1. Entanglement Classes and Measures for 4-qubits (as they emerge from “the entanglement description with nilpotent polynomials”) quant-ph/0508234  Aikaterini Mandilara: Lab Aime Cotton, CNRS, Orsay, France. Vladimir Akulin: Lab Aime Cotton, CNRS, Orsay, France Andrei Smilga: Subatech, Nantes, France Lorenza Viola: Dartmouth College, U.S. Eilat, Feb 2006

2. Outline: 1.Writing a quantum state as a nilpotent polynomial. Nilpotential. Tanglemeter. 2.Entanglement classes (sl-orbits)  sl-tanglemeter. 3.Entanglement measures  coefficients of the tanglemeter. 4.Conclusions. Open questions. Su-orbit

3. From quantum states to nilpotential Nilpotential:  Extensive property: + Product states become sum  Dynamics:

4. …. 1 2 3 4 …. n A state/nilpotential of N qubits An orbit of states All the states in the orbit Should have the same Entanglement description SU(2) SU(2) SU(2) SU(2) …… SU(2) 3 parameters each one How many parameters for the orbit marker? Tanglemeter Physical condition: Maximize Method: use feedback in dynamical equations From nilpotential to tanglemeter

5. More general, non-unitary, reversible, local operations LOCC operations= local operations assisted by classical communication SLOCC= stochastic LOCC (Bennet et al, PRA 63, 012307) * If ignore the normalization & divide by det(M): SLOCC described by SL(2,C) Three qubits can be entangled in two inequivalent ways : W. Dur et al,PRA 62, 062314, (2000) Four qubits can be entangled in nine different ways: F. Verstraete et al, PRA 65: 052112 (2002). generators: as Indirect measurement Entanglement Classes = set of states which are equivalent under local SLOCC operations nonselective selective

6. sl-tanglemeter  Entanglement Classes …. 1 2 3 4 …. n A state/nilpotential of N qubits An sl-orbit of states Merging different su-orbits together. SL(2,C) SL(2,C) SL(2,C) SL(2,C).. SL(2,C) 6 parameters each one How many parameters for the orbit marker? sl-orbit marker In general.. Sl-Tanglemeter.. Physical condition:? Method: use feedback in dynamical equations

9. 2 qubits 3 qubits su orbits tanglemeter= su-orbit marker sl orbits (entanglement classes) sl-tanglemeter= sl-orbit marker 4 qubits family of general orbits A. Miyake 03

10. Entanglement Measures SU- Measures SL-Measures In order to compare different su-orbits in the same sl-orbit or different sl-orbits in the same general family of orbits (Give 0 for separable state) and 1 for maximally entangled state of the sl-orbit Invariant under local SU operations and nonincreasing under LOCC transformations Give 1 for the maximally entangled state of the family of the sl-orbits Invariant under local SLOCC operations 2 ways to construct invariants: Polynomial invariants on the amplitudes of the states Invariant coefficients of the tanglemeter But, which su-invariants are decreasing under LOCC? The poly-inv. which are sl-invariants

11. sl-tanglemeter for 4 qubits SU- Measures SL-Measures Polynomial invariants : Tanglemeter’s coefficients : Only to be used in the states Belonging to the states above We start we the normalized state We apply sl-transformations to put in the sl- canonic form. The normalization of the state give us a measure on nonunitarity/distance of the Initial state to the maximal entangled state.

12. Conclusions: With sl-tanglemeter we can at least identify the most general class of entanglement for N qubits. It can be generalized to ensembles of quDits. Investigate a little bit more in the special classes and their applications. We introduced the idea of sl-invariant measures that extends the idea of su-measures. Tanglemeter’s coefficients can serve as invariants for construction of measures.

13. Acknowledgements My advisor in WashU: My advisor in WashU: J. W. Clark J. W. Clark The coworkers on this project: The coworkers on this project: V. M. Akulin V. M. Akulin A. V. Smilga A. V. Smilga Lorenza Viola Lorenza Viola Prof. G. Kurizki Prof. G. Kurizki