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Feb 2013

Volume 54, Issue 2, Articles (02xxxx)

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J. Math. Phys. 54, 021506 (2013); http://dx.doi.org/10.1063/1.4790887 (24 pages)

Gui-Qiang Chen, Vaibhav Kukreja, and Hairong Yuan
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back to top Quantum Information and Computation

Dimensions, lengths, and separability in finite-dimensional quantum systems

Lin Chen and Dragomir Ž. Ðoković

J. Math. Phys. 54, 022201 (2013); http://dx.doi.org/10.1063/1.4790405 (13 pages)

Online Publication Date: 6 February 2013

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Many important sets of normalized states in a multipartite quantum system of finite dimension d, such as the set S of all separable states, are real semialgebraic sets. We compute dimensions of many such sets in several low-dimensional systems. By using dimension arguments, we show that there exist separable states which are not convex combinations of d or less pure product states. For instance, such states exist in bipartite M⊗N systems when (M − 2)(N − 2) > 1. This solves an open problem proposed by DiVincenzo, Terhal and Thapliyal about 12 years ago. We prove that there exist a separable state ρ and a pure product state, whose mixture has smaller length than that of ρ. We show that any real ρ∈S, which is invariant under all partial transpose operations, is a convex sum of real pure product states. In the case of the 2⊗N system, the number r of product states can be taken to be r = rank ρ. We also show that the general multipartite separability problem can be reduced to the case of real states. Regarding the separability problem, we propose two conjectures describing S as a semialgebraic set, which may eventually lead to an analytic solution in some low-dimensional systems such as 2⊗4, 3⊗3, and 2⊗2⊗2.
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03.65.Ud Entanglement and quantum nonlocality (e.g. EPR paradox, Bell's inequalities, GHZ states, etc.)
02.10.Ab Logic and set theory

Bipartite entanglement, spherical actions, and geometry of local unitary orbits

Alan Huckleberry, Marek Kuś, and Adam Sawicki

J. Math. Phys. 54, 022202 (2013); http://dx.doi.org/10.1063/1.4791681 (19 pages)

Online Publication Date: 21 February 2013

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We use the geometry of the moment map to investigate properties of pure entangled states of composite quantum systems. The orbits of equally entangled states are mapped by the moment map onto coadjoint orbits of local transformations (unitary transformations which do not change entanglement). Thus, the geometry of coadjoint orbits provides a partial classification of different entanglement classes. To achieve the full classification, a further study of fibers of the moment map is needed. We show how this can be done effectively in the case of the bipartite entanglement by employing Brion's theorem. In particular, we presented the exact description of the partial symplectic structure of all local orbits for two bosons, fermions, and distinguishable particles putting a special emphasis on the generality of the approach allowing one to consider all three cases in completely parallel manners.
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03.65.Ud Entanglement and quantum nonlocality (e.g. EPR paradox, Bell's inequalities, GHZ states, etc.)
05.30.Fk Fermion systems and electron gas
05.30.Jp Boson systems
02.30.Uu Integral transforms
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