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arXiv:quant-ph/0309062 · Physical Review A

Characterization of pure quantum states of multiple qubits using the Groverian entanglement measure

Yishai Shimoni, Daniel Shapira, Ofer Biham

Published Jun 2, 2004• 47 citations
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Abstract

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.

DOI
10.1103/physreva.69.062303

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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"

Preeti Parashar, Swapan Rana

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

•Jan 1, 2009•0 citations
The Groverian entanglement measure G(\ensuremathψ)G(\ensuremath{\psi})G(\ensuremathψ) is applied to characterize pure quantum states \ensuremath∣\ensuremathψ⟩\ensuremath{\mid}\ensuremath{\psi}⟩\ensuremath∣\ensuremathψ⟩ of multiple qubits. This is an operational measure of entanglement in the sense that it quantifies the utility of the state \ensuremath∣\ensuremathψ⟩\ensuremath{\mid}\ensuremath{\psi}⟩\ensuremath∣\ensuremathψ⟩ 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(\ensuremathψ)G(\ensuremath{\psi})G(\ensuremathψ) 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.
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.
Grover's quantum search algorithm is analyzed for the case in which the initial state is an arbitrary pure quantum state $|\ensuremath{\varphi}〉$ of n qubits. It is shown that the optimal time to perform the measurement is independent of $|\ensuremath{\varphi}〉,$ namely, it is identical to the optimal time in the original algorithm in which $|\ensuremath{\varphi}〉=|0〉,$ with the same number of marked states, r. The probability of success Ps{P}_{\mathrm{s}}Ps​ is obtained in terms of the amplitudes of the state $|\ensuremath{\varphi}〉$ and is shown to be independent of r. A class of states, which includes fixed points and cycles of the Grover iteration operator, is identified. The relevance of these results in the context of using the success probability as an entanglement measure is discussed. In particular, the Groverian entanglement measure, previously limited to a single marked state, is generalized to the case of several marked states.
The Groverian entanglement measure of pure quantum states of nnn qubits is generalized to the case in which the qubits are divided into any $p\ensuremath{\leqslant}n$ 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.
Evolution of entanglement with the proceeding of quantum algorithms affects the outcome of the algorithm. Particularly, the performance of Grover's search algorithm gets worsened if the initial state of the algorithm is an entangled one. The success probability of search can be seen as an operational measure of entanglement. This paper demonstrates an entanglement measure based on the performance of Grover's search algorithm for three and five qubit systems. We also show that although the overall pattern shows growth of entanglement, its rise to a maximum and then consequent decay, the presence of local fluctuation within each iterative step is likely.
Quantum Algorithms have long captured the imagination of computer scientists and physicists primarily because of the speed up achieved by them over their classical counterparts using principles of quantum mechanics. Entanglement is believed to be the primary phenomena behind this speed up. However their precise role in quantum algorithms is yet unclear. In this article, we explore the nature of entanglement in the Grover's search algorithm. This algorithm enables searching of elements from an unstructured database quadratically faster than the best known classical algorithm. Geometric measure of entanglement has been used to quantify and analyse entanglement across iterations of the algorithm. We reveal how the entanglement varies with increase in the number of qubits and also with the number of marked or solution states. Numerically, it is seen that the behaviour of the maximum value of entanglement is monotonous with the number of qubits. Also, for a given value of the number of qubits, a change in the marked states alters the amount of entanglement. The amount of entanglement in the final state of the algorithm has been shown to depend solely on the nature of the marked states. Explicit analytical expressions are given showing the variation of entanglement with the number of iterations and the global maximum value of entanglement attained across all iterations of the algorithm.