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### Entanglement Classes and Measures for 4 Qubits Using Nilpotent Polynomials ###

This paper explores the characterization of quantum states of four qubits through nilpotent polynomial descriptions, introducing concepts like the sl-tanglemeter to classify entanglement classes and develop measures for quantum states. We delve into the dynamics of nilpotent states, provide extensive properties of quantum entanglement, and propose methods for using feedback in dynamical equations. The framework allows for the identification of generalized classes of entanglement, enabling comparisons between different SU-orbits and SL-orbits within the context of local operations and classical communication (LOCC). ###

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### Entanglement Classes and Measures for 4 Qubits Using Nilpotent Polynomials ###

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

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

  4. From nilpotential to tanglemeter 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

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

  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? In general.. Sl-Tanglemeter.. sl-orbit marker Physical condition:? Method: use feedback in dynamical equations

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

  8. Entanglement Measures In order to compare different su-orbits in the same sl-orbit or different sl-orbits in the same general family of orbits SU- Measures SL-Measures • (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 Polynomial invariants on the amplitudes of the states 2 ways to construct invariants: Invariant coefficients of the tanglemeter But, which su-invariants are decreasing under LOCC? The poly-inv. which are sl-invariants

  9. sl-tanglemeter for 4 qubits SU- Measures SL-Measures Polynomial invariants: Only to be used in the states Belonging to the states above Tanglemeter’s coefficients: • 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.

  10. 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.

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

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