◽

Cited By ~ 7

Author(s):

Xiangkai Sun ◽

Hongyong Fu ◽

Jing Zeng

Keyword(s):

Optimality Conditions ◽

Optimization Problem ◽

Constraint Qualification ◽

Optimization Problems ◽

Optimization Technique ◽

Optimal Solution ◽

Sufficient Optimality Conditions ◽

Convex Constraint ◽

Approximate Optimality Conditions ◽

Necessary And Sufficient

This paper deals with robust quasi approximate optimal solutions for a nonsmooth semi-infinite optimization problems with uncertainty data. By virtue of the epigraphs of the conjugates of the constraint functions, we first introduce a robust type closed convex constraint qualification. Then, by using the robust type closed convex constraint qualification and robust optimization technique, we obtain some necessary and sufficient optimality conditions for robust quasi approximate optimal solution and exact optimal solution of this nonsmooth uncertain semi-infinite optimization problem. Moreover, the obtained results in this paper are applied to a nonsmooth uncertain optimization problem with cone constraints.

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An elementary introduction to the geometry of quantum states with pictures

Reviews in Mathematical Physics ◽

10.1142/s0129055x20300010 ◽

2019 ◽

Vol 32 (02) ◽

pp. 2030001 ◽

Cited By ~ 1

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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