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arXiv:0709.4292 · Physical Review A

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

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

Published Jun 12, 2008• 25 citations
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Abstract

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.

DOI
10.1103/physreva.77.062317

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Groverian and geometric entanglement measures of the nnn-party pure state are expressed by the (n\ensuremath−1)(n\ensuremath{-}1)(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)(n\ensuremath{-}1)(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.
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.
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 thesis includes the original results of our articles [30, 37, 40, 42, 51, 53, 75]. A method is developed to compute analytically entanglement measures of three-qubit pure states. Owing to it closed-form expressions are presented for the geometric measure of entanglement for three-qubit states that are linear combinations of four orthogonal product states. It turns out that the geometric measure for these states has three different expressions depending on the range of definition in parameter space and the Hilbert space of three-qubits consists of three different entangled regions. The states that lie on joint surfaces separating different entangled regions, designated as shared states, have particularly interesting features and are dual quantum channels for the perfect teleportation and superdense coding [42]. A powerful method is developed to compute analytically multipartite entanglement measures. The method uses the duality concept and creates a bijection between highly entangled quantum states and their nearest separable states. The bijection gives explicitly the geometric entanglement measure of arbitrary generalized W states of n qubits [30, 75]. Additionaly, the behavior of the geometric entanglement measure of many-qubit W states is analyzed and an interpolating formula is derived [51]. Generalized Schmidt decomposition of pure three-qubit states has four positive and one complex coefficients. In contrast to the bipartite case, they are not arbitrary and the largest Schmidt coefficient restricts severely other coefficients. It is derived a non-strict inequality between three-qubit Schmidt coefficients, where the largest coefficient defines the least upper bound for the three nondiagonal coefficients or, equivalently, the three nondiagonal coefficients together define the greatest lower bound for the largest coefficient [53].
A method is developed to derive algebraic equations for the geometric measure of entanglement of three-qubit pure states. The equations are derived explicitly and solved in the cases of most interest. These equations allow one to derive analytic expressions of the geometric entanglement measure in a wide range of three-qubit systems, including the general class of W states and states which are symmetric under the permutation of two qubits. The nearest separable states are not necessarily unique, and highly entangled states are surrounded by a one-parametric set of equally distant separable states. A possibility for physical applications of the various three-qubit states to quantum teleportation and superdense coding is suggested from the aspect of entanglement.
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.