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arXiv:quant-ph/9810093 · International Journal of Modern Physics C

BUILDING PARTIALLY ENTANGLED STATES WITH GROVER'S AMPLITUDE AMPLIFICATION PROCESS

Hiroo Azuma

Published May 1, 2000• 7 citations
arXivHTMLPDF

Abstract

In the present study, we discuss how to build some partially entangled states of n two-state quantum systems (qubits). The optimal partially entangled state with a high degree of symmetry is considered to be useful for overcoming a shot noise limit of Ramsey spectroscopy under some decoherence. This state is invariant under permutation of any two qubits, and inversion between the ground state |0> and an excited state |1> for each qubit. We show that by using selective phase shifts in certain basis vectors and Grover's inversion about average operations, we can construct this high symmetric entangled state by (polynomial in n) ×2n/2 successive unitary transformations that are applied on two or three qubits.

DOI
10.1142/s0129183100000407

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In the present study, we discuss how to build some partially entangled states of n two-state quantum systems (qubits). The optimal partially entangled state with a high degree of symmetry is considered to be useful for overcoming a shot noise limit of Ramsey spectroscopy under some decoherence. This state is invariant under permutation of any two qubits, and inversion between the ground state |0> and an excited state |1> for each qubit. We show that by using selective phase shifts in certain basis vectors and Grover's inversion about average operations, we can construct this high symmetric entangled state by (polynomial in n) ×2n/2 successive unitary transformations that are applied on two or three qubits.
How to build partially entangled states defined with functions whose all collisions are even are discussed. These states include a partially entangled state with a high degree of symmetry that is invariant under permutation of any two qubits and inversion between the ground state |0> and an excited state |1> for each qubit. An optimized high symmetric partially entangled state is considered to be useful for overcoming a shot noise limit of the Ramsey spectroscopy. We show that using selective rotations and Grover's inversion about average operations, we can construct this high symmetric entangled state by a network of poly(n) x 2^{n/2} quantum basic gates, where n is a number of qubits.
The optimal precision of frequency measurements in the presence of decoherence is discussed. We analyze different preparations of nnn two-level systems as well as different measurement procedures. We show that standard Ramsey spectroscopy on uncorrelated atoms and optimal measurements on maximally entangled states provide the same resolution. The best resolution is achieved using partially entangled preparations with a high degree of symmetry.
Given two arbitrary pure states $ |\\phi>$ and $ |\\psi>$ of qubits or higher level states, we provide arguments in favor of states of the form $ \\frac{1}{\\sqrt{2}}(|\\psi> |\\phi> + i |\\phi> |\\psi>) $ instead of symmetric or anti-symmetric states, as natural candidates for optimally entangled states constructed from these states. We show that such states firstly have on the average a high value of concurrence, secondly can be constructed by a universal unitary operator independent of the input states. We also show that these states are the only ones which can be produced with perfect fidelity by any quantum operation designed for intertwining two pure states with a relative phase. A probabilistic method is proposed for producing any pre-determined relative phase into the combination of any two arbitrary states.
In this work, a variation of the problem originally solved by Verstraete, Audenaert, and De Moor [Phys. Rev. A 64, 012316 (2001)] on what is the maximum entanglement that can be created in a two-qubit system by a global unitary transformation is considered and solved when permutation invariance in the state is imposed. The additional constraint of permutation symmetry appears naturally in the context of bosonic systems or spin states. We also characterise symmetric two-qubit states that remain separable after any global unitary transformation, called symmetric absolutely separable states (SAS), or absolutely classical for spin states. This allows us to determine the maximal radius of a ball of SAS states around the maximally mixed state in the symmetric sector, and the minimal radius of a ball that includes the set of SAS states. For three-qubit systems, a necessary condition for absolute separability of symmetric states is given, which leads us to upper bounds on the ball radii similar to those studied for the two-qubit system.
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.
Abstract In this article, we revisit the question of analysing the efficiencies of partially entangled states in three-qubit classes under real conditions. Our results show some interesting observations regarding the efficiencies and correlations of partially entangled states. Surprisingly, we find that the efficiencies of many three-qubit partially entangled states exceed that of maximally entangled three-qubit states under real noisy conditions and applications of weak measurements. Our analysis, therefore, suggests that the efficiencies of partially entangled states are much more robust to noise than those of maximally entangled states at least for the GHZ (Greenberger–Horne–Zeilinger) class states, for certain protocols; i.e. less correlations in the initially prepared state may also lead to better efficiency and hence one need not always consider starting with a maximally entangled state with maximum correlations between the qubits. For a set of partially entangled states, we find that the efficiency is optimal, independent of the decoherence and state parameters, if the value of weak measurement parameter is very large. For other values of the weak measurement parameter, the robustness of the states depends on the decoherence and state parameters. Moreover, we further show that one can achieve higher efficiencies in a protocol by using non-optimal weak measurement strengths instead of optimal weak measurement strengths.