58 research outputs found
Singularity of type arising from four qubit systems
An intriguing correspondence between four-qubit systems and simple
singularity of type is established. We first consider an algebraic
variety of separable states within the projective Hilbert space
. Then, cutting with a specific
hyperplane , we prove that the -hypersurface, defined from the section
, has an isolated singularity of type ; it is also shown
that this is the "worst-possible" isolated singularity one can obtain by this
construction. Moreover, it is demonstrated that this correspondence admits a
dual version by proving that the equation of the dual variety of , which is
nothing but the Cayley hyperdeterminant of type ,
can be expressed in terms of the SLOCC invariant polynomials as the
discriminant of the miniversal deformation of the -singularity.Comment: 20 pages, 5 table
Entanglement of four-qubit systems: a geometric atlas with polynomial compass II (the tame world)
We propose a new approach to the geometry of the four-qubit entanglement
classes depending on parameters. More precisely, we use invariant theory and
algebraic geometry to describe various stratifications of the Hilbert space by
SLOCC invariant algebraic varieties. The normal forms of the four-qubit
classification of Verstraete {\em et al.} are interpreted as dense subsets of
components of the dual variety of the set of separable states and an algorithm
based on the invariants/covariants of the four-qubit quantum states is proposed
to identify a state with a SLOCC equivalent normal form (up to qubits
permutation).Comment: 49 pages, 16 figure
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