A complete set of covariants of the four qubit system

Briand, Emmanuel; Luque, Jean-Gabriel; Thibon, Jean‐Yves

doi:10.1088/0305-4470/36/38/309

Cited by 60 publications

(75 citation statements)

References 18 publications

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“…, d N are represented by the elements of the group GL(d j , C) [4,7]. While some authors related concurrence and three-tangle to hyperdeterminants [8,9], the relevance of determinant 1 SLOCC operations had not been realized and exploited until two seminal papers by Verstraete et al appeared [10,11]. In Ref.…”

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confidence: 99%

Multipartite-entanglement monotones and polynomial invariants

et al. 2012

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We show that a positive homogeneous function that is invariant under determinant 1 stochastic local operations and classical communication (SLOCC) transformations defines an N -qubit entanglement monotone if and only if the homogeneous degree is not larger than four. We then describe a common basis and formalism for the N -tangle and other known invariant polynomials of degree four. This allows us to elucidate the relation of the four-qubit invariants defined by Luque and Thibon [Phys. Rev. A 67, 042303 (2003)] and the reduced two-qubit density matrices of the states under consideration, thus giving a physical interpretation for those invariants. We demonstrate that this is a special case of a completely general law that holds for any multipartite system with bipartitions of equal dimension, e.g., for an even number of qudits.

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confidence: 99%

Multipartite-entanglement monotones and polynomial invariants

et al. 2012

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“…Both the classification of entanglement and its quantification are at a preliminary stage even for distinguishable particles (see however Briand et al, 2003Briand et al, , 2004Dür et al, 2000;Luque and Thibon, 2005;Mandilara et al, 2006;Miyake and Wadati, 2002;Siewert, 2005, 2006;Verstraete et al, 2002 and references therein). We restrict ourselves to those approaches which have been applied so far for the study of condensed matter systems discussed in this review.…”

Section: E Multipartite Entanglement Measuresmentioning

confidence: 99%

Entanglement in many-body systems

Amico¹,

et al. 2008

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The recent interest in aspects common to quantum information and condensed matter has prompted a flory of activity at the border of these disciplines that were far distant untill few years ago. Numerous interesting questions have been addressed so far. Here we review an important part of this field, the properties of the entanglement in many-body systems. We discuss the zero and finite temperature properties of entanglement in interacting spin, fermion and boson model systems. Both bipartite and multipartite entanglement will be considered. In equilibrium we show how entanglement is tightly connected to the characteristics of the phase diagram. The behavior of entanglement can be related, via certain witnesses, to thermodynamic quantities thus offering interesting possibilities for an experimental test. Out of equilibrium we discuss how to generate and manipulate entangled states by means of many-body Hamiltonians.

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“…Classical methods of invariant theory allows one to express the Hilbert series of algebras of covariants as a constant term (see (Briand et al 2003) for an example). Hence, the Hilbert series of unitary and special unitary invariants are…”

Section: Hilbert Seriesmentioning

confidence: 99%

“…In (Briand et al 2003), the algebra of G-covariants for 4 qubits has been investigated, and a complete set of generators has been obtained.…”

Section: Introductionmentioning

confidence: 99%

Unitary invariants of qubit systems

Luque¹,

Thibon²,

Toumazet³

2007

Math. Struct. in Comp. Science

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We give an algorithm allowing to construct bases of local unitary invariants of pure k-qubit states from the knowledge of polynomial covariants of the group of invertible local filtering operations. The simplest invariants obtained in this way are explicited and compared to various known entanglement measures. Complete sets of generators are obtained for up to four qubits, and the structure of the invariant algebras is discussed in detail. IntroductionFrom a mathematical point of view, Quantum Information Theory deals with finite dimensional Hilbert spaces, the state spaces of finite k-partite systems, which have the special formwhere V i is the finite dimensional state space of the ith part (or particle) of the system, most of the time assumed to be a qubit, which means that dim V i = 2. The interesting non-classical behaviors on which the theory is based already occur for two-qubit systems, for the so-called entangled states, those ψ ∈ V 1 ⊗ V 2 which cannot be written in the form v 1 ⊗v 2 . The properties of such states are the basis of the EPR paradox (Einstein et al. 1935), and since its discovery, the entanglement phenomenon has been thoroughly investigated by physicists, cf. (Bell 1966;Clauser et al. 1969;Aspect et al. 1982;Bennett and Wiesner 1992), and more recently by mathematicians, e.g., (Brylinski and Brylinsky 2002;Klyachko 2002; Meyer and Wallach 2002).There is, however, no general agreement on the definition of entanglement for systems with more than two parts. Klyachko has proposed (Klyachko 2002; Klyachko and Shumovsky 2003) to regard as entangled the states which are semi-stable for the action of the group of in-J.-G. Luque, J.Y. Thibon and F. Toumazet 2 vertible local filtering operations, also called SLOCC † ,in the sense of geometric invariant theory, which means those states on which at least one non trivial G-invariant polynomial does not vanish. The point of introducing geometric invariant theory is that this theory provides methods for characterizing such states without explicitly computing the invariants. In order to explore the significance of this property, the invariants have been explicited in the simplest cases (up to 4 qubits, and 3 qutrits, with partial results for 5 qubits, Luque and Thibon 2005;Verstraete et al. 2002)). One would also like to quantify entanglement. The non-locality properties of an entangled state does not change under unitary operations acting independently on each of its sub-systems. The idea of describing entanglement by means of local unitary invariants is explored in (Grassl et al. 1998), see also (Schlienz and Mahler 1996;Schlienz and Mahler 1995). However, except for the simplest systems, there are far too many orbits and a complete classification is out of reach.An intermediate possibility is to look at the G-orbits. The knowledge of the G-invariant polynomials is not sufficient to separate the G-orbits, and in general, one has to look for the covariants in the sense of classical invariant theory.In (Briand et al. 2003), the algebra of G-covariants f...

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A complete set of covariants of the four qubit system

Abstract: International audienc

Cited by 60 publications

References 18 publications

Multipartite-entanglement monotones and polynomial invariants

Multipartite-entanglement monotones and polynomial invariants

Entanglement in many-body systems

Unitary invariants of qubit systems

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