The measurement of quantum entanglement and enumeration of graph coverings

Hero, Michael W.; Willenbring, Jeb F.; Williams, Lauren Kelly

doi:10.48550/arxiv.0911.0222

Cited by 4 publications

(16 citation statements)

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“…(xiii) An interesting question is as to whether all pure-state indicator functions can be obtained without products of local entropies, but using only linear combinations of them. (xiv) We note that there are also recent attempts to study the general structure of LU invariant homogeneous polynomials [59][60][61][62][63]. Looking for convex roof extensions in the language of LU-invariant polynomials would be an interesting direction of research.…”

Section: Summary and Remarksmentioning

confidence: 99%

Partial separability revisited: Necessary and sufficient criteria

Szalay

Kökényesi

2012

Phys. Rev. A

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We extend the classification of mixed states of quantum systems composed of arbitrary number of subsystems of arbitrary dimensions. This extended classification is complete in the sense of partial separability and gives 1+18+1 partial separability classes in the tripartite case contrary to a former 1+8+1. Then we give necessary and sufficient criteria for these classes, which make it possible to determine to which class a mixed state belongs. These criteria are given by convex roof extensions of functions defined on pure states. In the special case of three-qubit systems, we define a different set of such functions with the help of the Freudenthal triple system approach of three-qubit entanglement.Comment: v3: 22 pages, 5 tables, 1 figure, minor corrections (typos), clarification in the Introduction. Accepted in Phys. Rev. A. Comments are welcom

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Section: Summary and Remarksmentioning

confidence: 99%

Partial separability revisited: Necessary and sufficient criteria

Szalay

Kökényesi

2012

Phys. Rev. A

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“…Following the strategy of refs. [11,2], in this section we introduce certain combinatorial objects which can be used to conveniently label a basis of I 1,((l)) .…”

Section: Integer Stochastic Matrices and Regular Bipartite Graphsmentioning

confidence: 99%

“…= l k = 1 special case corresponds to distinguishable particles, and the resulting graphs can be identified (after merging pairs of vertices along one of the colours) with the graph coverings of ref. [11].…”

Section: Systems Of Different Kinds Of Indistinguishable Particlesmentioning

confidence: 99%

On the algebra of local unitary invariants of pure and mixed quantum states

Vrana¹

2011

J. Phys. A: Math. Theor.

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We study the structure of the inverse limit of the graded algebras of local unitary invariant polynomials using its Hilbert series. For k subsystems, we conjecture that the inverse limit is a free algebra and the number of algebraically independent generators with homogenous degree 2m equals the number of conjugacy classes of index m subgroups in a free group on k − 1 generators.Similarly, we conjecture that the inverse limit in the case of k-partite mixed state invariants is free and the number of algebraically independent generators with homogenous degree m equals the number of conjugacy classes of index m subgroups in a free group on k generators. The two conjectures are shown to be equivalent.To illustrate the equivalence, using the representation theory of the unitary groups, we obtain all invariants in the m = 2 graded parts and express them in a simple form both in the case of mixed and pure states. The transformation between the two forms is also derived. Analogous invariants of higher degree are also introduced.

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“…The notion of entanglement of a composite quantum system is known to be invariant under unitary transformations on the subsystems, so the investigation of local unitary (LU) invariants is a natural way of studying quantum entanglement. In this paper, we give illustrations for the general results of Hero et al [1,2] and Vrana [3,4] on LU-invariant polynomials for pure quantum states. In [4], it has been pointed out that the inverse limit (in the local dimensions) of algebras of LU-invariant polynomials of finite-dimensional k-partite quantum systems is free, and an algebraically independent generating set for that has been given.…”

Section: Introductionmentioning

confidence: 98%