A Characterization of Global Entanglement

We define a set of 2n−1−1 entanglement monotones for n qubits and give a single measure of entanglement in terms of these. This measure is zero except on globally entangled (fully inseparable) states. This measure is compared to the Meyer–Wallach measure for two, three, and four qubits. We determine the four-qubit state, symmetric under exchange of qubit labels, which maximizes this measure. It is also shown how the elementary monotones may be computed as a function of observable quantities. We compute the magnitude of our measure for the ground state of the four-qubit superconducting experimental system investigated in [M. Grajcar et al., Phys. Rev. Lett. 96, 047006 (2006)], and thus confirm the presence of global entanglement in the ground state.

[1]  Goong Chen,et al.  Mathematics of Quantum Computation , 2002 .

[2]  Timothy F. Havel.  The Real Density Matrix , 2002, Quantum Inf. Process..

[3]  A. J. Scott.  Multipartite entanglement, quantum-error-correcting codes, and entangling power of quantum evolutions , 2003, quant-ph/0310137.

[4]  E. Farhi,et al.  A Quantum Adiabatic Evolution Algorithm Applied to Random Instances of an NP-Complete Problem , 2001, Science.

[5]  W. Wootters.  Entanglement of Formation of an Arbitrary State of Two Qubits , 1997, quant-ph/9709029.

[6]  G. Vidal.  On the characterization of entanglement , 1998 .

[7]  G. Vidal.  Efficient classical simulation of slightly entangled quantum computations. , 2003, Physical review letters.

[8]  A. Sudbery,et al.  Multipartite generalization of the Schmidt decomposition , 2000, quant-ph/0006125.

[9]  Method for direct observation of coherent quantum oscillations in a superconducting phase qubit , 2002, cond-mat/0208143.

[10]  Gilles Brassard,et al.  Quantum Cryptography , 2005, Encyclopedia of Cryptography and Security.

[11]  D. Meyer,et al.  Global entanglement in multiparticle systems , 2001, quant-ph/0108104.

[12]  Albert Einstein,et al.  Can Quantum-Mechanical Description of Physical Reality Be Considered Complete? , 1935 .

[13]  He-Shan Song,et al.  Global entanglement for multipartite quantum states , 2006, quant-ph/0603038.

[14]  Thierry Paul,et al.  Quantum computation and quantum information , 2007, Mathematical Structures in Computer Science.

[15]  P. Grangier,et al.  Experimental Tests of Realistic Local Theories via Bell's Theorem , 1981 .

[16]  A. Sudbery,et al.  Non-local properties of multi-particle density matrices , 1998, quant-ph/9801076.

[17]  A. Osterloh,et al.  Constructing N-qubit entanglement monotones from antilinear operators (4 pages) , 2004, quant-ph/0410102.

[18]  Hans-J. Briegel,et al.  Local invariants for multi-partite entangled states allowing for a simple entanglement criterion , 2004, Quantum Inf. Comput..

[19]  J. Cirac,et al.  Three qubits can be entangled in two inequivalent ways , 2000, quant-ph/0005115.

[20]  A. Osterloh,et al.  ENTANGLEMENT MONOTONES AND MAXIMALLY ENTANGLED STATES IN MULTIPARTITE QUBIT SYSTEMS , 2005, quant-ph/0506073.

[21]  H. Meyer,et al.  Four-qubit device with mixed couplings. , 2005, Physical review letters.

[22]  J. Bell.  On the Problem of Hidden Variables in Quantum Mechanics , 1966 .

[23]  Nolan R. Wallach.  The Hilbert Series of Measures of Entanglement for 4 Qubits , 2005 .

[24]  Christian Kurtsiefer,et al.  Experimental detection of multipartite entanglement using witness operators. , 2004, Physical review letters.