We shall now show evidence to support the validity of this approximation. We first note that in the case of an even period, the least significant bit has the same value for all the basis states in the superposition, which makes the state reducible. When factoring out the last qubit, the rest of the qubits constitute an arithmetic progression themselves, making them a periodic state with half the period. The value of the last bit depends on the parity of the shift: Therefore, the problem of finding the entanglement of a periodic state with an even period can be reduced to the same problem with a state that has one less qubit and half the period. We shall concern ourselves from now on only with states that have odd periods. One important consequence of r being odd is that r is coprime to 2 m for every m . This means that the values of the m th significant bit in all the basis states go through a 2 $^{m}$-long cycle with an equal number of zeros and ones. If the number of cycles is whole then the total numbers of zeros and ones in the m th bit are equal as well. However, in most cases 2 m does not divide A , the cycle is truncated and the equality is only approximate. Furthermore, the values of the n least significant bits make out an arithmetic progression with common difference r in respect to addition modulo 2 $^{n}$, which means they also go through all their possible values in a cyclic manner. Therefore, the values of the different bits are in general uncorrelated. This means that for any basis state | k$_{0}$ 〉 , the number of states in S that have n bits in common with it reduces approximately by half when n is increased by 1. This is because for each qubit m , about half of all the states have the same value for m as | k$_{0}$ 〉 , and the same is true for any subset that is determined by the values of n qubits. We conclude that for a periodic state there is no basis state | k$_{0}$ 〉 that more than any other basis state, has basis states in S within a small Hamming distance from it. This again suggests that the approximated formula for P$_{max}$ we have presented is valid for periodic states, as opposed to the simple W state. An important observation concerning Eq. (27) is that for the descending branch, where the nearest product state | ϕ 〉 is taken to be some basis state k ∈ S , it is at least a local maximum of the overlap function f q $_{r,l}$. To see this, note first that there is no basis state k ′ ∈ S that differs from k on exactly one bit. If there was such a state, the difference k − k ′ would be a power of 2, but this difference for a periodic state must be a multiplicand of r , 13 | ψ q 2 r,l 〉 = | ψ q − 1 r, ⌊ l/ 2 ⌋ 〉 ⊗ | l mod 2 〉 . (29)