Connections of geometric measure of entanglement of pure symmetric states to quantum state estimation

Journal Article · Sat Jan 15 00:00:00 EST 2011 · Physical Review. A

DOI:https://doi.org/10.1103/PHYSREVA.83.012305· OSTI ID:21529040

Lin, Chen ^[1]; Huangjun, Zhu ^[1]; Wei, Tzu-Chieh ^[2]

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1. Centre for Quantum Technologies, National University of Singapore, Singapore 117543 (Singapore)
2. Department of Physics and Astronomy, University of British Columbia, Vancouver, British Columbia, V6T 1Z1 (Canada)

We study the geometric measure of entanglement (GM) of pure symmetric states related to rank 1 positive-operator-valued measures (POVMs) and establish a general connection with quantum state estimation theory, especially the maximum likelihood principle. Based on this connection, we provide a method for computing the GM of these states and demonstrate its additivity property under certain conditions. In particular, we prove the additivity of the GM of pure symmetric multiqubit states whose Majorana points under Majorana representation are distributed within a half sphere, including all pure symmetric three-qubit states. We then introduce a family of symmetric states that are generated from mutually unbiased bases and derive an analytical formula for their GM. These states include Dicke states as special cases, which have already been realized in experiments. We also derive the GM of symmetric states generated from symmetric informationally complete POVMs (SIC POVMs) and use it to characterize all inequivalent SIC POVMs in three-dimensional Hilbert space that are covariant with respect to the Heisenberg-Weyl group. Finally, we describe an experimental scheme for creating the symmetric multiqubit states studied in this article and a possible scheme for measuring the permanence of the related Gram matrix.

• OSTI ID: 21529040
• Journal Information: Physical Review. A, Vol. 83, Issue 1; Other Information: DOI: 10.1103/PhysRevA.83.012305; (c) 2011 American Institute of Physics; ISSN 1050-2947
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
HILBERT SPACE
MAJORANA THEORY
MATRICES
MAXIMUM-LIKELIHOOD FIT
MEASURE THEORY
QUANTUM ENTANGLEMENT
QUANTUM STATES
QUBITS
SYMMETRY
THREE-DIMENSIONAL CALCULATIONS
BANACH SPACE
INFORMATION
MATHEMATICAL SOLUTIONS
MATHEMATICAL SPACE
MATHEMATICS
NUMERICAL SOLUTION
QUANTUM INFORMATION
SPACE