The Hilbert Series of Measures of Entanglement for 4 Qubits

C. Witte, M. Trucks

Masanao Ozawa

W. A. Majewski, Joint work with M. Ohya T. Matsuoka

Jean-Gabriel Luque, Jean‐Yves Thibon

1. For mathematical point of view, Quantum Information Theory deals with finite dimensional Hilbert spaces, the state spaces of finite k-bipartite systems. In the case of pure k-bipartite systems, this space has the special form 2. H=V1 ⊗ ... ⊗ Vk, 3. where Vi denotes a finite dimensional state space of the ith particle of the system (most of the time, one assumes that it is a qubit, i.e. dim Vi = 2). The interesting non-classical behavior already occurs for two-qubit systems, for the so-called entangled states: the states which cannot be written in the form v1 ⊗ v2. There is no general agreement on the definition of entanglement for system with more than two parts. Klyachko has proposed to regard as entangled the states which are semi-stable for the action of the SLOCC group (G=SL(V1) × ... × SL(Vk)) in the sense of geometric invariant theory. Geometric invariant theory provides 4. methods for characterizing such states without explicitly computing the invariants. In order to test the significance of this property, we have explicitly computed the invariants in the simplest cases. 5. The non-locality properties of an entangled states does not change under unitary operations acting independently on each of its subsystem. The idea to construct entanglement measure by means of unitary invariant have been explored for example by Grassl et al. (1998), Schlienz and Mahler (1996). We have proved in 2006, that such invariants can be deduced from SLOCC-covariants. The knowledge of all the covariants and hence, the classification of the SLOCC orbits seems to be relevant in this context. However, except for simplest systems there are far too many orbits and the complete classification is out of reach. 6. 7. The aim of the series of four lectures given by Jean-Gabriel Luque and Jean-Yves Thibon is to explain a subproblem: how to compute the Hilbert series of the algebraic invariants for the pure k-qubit systems. 8. 9. In the two first lectures, Jean-Gabriel Luque will give a brief account on the symmetric functions and the characters of the symmetric groups. The first two talks will be meant for an audience of beginner in the area of symmetric functions. In the 10. two remaining lectures, Jean-Yves Thibon will explore the links between symmetric functions and Kronecker product (Lecture 3) in order to explain how to compute the Hilbert series (Lecture 4).

Pinilla, P. A., Jagdish R. Luthra

Based on the geometry of entangled three and two qubit states, we present the connection between the entanglement measure of the three-qubit state defined using the last Hopf fibration and the entanglement measures known as two- and three-tangle. Moreover, the generalization of the geometric representation of four qubit state and a potential entanglement measure is studied using sedenions for the simplification of the Hilbert space S^31 of the four qubit system. An entanglement measure is proposed and the degree of entanglement is calculated for specific states. The difficulties of a possible generalization are discussed.

João Luzeilton de Oliveira, David Sena Oliveira, Rubens Viana Ramos