Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:quant-ph/0609064

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

Arti Chamoli, Chandra Mohan Bhandari

Published Sep 8, 2006• 0 citations
arXivHTMLPDF

Abstract

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.

DOI
10.48550/arxiv.quant-ph/0609064

Related papers

Success Rate and Entanglement Evolution in Search Algorithm

Arti Chamoli, Chandra Mohan Bhandari

Evolution of entanglement with the processing 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. Biham et al [14] have shown that the success probability of search algorithm can be seen as an operational measure of entanglement. Following the same line of thought, analytical expressions for entanglement measure for three and five qubit system have been derived, which reveals that entanglement measure based on Grover's search algorithm is valid for any state with real coefficients for even number of qubits.

arXiv:quant-ph/0702221•Feb 24, 2007•2 citations

Evolution of Entanglement in Groverian Search Algorithm: n-qudit System

Arti Chamoli, Chandra Mohan Bhandari

Entanglement plays a crucial role in quantum processes particularly those pertaining to quantum information and computation. An analytical expression for an entanglement measure defined in terms of success rate of Grover's search algorithm has been obtained for qutrit systems with real coefficients and the calculated results agree well with the conventional entropy based measure. The entanglement measure is further generalised for qudit (d-level) systems with real coefficients.

arXiv:quant-ph/0702222•Feb 24, 2007•1 citation

Analysis of Grover’s quantum search algorithm as a dynamical system

Ofer Biham, Daniel Shapira, Yishai Shimoni

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 ${P}_{\mathrm{s}}$ 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.

arXiv:quant-ph/0307141•Physical Review A•Aug 29, 2003•15 citations

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

Groverian measure of entanglement for mixed states

Daniel Shapira, Yishai Shimoni, Ofer Biham

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 $n$ qubits is obtained by a purification procedure into a pure state of $2n$ 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.

arXiv:quant-ph/0508108•Physical Review A•Apr 5, 2006•32 citations

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

Yishai Shimoni, Daniel Shapira, Ofer Biham

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.

arXiv:quant-ph/0309062•Physical Review A•Jun 2, 2004•47 citations
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.
Evolution of entanglement with the processing 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. Biham et al [14] have shown that the success probability of search algorithm can be seen as an operational measure of entanglement. Following the same line of thought, analytical expressions for entanglement measure for three and five qubit system have been derived, which reveals that entanglement measure based on Grover's search algorithm is valid for any state with real coefficients for even number of qubits.
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, 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.