the maximal Hamming distance of 2 n . In analogy, the GHZ state has a maximal Hamming distance between its two basis states. In both states there is no specific choice of k$_{0}$ for which all the basis states are within a small Hamming distance from it. This observation suggests that in general, for ES states that include basis states with large Hamming distances from each other, taking a small variation around some basis state does not aid in finding the nearest product state. Returning to periodic states, the periodic state | ψ q r,l 〉 is an ES state with: The overlap function in Eq. (13) for this special case will be denoted by The approximated formula for P$_{max}$ , which is the square of the maximum of the preceding function, becomes B. Application to periodic states S = { l, l + r, . . . , l + ( A − 1) r } ; A = | S | = ⌈ Q − l r ⌉ . (25) f q $_{r,l}$( x$_{1}$, . . . , x$_{q}$ ) = 1 √$_{A}$ ∑ j 1 ...j$_{q}$ ∈ S q ∏ m =1 C m $_{j$_{m}$}$. (26) P q r,l =    1 A A ≤ √$_{Q}$ A Q A > √$_{Q}$ (27) q G q r =    − ln 1 r r < √$_{Q}$ − ln r Q r ≥ √$_{Q}$ (28) Approximating A ≈ Q/r , we reach a simpler formula for G $_{r}$, which is an approximation of the logarithmic Groverian entanglement of periodic states G ( ψ q $_{r,l}$) : Looking at G q r as a function of r (Fig. 2) we see that it consists of two branches: (ii) A descending branch , which starts at the maximum and descents back to 0 as r reaches Q . This branch covers the states with large periods, which have a small number of basis states, and for which a basis state is the presumed nearest product state. (i) An ascending branch , which starts at 0 when r = 1 , and rises to a maximum at r = √$_{Q}$$_{. This branch covers the states with small periods, which have a large number of}$ basis states, and for which the complete ES state is the presumed nearest product state. 12