The moduli space of three-qutrit states

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

doi:10.1063/1.1809255

Cited by 62 publications

(84 citation statements)

References 28 publications

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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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Section: E Multipartite Entanglement Measuresmentioning

confidence: 99%

Entanglement in many-body systems

Amico¹,

et al. 2008

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“…See Sec. VI of [25] and Sec. 4.1 of [26] for other occurences of the group G 648 in relation to the geometry of the 27 lines on a smooth cubic surface.…”

Section: The Two-qutrit Systemmentioning

confidence: 99%

Pauli graphs when the Hilbert space dimension contains a square: Why the Dedekind psi function?

Planat¹

2010

J. Phys. A: Math. Theor.

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Abstract. We study the commutation relations within the Pauli groups built on all decompositions of a given Hilbert space dimension q, containing a square, into its factors. Illustrative low dimensional examples are the quartit (q = 4) and two-qubit (q = 22 ) systems, the octit (q = 8), qubit/quartit (q = 2 × 4) and three-qubit (q = 2 3 ) systems, and so on. In the single qudit case, e.g. q = 4, 8, 12, . . ., one defines a bijection between the σ(q) maximal commuting sets [with σ[q) the sum of divisors of q] of Pauli observables and the maximal submodules of the modular ring Z 2 q , that arrange into the projective line P 1 (Z q ) and a independent set of size σ(q)−ψ(q) [with ψ(q) the Dedekind psi function]. In the multiple qudit case, e.g. q = 2 2 , 2 3 , 3 2 , . . ., the Pauli graphs rely on symplectic polar spaces such as the generalized quadrangles GQ(2, 2) (if q = 22 ) and GQ(3, 3) (if q = 3 2 ). More precisely, in dimension p n (p a prime) of the Hilbert space, the observables of the Pauli group (modulo the center) are seen as the elements of the 2n-dimensional vector space over the field F p . In this space, one makes use of the commutator to define a symplectic polar space W 2n−1 (p) of cardinality σ(p 2n−1 ), that encodes the maximal commuting sets of the Pauli group by its totally isotropic subspaces. Building blocks of W 2n−1 (p) are punctured polar spaces (i.e. a observable and all maximum cliques passing to it are removed) of size given by the Dedekind psi function ψ(p 2n−1 ). For multiple qudit mixtures (e.g. qubit/quartit, qubit/octit and so on), one finds multiple copies of polar spaces, ponctured polar spaces, hypercube geometries and other intricate structures. Such structures play a role in the science of quantum information.

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“…Since this include the 4-qubit system distributed over 3 parties, which is the case in e.g., entanglement swapping, our results will clarify what kinds of essentially different multipartite entanglement there exist in this situation, and give better understanding for multi-party LOCC protocols. More specifically, we will address the stochastic LOCC (SLOCC) classification of entanglement [1,4,5,6,7,8,9,10,11,12,13], which is a coarsegrained classification under LOCC. Let us consider the single copy of a multipartite pure state |Ψ on the Hilbert space H = C k1 ⊗ · · · ⊗ C k l (precisely, in abuse of the notation, we would denote a ray on its complex projective space CP …”

Section: Introductionmentioning

confidence: 99%

Multipartite entanglement in2×2×nquantum systems

Miyake

Verstraete

2004

Phys. Rev. A

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We classify multipartite entangled states in the Hilbert space H = C 2 ⊗ C 2 ⊗ C n (n ≥ 4), for example the 4-qubit system distributed over 3 parties, under local filtering operations. We show that there exist nine essentially different classes of states, giving rise to a five-graded partially ordered structure, including the celebrated Greenberger-Horne-Zeilinger (GHZ) and W classes of three qubits. In particular, all 2×2×n-states can be deterministically prepared from one maximally entangled state, and some applications like entanglement swapping are discussed.

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The moduli space of three-qutrit states

Cited by 62 publications

References 28 publications

Entanglement in many-body systems

Entanglement in many-body systems

Pauli graphs when the Hilbert space dimension contains a square: Why the Dedekind psi function?

Multipartite entanglement in2×2×nquantum systems

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