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Title: Groverian entanglement measure of pure quantum states with arbitrary partitions

Abstract

The Groverian entanglement measure of pure quantum states of n qubits is generalized to the case in which the qubits are divided into any p{<=}n parties. The entanglement between these parties is evaluated numerically using an efficient parametrization. To demonstrate this measure we apply it to symmetric states such as the Greenberg-Horne-Zeiliner state and the W state. Interestingly, this measure is equivalent to an entanglement measure introduced earlier [H. Barnum and N. Linden, J. Phys. A 34, 6787 (2001)], using different considerations.

Authors:
;  [1]
  1. Racah Institute of Physics, The Hebrew University, Jerusalem 91904 (Israel)
Publication Date:
OSTI Identifier:
20982075
Resource Type:
Journal Article
Resource Relation:
Journal Name: Physical Review. A; Journal Volume: 75; Journal Issue: 2; Other Information: DOI: 10.1103/PhysRevA.75.022308; (c) 2007 The American Physical Society; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ENERGY LEVELS; QUANTUM COMPUTERS; QUANTUM ENTANGLEMENT; QUANTUM INFORMATION; QUANTUM MECHANICS; QUBITS

Citation Formats

Shimoni, Yishai, and Biham, Ofer. Groverian entanglement measure of pure quantum states with arbitrary partitions. United States: N. p., 2007. Web. doi:10.1103/PHYSREVA.75.022308.
Shimoni, Yishai, & Biham, Ofer. Groverian entanglement measure of pure quantum states with arbitrary partitions. United States. doi:10.1103/PHYSREVA.75.022308.
Shimoni, Yishai, and Biham, Ofer. Thu . "Groverian entanglement measure of pure quantum states with arbitrary partitions". United States. doi:10.1103/PHYSREVA.75.022308.
@article{osti_20982075,
title = {Groverian entanglement measure of pure quantum states with arbitrary partitions},
author = {Shimoni, Yishai and Biham, Ofer},
abstractNote = {The Groverian entanglement measure of pure quantum states of n qubits is generalized to the case in which the qubits are divided into any p{<=}n parties. The entanglement between these parties is evaluated numerically using an efficient parametrization. To demonstrate this measure we apply it to symmetric states such as the Greenberg-Horne-Zeiliner state and the W state. Interestingly, this measure is equivalent to an entanglement measure introduced earlier [H. Barnum and N. Linden, J. Phys. A 34, 6787 (2001)], using different considerations.},
doi = {10.1103/PHYSREVA.75.022308},
journal = {Physical Review. A},
number = 2,
volume = 75,
place = {United States},
year = {Thu Feb 15 00:00:00 EST 2007},
month = {Thu Feb 15 00:00:00 EST 2007}
}
  • The Groverian entanglement measure G({psi}) is applied to characterize pure quantum states vertical bar {psi}> of multiple qubits. This is an operational measure of entanglement in the sense that it quantifies the utility of the state vertical bar {psi}> as an initial state for the search algorithm. A convenient parametrization is presented, which allows us to calculate the Groverian measure analytically for certain states of high symmetry. A numerical procedure is used in order to calculate it for arbitrary pure states of multiple qubits. Using the Groverian measure to evaluate the entanglement produced by quantum algorithms may provide useful insightmore » into the role of entanglement in making quantum algorithms powerful. Here we calculate G({psi}) for the intermediate states generated during the evolution of Grover's algorithm for various initial states and for different sets of marked states. It is shown that Grover's iterations generate highly entangled states in intermediate stages of the quantum search process, even if the initial state and the target state are product states.« less
  • 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 generatedmore » 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.« less
  • We present a multipartite entanglement measure for N-qudit pure states, using the norm of the correlation tensor which occurs in the Bloch representation of the state. We compute this measure for an important class of N-qutrit pure states, namely, general GHZ states. We prove that this measure possesses all the essential and many desirable properties expected of a good entanglement measure, including monotonicity. We also discuss the feasibility of the experimental evaluation of this measure for an N-qutrit system.
  • We provide methods for computing the geometric measure of entanglement for two families of pure states with both experimental and theoretical interests: symmetric multiqubit states with non-negative amplitudes in the Dicke basis and symmetric three-qubit states. In addition, we study the geometric measure of pure three-qubit states systematically in virtue of a canonical form of their two-qubit reduced states and derive analytical formulas for a three-parameter family of three-qubit states. Based on this result, we further show that the W state is the maximally entangled three-qubit state with respect to the geometric measure.
  • The degree to which a pure quantum state is entangled can be characterized by the distance or angle to the nearest unentangled state. This geometric measure of entanglement, already present in a number of settings [A. Shimony, Ann. NY. Acad. Sci. 755, 675 (1995); H. Barnum and N. Linden, J. Phys. A: Math. Gen. 34, 6787 (2001)], is explored for bipartite and multipartite pure and mixed states. The measure is determined analytically for arbitrary two-qubit mixed states and for generalized Werner and isotropic states, and is also applied to certain multipartite mixed states. In particular, a detailed analysis is givenmore » for arbitrary mixtures of three-qubit Greenberger-Horne-Zeilinger, W, and inverted-W states. Along the way, we point out connections of the geometric measure of entanglement with entanglement witnesses and with the Hartree approximation method.« less