It turns out that the approximated formula presented above is not valid for all the ES states. A state that violates this formula was analyzed in Ref. [6]. This is the q -qubit W state, which consists of all the basis states that have q − 1 zeros and a single 1: which yields P$_{max}$ ( W ) = [( q − 1) /q ] q − $^{1}$. This is not only different from P q $_{S}$; it is also asymp- totically different, as 10 P q S =    1 | S | | S | ≤ √$_{Q}$ | S | Q | S | > √$_{Q}$ (19) q | W 〉 = 1 √ q q − 1 ∑ m $_{=0}$| 2 $^{m}$〉 . (20) | ϕ 〉 = (√ q − 1 q | 0 〉 + √ 1 q | 1 $_{〉}$) ⊗ q , (21) We note that P q S is a lower bound on P$_{max}$ ( ψ q $_{S}$) . We can also present a crude argument to support the claim that P q S is a good approximation for P$_{max}$ ( ψ q $_{S}$) . For a general product state | ϕ 〉 , consider expanding it to a superposition of basis states. In this expansion, we would like to maximize the number of basis states | k 〉 that have corresponding basis states in | ψ q S 〉 (namely, for which k ∈ S ). The more such basis states, the larger the overlap will be (ignoring, for now, the amplitudes of the states). In the case of a small S , a single basis state is a good guess for | ϕ 〉 , since trying to vary any of the x$_{m}$ ’s away from the edges of their range will add a lot of basis states to the expansion, and most of them will not be members of S , consequently decreasing P . In the case of a large S , a product state is desired with a lot of basis states in its expansion, since there are a lot of members in S , and in that case the complete ES state is hard to beat. Since for the simple W state | S | = q , and q ≤ √$_{Q}$ for a large enough q (practically q ≥ 4 is sufficient), our guess yields one of the basis states as the nearest product state and P q S = 1 /q . However, it turns out that the real nearest product state is 4. A counterexample