Quantum entanglement and fixed-point bifurcations

Hines, Andrew P.; McKenzie, Ross H.; Milburn, G. J.

doi:10.1103/physreva.71.042303

Cited by 74 publications

(92 citation statements)

References 38 publications

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“…A maximum at the critical point was found, which is consistent with a theoretical conjecture in [209]. Furthermore, the von Neumann entropy of the ground state scales logarithmically with the block size L of the bipartite decomposition [208].…”

Section: Lipkin-meshkov-glick Modelsupporting

confidence: 75%

Classification of Entanglement in Symmetric States

Aulbach

2012

Int. J. Quantum Inform.

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Quantum states that are symmetric with respect to permutations of their subsystems appear in a wide range of physical settings, and they have a variety of promising applications in quantum information science. In this thesis, the entanglement of symmetric multipartite states is categorized, with a particular focus on the pure multi-qubit case and the geometric measure of entanglement. An essential tool for this analysis is the Majorana representation, a generalization of the single-qubit Bloch sphere representation, which allows for a unique representation of symmetric n-qubit states by n points on the surface of a sphere. Here this representation is employed to search for the maximally entangled symmetric states of up to 12 qubits in terms of the geometric measure, and an intuitive visual understanding of the upper bound on the maximal symmetric entanglement is given. Furthermore, it will be seen that the Majorana representation facilitates the characterization of entanglement equivalence classes such as stochastic local operations and classical communication (SLOCC) and the degeneracy configuration (DC). It is found that SLOCC operations between symmetric states can be described by the Möbius transformations of complex analysis, which allows for a clear visualization of the SLOCC freedoms and facilitates the understanding of SLOCC invariants and equivalence classes. In particular, explicit forms of representative states for all symmetric SLOCC classes of up to five qubits are derived. Well-known entanglement classification schemes such as the four qubit entanglement families or polynomial invariants are reviewed in the light of the results gathered here, which leads to sometimes surprising connections. Some interesting links and applications of the Majorana representation to related fields of mathematics and physics are also discussed.

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Section: Lipkin-meshkov-glick Modelsupporting

confidence: 75%

Classification of Entanglement in Symmetric States

Aulbach

2012

Int. J. Quantum Inform.

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“…See Figure 2b. Irrespective of the nature of the bifurcation, it has been observed in the classical analysis [21,22] that fixed points can be used to identify quantum phase transitions. This model therefore becomes a promising candidate to study.…”

Section: Classical Analysismentioning

confidence: 99%

Two-site Bose-Hubbard model with nonlinear tunneling: Classical and quantum analysis

et al. 2017

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The extended Bose-Hubbard model for a double-well potential with atom-pair tunneling is studied. Starting with a classical analysis we determine the existence of three different quantum phases: selftrapping, phase-locking and Josephson states. From this analysis we built the parameter space of quantum phase transitions between degenerate and non-degenerate ground states driven by the atom-pair tunneling. Considering only the repulsive case, we confirm the phase transition by the measure of the energy gap between the ground state and the first excited state. We study the structure of the solutions of the Bethe ansatz equations for a small number of particles. An inspection of the roots for the ground state suggests a relationship to the physical properties of the system. By studying the energy gap we find that the profile of the roots of the Bethe ansatz equations is related to a quantum phase transition.

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“…When S a = S b , there is only one bifurcation point at J ⊥ = J z between the fixed points (1) and (2); there is also only one bifurcation point at J ⊥ = −J z between the fixed points (1) and (3). Each of these bifurcation points corresponds to a maximally entangled quantum ground state.…”

Section: Bifurcation and Entanglementmentioning

confidence: 98%

Binary mixture of pseudo-spin-12Bose gases with interspecies spin exchange: From classical fixed points and ground states to quantum ground states

Shi

2011

Phys. Rev. A

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We consider the effective spin Hamiltonian describing a mixture of two species of pseudo-spin- 2Bose gases with interspecies spin exchange. First we analyze the stability of the fixed points of the corresponding classical dynamics, of which the signature is found in quantum dynamics with a disentangled initial state. Focusing on the case without an external potential, we find all the ground states by taking into account quantum fluctuations around the classical ground state in each parameter regime. The nature of entanglement and its relation with classical bifurcation is investigated. When the total spins of the two species are unequal, the maximal entanglement at the parameter point of classical bifurcation is possessed by the excited state corresponding to the classical fixed point which bifurcates, rather than by the ground state.

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Quantum entanglement and fixed-point bifurcations

Cited by 74 publications

References 38 publications

Classification of Entanglement in Symmetric States

Classification of Entanglement in Symmetric States

Two-site Bose-Hubbard model with nonlinear tunneling: Classical and quantum analysis

Binary mixture of pseudo-spin-12Bose gases with interspecies spin exchange: From classical fixed points and ground states to quantum ground states

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