Comment on “ Groverian Entanglement Measure and Evolution of Entanglement in Search Algorithm for n ( = 3 , 5 ) - Qubit Systems with Real Coefficients ” ( Volume 6 ,

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〉) = 0 for some entangled states. 1 What is this communication about? In their paper “Groverian Entanglement Measure and Evolution of Entanglement in Search Algorithm for n(= 3, 5)-Qubit Systems with Real Coefficients” [1], Arti Chamoli and C. M. Bhandari have given an explicit formula for calculating the ‘Groverian Measure’ of entaglement G(|ψ〉) in terms of the state coefficients. The derivation of the results involves some maximization process which is very difficult to solve analytically and we will show that unfortunately the authors have missed some crucial technical points which has led to incorrect results. 1.1 The result gives G(|ψ〉) = 0 for some entangled states. Let us consider the 3-qubit generalized Greenberger-Horne-Zeilinger (GGHZ) state given by |ψGGHZ〉 = a000|000〉 + a111|111〉, a2000 + a2111 = 1. Then using the formula provided in the paper, we get Pmax(|ψGGHZ〉) = 1 ⇒ G(|ψGGHZ〉) = 0, which contradicts the fact that |ψGGHZ〉 is an (genuinely) entangled state. Indeed it is well known [2] that Pmax(|ψGGHZ〉) = max { a2000, a 2 111 } . (1) Similarly, considering the 5-qubit GGHZ state, the result leads to a contradiction. Physics and Applied Mathematics Unit Indian Statistical Institute 203 B T Road Kolkata 700 108 India E-mail: [email protected], swapan [email protected]