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arXiv:quant-ph/0702222

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

Arti Chamoli, Chandra Mohan Bhandari

Published Feb 24, 2007• 1 citation
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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 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.

DOI
10.48550/arxiv.quant-ph/0702222

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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.
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.
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.
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.