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Quantum Information and Computation

Three-qubit groverian measure

JungEylee, HwangMi-Ra, ParkDaekil, TamaryanLevon, TamaryanSayatnova

Published Nov 1, 2008• 0 citations
Publisher

Abstract

The Groverian measures are analytically computed in various types of three-qubit states. The finalresults are also expressed in terms of local-unitary invariant quantities in each type. This fact r...
DOI
10.5555/2016985.2016988

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Three-qubit Groverian measure

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The Groverian measures are analytically computed in various types of three-qubit states. The final results are also expressed in terms of local-unitary invariant quantities in each type. This fact reflects the manifest local-unitary invariance of the Groverian measure. It is also shown that the analytical expressions for various types have correct limits to other types. For some types (type 4 and type 5) we failed to compute the analytical expression of the Groverian measure in this paper. However, from the consideration of local-unitary invariants we have shown that the Groverian measure in type 4 should be independent of the phase factor $\varphi$, which appear in the three-qubit state $|\psi \rangle$. This fact with geometric interpretation on the Groverian measure may enable us to derive the analytical expressions for general arbitrary three-qubit states in near future.

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Comment on "Groverian entanglement measure and evolution of entanglement in search algorithm for n(= 3, 5)-qubit systems with real coefficients"

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We point out that the main results—the analytic expressions for the Groverian measure of entanglement, in the above mentioned paper are erroneous. The technical mistake of the paper is discussed. It is shown by an explicit example that the formula for calculating the Groverian measure yields G(|ψ� ) = 0f or some entangled states. Keyword Groverian measure of entanglement

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We present a classification of three-qubit states based in their three-qubit and reduced two-qubit entanglements. For pure states these criteria can be easily implemented, and the different types can be related with sets of equivalence classes under Local Unitary operations. For mixed states characterization of full tripartite entanglement is not yet solved in general; some partial results will be presented here. I.

•Jul 12, 2007•0 citations
The Groverian measures are analytically computed in various types of three-qubit states. The final results are also expressed in terms of local-unitary invariant quantities in each type. This fact reflects the manifest local-unitary invariance of the Groverian measure. It is also shown that the analytical expressions for various types have correct limits to other types. For some types (type 4 and type 5) we failed to compute the analytical expression of the Groverian measure in this paper. However, from the consideration of local-unitary invariants we have shown that the Groverian measure in type 4 should be independent of the phase factor φ\varphiφ, which appear in the three-qubit state ∣ψ⟩|\psi \rangle∣ψ⟩. This fact with geometric interpretation on the Groverian measure may enable us to derive the analytical expressions for general arbitrary three-qubit states in near future.
The Groverian entanglement measure, introduced earlier for pure quantum states of multiple qubits [O. Biham, M.A. Nielsen, and T. Osborne, Phys. Rev. A 65, 062312 (2002)], is generalized to the case of mixed states. The Groverian measure of a mixed state of nnn qubits is obtained by a purification procedure into a pure state of 2n2n2n qubits, followed by an optimization process, before the resulting state is fed into Grover's search algorithm. It is expressed in terms of the maximal success probability of the algorithm and in this sense provides an operational measure of entanglement.
The Groverian entanglement measure $G(\ensuremath{\psi})$ is applied to characterize pure quantum states $\ensuremath{\mid}\ensuremath{\psi}⟩$ of multiple qubits. This is an operational measure of entanglement in the sense that it quantifies the utility of the state $\ensuremath{\mid}\ensuremath{\psi}⟩$ 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 insight into the role of entanglement in making quantum algorithms powerful. Here we calculate $G(\ensuremath{\psi})$ 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.