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arXiv:quant-ph/0609010

Evolution Of Entanglement In Groverian Search Algorithm: Two-Qutrit System

Arti Chamoli, Chandra Mohan Bhandari

Published Sep 2, 2006• 0 citations
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Abstract

Entanglement plays a crucial role in quantum processes particularly those pertaining to quantum information and computation. An analytical expression for entanglement measure defined in terms of success rate of Grover's search algorithm has been obtained for a two-qutrit system and the calculated results agree well with the conventional entropy based measure. Qutrit system are of special interest because the Hilbert space dimensionality (for a given number of qudits-d-dimensional system)is optimal for d=3 and this may be of significance in the enhancement of computing power.

DOI
10.48550/arxiv.quant-ph/0609010

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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
Entanglement plays a crucial role in quantum processes particularly those pertaining to quantum information and computation. An analytical expression for entanglement measure defined in terms of success rate of Grover's search algorithm has been obtained for a two-qutrit system and the calculated results agree well with the conventional entropy based measure. Qutrit system are of special interest because the Hilbert space dimensionality (for a given number of qudits-d-dimensional system)is optimal for d=3 and this may be of significance in the enhancement of computing power.
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.
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.
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.
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.