Entanglement, local measurements and symmetry

Klyachko, Alexander A.; Shumovsky, A.S.

doi:10.1088/1464-4266/5/3/364

Cited by 16 publications

(44 citation statements)

References 27 publications

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“…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 † ,…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…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-group of local special unitary transformations. We shall also be interested in arbitrary local unitary transformations, which form the group U = G LUT = U (2) k .…”

Section: Introductionmentioning

confidence: 99%

“…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.…”

mentioning

confidence: 99%

“…(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).…”

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confidence: 99%

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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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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

“…(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).…”

mentioning

confidence: 99%

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Unitary invariants of qubit systems

Luque¹,

Thibon²,

Toumazet³

2007

Math. Struct. in Comp. Science

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Entanglement of a Single Spin-1 Object: An Example of Ubiquitous Entanglement

et al. 2007

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Using a single spin-1 object as an example, we discuss a recent approach to quantum entanglement. [A.A. Klyachko and A.S. Shumovsky, J. Phys: Conf. Series 36, 87 (2006), E-print quant-ph/0512213]. The key idea of the approach consists in presetting of basic observables in the very definition of quantum system. Specification of basic observables defines the dynamic symmetry of the system. Entangled states of the system are then interpreted as states with maximal amount of uncertainty of all basic observables. The approach gives purely physical picture of entanglement. In particular, it separates principle physical properties of entanglement from inessential. Within the model example under consideration, we show relativity of entanglement with respect to dynamic symmetry and argue existence of single-particle entanglement. A number of physical examples are considered.

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Entanglement and its operational measure

2006

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An operational representation of concurrence measuring the entanglement of bipartite systems by means of averages of basic observables is discussed. We prove the validity of this representation for bipartite systems with any dimension of a single-party Hilbert space. We show that Wigner-Yanase "skew" information gives a reasonable estimation of the amount of entanglement (in ebits) carried by mixed two-qubit states. ©2006 Springer Science+Business Media, Inc

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Entanglement, local measurements and symmetry

Cited by 16 publications

References 27 publications

Unitary invariants of qubit systems

Unitary invariants of qubit systems

Entanglement of a Single Spin-1 Object: An Example of Ubiquitous Entanglement

Entanglement and its operational measure

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