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21
Global Entanglement in Multiparticle Systems
 Journal of Mathematical Physics
"... We define a polynomial measure of multiparticle entanglement which is scalable, i.e., which applies to any number of spin1 2 particles. By evaluating it for three particle states, for eigenstates of the one dimensional Heisenberg antiferromagnet and on quantum error correcting code subspaces, we il ..."
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Cited by 20 (3 self)
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We define a polynomial measure of multiparticle entanglement which is scalable, i.e., which applies to any number of spin1 2 particles. By evaluating it for three particle states, for eigenstates of the one dimensional Heisenberg antiferromagnet and on quantum error correcting code subspaces, we illustrate the extent to which it quantifies global entanglement. We also apply it to track the evolution of entanglement during a quantum computation.
Entanglement detection
 Physics Reports
"... How can one prove that a given state is entangled? In this paper we review different methods that have been proposed for entanglement detection. We first explain the basic elements of entanglement theory for two or more particles and then entanglement verification procedures such as Bell inequalitie ..."
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Cited by 14 (0 self)
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How can one prove that a given state is entangled? In this paper we review different methods that have been proposed for entanglement detection. We first explain the basic elements of entanglement theory for two or more particles and then entanglement verification procedures such as Bell inequalities, entanglement witnesses, the determination of nonlinear properties of a quantum state via measurements on several copies, and spin squeezing inequalities. An emphasis is given to the theory and application of entanglement witnesses. We also discuss several experiments, where some of the presented methods have been implemented.
Freudenthal triple classification of threequbit entanglement
, 2009
"... We show that the threequbit entanglement classes: (0) Null, (1) Separable ABC, (2a) Biseparable ABC, (2b) Biseparable BCA, (2c) Biseparable CAB, (3) W and (4) GHZ correspond respectively to ranks 0, 1, 2a, 2b, 2c, 3 and 4 of a Freudenthal triple system defined over the Jordan algebra C ⊕ C ⊕ C ..."
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Cited by 13 (3 self)
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We show that the threequbit entanglement classes: (0) Null, (1) Separable ABC, (2a) Biseparable ABC, (2b) Biseparable BCA, (2c) Biseparable CAB, (3) W and (4) GHZ correspond respectively to ranks 0, 1, 2a, 2b, 2c, 3 and 4 of a Freudenthal triple system defined over the Jordan algebra C ⊕ C ⊕ C. We also compute the corresponding SLOCC orbits.
Lectures on Tensor Network States
"... The most updated version of these notes will be kept on the webpage listed above. Feedback welcome. Other University webpages storing a copy of these notes will not be updated. ..."
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Cited by 5 (4 self)
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The most updated version of these notes will be kept on the webpage listed above. Feedback welcome. Other University webpages storing a copy of these notes will not be updated.
Black Holes, Qubits and Octonions
, 2008
"... We review the recently established relationships between black hole entropy in string theory and the quantum entanglement of qubits and qutrits in quantum information theory. The first example is provided by the measure of the tripartite entanglement of three qubits (Alice, Bob and Charlie), known a ..."
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Cited by 4 (0 self)
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We review the recently established relationships between black hole entropy in string theory and the quantum entanglement of qubits and qutrits in quantum information theory. The first example is provided by the measure of the tripartite entanglement of three qubits (Alice, Bob and Charlie), known as the 3tangle, and the entropy of the 8charge ST U black hole of N = 2 supergravity, both of which are given by the [SL(2)] 3 invariant hyperdeterminant, a quantity first introduced by Cayley in 1845. Moreover the classification of threequbit entanglements is related to the classification of N = 2 supersymmetric ST U black holes. There are further relationships between the attractor mechanism and local distillation protocols and between supersymmetry and the suppression of bit flip errors. At the microscopic level, the black holes are described by intersecting D3branes whose wrapping around the six compact dimensions T 6 provides the stringtheoretic interpretation of the charges and we associate the threequbit basis vectors, ABC〉 (A, B, C = 0 or 1), with the corresponding 8 wrapping cycles. The black hole/qubit correspondence extends to the 56 charge N = 8 black holes and the tripartite entanglement of seven qubits where the measure is provided by Cartan’s E7 ⊃ [SL(2)] 7 invariant. The qubits are naturally described by the seven vertices ABCDEF G of the Fano plane, which provides the multiplication table of the seven imaginary octonions,
2012 Convexity of momentum map, Morse index, and quantum entanglement arXiv:1208.0556
"... We analyze form the topological perspective the space of all SLOCC (Stochastic Local Operations with Classical Communication) classes of pure states for composite quantum systems. We do it for both distinguishable and indistinguishable particles. In general, the topology of this space is rather comp ..."
