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Dragomir Z. Djokovic

Publications and source records attributed to Dragomir Z. Djokovic.

62 records · Page 4Linked to original sources
arXiv2007

Normal Forms and Tensor Ranks of Pure States of Four Qubits

We examine the SLOCC classification of the (non-normalized) pure states of four qubits obtained by F. Verstraete et al. The rigorous proofs of their basic results are provided and necessary corrections implemented. We use Invariant Theory to solve the problem of equivalence of pure states under SLOCC transformations of determinant 1 and qubit permutations. As a byproduct, we produce a new set of generators for the invariants of the Weyl group of type F_4. We complete the determination of the tensor ranks of 4-qubit pure states initiated by J.-L. Brylinski. As a result we obtain a simple algorithm for computing these ranks. We obtain also a very simple classification of pure states of rank at most 3.

quant-ph↗
arXiv2007

Dimensions of generic local orbits of multipartite quantum systems

We consider the action of the group of local unitary transformations, U(m) x U(n), on the set of (mixed) states W of the bipartite m x n quantum system. We prove that the generic U(m) x U(n)--orbits in W have dimension m^2+n^2-2. This problem was mentioned (and left open) by Kus and Zyczkowski in their paper Geometry of entangled states. The proof can be extended to the case of arbitrary finite-dimensional multipartite quantum systems.

quant-ph↗
arXiv2006

Poincare series of some pure and mixed trace algebras of two generic matrices

We work over a field K of characteristic zero. The Poincare series for the algebra C_{n,2} of GL_n-invariants and the algebra T_{n,2} of GL_n-concomitants of two generic n x n matrices x and y are presented for n less than or equal 6. Both simply graded and bigraded cases are included. The cases for n at most 4 were known previously. If n=5 or 6, we show that C_{n,2} has no bigraded system of parameters. For the algebra C_{4,2} and C_{5,2} we construct a minimal set of generators and give an application to Specht's theorem on unitary similarity of two complex matrices. Five conjectures are proposed concerning the numerators and denominators of various Poincare series mentioned above. Some heuristic formulas and open problems are stated.

math.AC↗
arXiv2006

Multigraded Poincare series for mixed states of two qubits and the boundary of the set of separable states

Let M be the set of mixed states and S the set of separable states of the two-qubit system, and G = SU(2) x SU(2) the group of local unitary transformations (ignoring the overall phase factor). We compute the multigraded Poincare series for the algebra of G-invariant polynomial functions on the affine space of all Hermitian operators of trace 1. We check that this series is consistent with the list of invariants computed by Makhlin. By using the recent result of Augusiak et al., we show that the boundary of S decomposes naturally into two pieces. We prove that the part of this boundary which is contained in the relative interior of M is a smooth manifold.

quant-ph↗
arXiv2006

Poincare series for local unitary invariants of mixed states of the qubit-qutrit system

We consider the mixed states of the bipartite quantum system with the first party a qubit and the second a qutrit. The group of local unitary transformations of the system, ignoring the overall phase factor, is the direct product G of SU(2) and SU(3). We compute the simply graded Poincare series of the algebra of G-invariant polynomial functions on the set of mixed states of the system, and construct several low degree invariants.

quant-ph↗