Stable Hilbert series as related to the measurement of quantum entanglement

Hero, Michael W.; Willenbring, Jeb F.

doi:10.1016/j.disc.2009.06.021

Cited by 21 publications

(38 citation statements)

References 5 publications

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“…(xxvi) Since the convex roof extensions of semialgebraic functions are known to be semialgebraic functions [119,120], it can be useful to use LU-invariant homogeneous polynomials [121][122][123][124][125] for the role of entanglement measures in Level I of the construction, which leads to semialgebraic functions in Level II. This holds, in particular, if one sets out from Tsallis entropy for integer q ≥ 2 in the construction (82)-(88a)-(99d)- (108).…”

Section: B On the Quantification Of Multipartite Entanglementmentioning

confidence: 99%

Multipartite entanglement measures

2015

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The main concern of this paper is how to define proper measures of multipartite entanglement for mixed quantum states. Since the structure of partial separability and multipartite entanglement is getting complicated if the number of subsystems exceeds two, one cannot expect the existence of an ultimate scalar entanglement measure, which grasps even a small part of the rich hierarchical structure of multipartite entanglement, and some higher-order structure characterizing that is needed. In this paper we make some steps in this direction.First, we reveal the lattice-theoretic structure of the partial separability classification, introduced earlier [Sz. Szalay and Z. Kökényesi, Phys. Rev. A 86, 032341 (2012)]. It turns out that, mathematically, the structure of the entanglement classes is the up-set lattice of the structure of the different kinds of partial separability, which is the down-set lattice of the lattice of the partitions of the subsystems. It also turns out that, physically, this structure is related to the LOCC convertibility: If a state from a class can be mapped into another one, then that class can be found higher in the hierarchy.Second, we introduce the notion of multipartite monotonicity, expressing that a given set of entanglement monotones, while measuring the different kinds of entanglement, shows also the same hierarchical structure as the entanglement classes. Then we construct such hierarchies of entanglement measures, and we propose a physically well-motivated one, being the direct multipartite generalization of the entanglement of formation based on the entanglement entropy, motivated by the notion of statistical distinguishability. The multipartite monotonicity shown by this set of measures motivates us to consider the measures to be the different manifestations of some "unified" notion of entanglement.

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Section: B On the Quantification Of Multipartite Entanglementmentioning

confidence: 99%

Multipartite entanglement measures

2015

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“…For small numbers of qubits (up to four), finite generating sets are explicitly known [40,47] (although there was a misprint in [47] that was corrected in [8]). Work has been done for higher numbers of qubits [15,16,33]. In Theorem 5.6, we classify all invariants for this action for any number of qubits.…”

Section: Organization Of the Papermentioning

confidence: 99%

A Complete Set of Invariants for LU-Equivalence of Density Operators

Turner¹,

Morton²

2017

SIGMA

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Abstract. We show that two density operators of mixed quantum states are in the same local unitary orbit if and only if they agree on polynomial invariants in a certain Noetherian ring for which degree bounds are known in the literature. This implicitly gives a finite complete set of invariants for local unitary equivalence. This is done by showing that local unitary equivalence of density operators is equivalent to local GL equivalence and then using techniques from algebraic geometry and geometric invariant theory. We also classify the SLOCC polynomial invariants and give a degree bound for generators of the invariant ring in the case of n-qubit pure states. Of course it is well known that polynomial invariants are not a complete set of invariants for SLOCC.

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“…. = λ k = (1) the groups reduce to G m = S k m and H m = S m , and we have that d m equals the number of orbits of S k m under S m × S m acting via left and right multiplication, or equivalently, the number of orbits of S k−1 m under S m acting by simultaneous conjugation [10].…”

Section: Hilbert Series Of the Algebra Of Lu-invariantsmentioning

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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Stable Hilbert series as related to the measurement of quantum entanglement

Cited by 21 publications

References 5 publications

Multipartite entanglement measures

Multipartite entanglement measures

A Complete Set of Invariants for LU-Equivalence of Density Operators

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

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