content_image/366869.jpg content_image/366870.jpg content_image/366871.jpg content_image/366872.jpg
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. 

## B. Application to periodic states 

Returning to periodic states, the periodic state | ψ q r,l 〉 is an ES state with: 

<img src='content_image/366869.jpg'>

The overlap function in Eq. (13) for this special case will be denoted by 

<img src='content_image/366870.jpg'>

The approximated formula for P$_{max}$ , which is the square of the maximum of the preceding function, becomes 

Approximating A ≈ Q/r , we reach a simpler formula for G q $_{r}$, 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: 

<img src='content_image/366871.jpg'>

<img src='content_image/366872.jpg'>

(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. 

(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. 