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We analyze form the topological perspective the space of all SLOCC (Stochastic Local Operations with Classical Communication) classes of pure states for composite quantum systems. We do it for both distinguishable and indistinguishable particles. In general, the topology of this space is rather complicated as it is a nonHausdorff space. Using geometric invariant theory (GIT) and momentum map geometry we propose a way to divide the space of all SLOCC classes into mathematically and physically meaningful families. Each family consists of possibly many ‘asymptotically ’ equivalent SLOCC classes. Moreover, each contains exactly one distinguished SLOCC class on which the total variance (a well defined measure of entanglement) of the state Var[v] attains maximum. We provide an algorithm for finding critical sets of Var[v], which makes use of the convexity of the momentum map and allows classification of such defined families of SLOCC classes. The number of families is in general infinite. We introduce an additional refinement into finitely many groups of families using the recent developments in the momentum map geometry known as Ness stratification. We also discuss how to define it equivalently using the convexity of the momentum map applied to SLOCC classes. Moreover, we note that the Morse index at the critical set of the total variance of state has an interpretation of number of nonSLOCC directions in which entanglement increases and calculate it for several exemplary systems. Finally, we introduce the SLOCCinvariant measure of entanglement as a square root of the total variance of state at the critical point and explain its geometric meaning. 1.
POINCARÉ SERIES FOR LOCAL UNITARY INVARIANTS OF MIXED STATES OF THE QUBITQUTRIT SYSTEM
, 2006
"... Abstract. We consider the mixed states of the bipartite quantum system with the first party a qubit and the second a qutrit. The Hilbert space of this system is the tensor product H = C 2 ⊗ C 3 and the group of local unitary transformations, ignoring the overall phase factor, is the direct product G ..."
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Abstract. We consider the mixed states of the bipartite quantum system with the first party a qubit and the second a qutrit. The Hilbert space of this system is the tensor product H = C 2 ⊗ C 3 and the group of local unitary transformations, ignoring the overall phase factor, is the direct product G = SU(2) ×SU(3). Let P be the algebra of real polynomial functions on the affine space of all hermitian operators of trace 1 on H. The polynomials f ∈ P which are invariant under G form the subalgebra P G ⊆ P. We compute the simply graded Poincaré series of this subalgebra and construct several low degree invariants. 1.
June 2001 quantph/0108104 GLOBAL ENTANGLEMENT
"... We define a polynomial measure of multiparticle entanglement which is scalable, i.e., which applies to any number of spin particles. By evaluating it for three particle states, for eigenstates of the one dimensional Heisenberg antiferromagnet and on quantum error correcting code subspaces, we il ..."
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We define a polynomial measure of multiparticle entanglement which is scalable, i.e., which applies to any number of spin particles. By evaluating it for three particle states, for eigenstates of the one dimensional Heisenberg antiferromagnet and on quantum error correcting code subspaces, we illustrate the extent to which it quantifies global entanglement. We also apply it to track the evolution of entanglement during a quantum computation.
Multipartite generalisation of the Schmidt
, 2000
"... We find a canonical form for pure states of a general multipartite system, in which the constraints on the coordinates (with respect to a factorisable orthonormal basis) are simply that certain ones vanish and certain others are real. For identical particles they are invariant under permutations of ..."
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We find a canonical form for pure states of a general multipartite system, in which the constraints on the coordinates (with respect to a factorisable orthonormal basis) are simply that certain ones vanish and certain others are real. For identical particles they are invariant under permutations of the particles. As an application, we find the dimension of the generic local equivalence class.