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

Groverian Measure in Multi-Qubit Systems

Eylee Jung, Mi-Ra Hwang, Hung-Soo Kim, Minsoo Kim, DaeKil Park, Jin-Woo Son, Sayatnova Tamaryan

Published Sep 27, 2007• 1 citation
OpenAlex

Abstract

How much information (n-1)-party reduced state possesses about n-party pure state quantum correlation? It is shown that any (n-1)-party reduced state uniquely determines the Groverian and Geometric entanglement measures of the original n-party pure state. The theorem has two important consequences. First, if two pure n-qubit states have (n-1)-qubit reduced states equivalent under local unitary(LU) transformations, then they have equal Groverian and Geometric entanglement measures. Second, both measures have an upper bound for pure states. However, this upper bound is reached only for two qubit systems.

Related papers

Reduced state uniquely defines the Groverian measure of the original pure state

Eylee Jung, Mi-Ra Hwang, Hung-Soo Kim +4 more

Groverian and geometric entanglement measures of the $n$-party pure state are expressed by the $(n\ensuremath{-}1)$-party reduced state density operator directly. This main theorem derives several important consequences. First, if two pure $n$-qudit states have reduced states of $(n\ensuremath{-}1)$-qudits, which are equivalent under local unitary transformations, then they have equal Groverian and geometric entanglement measures. Second, both measures have an upper bound for pure states. However, this upper bound is reached only for two-qubit systems. Third, it converts effectively the nonlinear eigenvalue problem for the three-qubit Groverian measure into linear eigenvalue equations. Some typical solutions of these linear equations are written explicitly and the features of the general solution are discussed in detail.

arXiv:0709.4292•Physical Review A•Jun 12, 2008•25 citations

Groverian entanglement measure of pure quantum states with arbitrary partitions

Yishai Shimoni, Ofer Biham

The Groverian entanglement measure of pure quantum states of $n$ qubits is generalized to the case in which the qubits are divided into any $p\ensuremath{\leqslant}n$ 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.

arXiv:quant-ph/0702179•Physical Review A•Feb 8, 2007•9 citations

Three-qubit Groverian measure

Eylee Jung, Mi-Ra Hwang, D. Park +2 more

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.

arXiv:0803.3311•Quantum Information and Computation•Nov 1, 2008•8 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

Entanglement bound for multipartite pure states based on local measurements

Li-zhen Jiang, Xiaoyu Chen, Tian-Yu Ye

An entanglement bound based on local measurements is introduced for multipartite pure states. It is the upper bound of the geometric measure and the relative entropy of entanglement. It is the lower bound of the minimal-measurement entropy. For pure bipartite states, the bound is equal to the entanglement entropy. The bound is applied to pure tripartite qubit states and the exact tripartite relative entropy of entanglement is obtained for a wide class of states.

arXiv:1104.3280•Physical Review A•Oct 5, 2011•2 citations
How much information (n-1)-party reduced state possesses about n-party pure state quantum correlation? It is shown that any (n-1)-party reduced state uniquely determines the Groverian and Geometric entanglement measures of the original n-party pure state. The theorem has two important consequences. First, if two pure n-qubit states have (n-1)-qubit reduced states equivalent under local unitary(LU) transformations, then they have equal Groverian and Geometric entanglement measures. Second, both measures have an upper bound for pure states. However, this upper bound is reached only for two qubit systems.
Groverian and geometric entanglement measures of the nnn-party pure state are expressed by the $(n\ensuremath{-}1)$-party reduced state density operator directly. This main theorem derives several important consequences. First, if two pure nnn-qudit states have reduced states of $(n\ensuremath{-}1)$-qudits, which are equivalent under local unitary transformations, then they have equal Groverian and geometric entanglement measures. Second, both measures have an upper bound for pure states. However, this upper bound is reached only for two-qubit systems. Third, it converts effectively the nonlinear eigenvalue problem for the three-qubit Groverian measure into linear eigenvalue equations. Some typical solutions of these linear equations are written explicitly and the features of the general solution are discussed in detail.
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.
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.