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⌈ ( Q − l ) /r ⌉ ≈ Q/r . 

## V. ENTANGLEMENT INDUCED BY THE QFT 

We have derived an approximated formula for the entanglement of periodic states with the hope of better understanding why the QFT does not increase their entanglement. First we note that this is indeed a special property of periodic states, since as illustrated in Fig. 3, the QFT operator in general changes the entanglement of quantum states. Looking again at Eq. (8), we can see that the QFT approximately takes a periodic state with period r and shift l to a periodic state with period Q/r and zero shift, up to relative phases. This was shown more rigorously in Eq. (7), where it was also shown that the relative phases previously ignored take a very special form. To explain this, we define a generalization of the regular ES state, by adding relative phases that depend on a parameter p : 

In this phased ES state the relative phase of each basis state is proportional to the index of the state. An interesting property of this state is that it can be obtained from the corresponding regular ES state by local unitary operations, making it locally equivalent to the ES state: 

<img src='content_image/350267.jpg'>

<img src='content_image/350268.jpg'>

Naturally, this means that both states have the same entanglement. As seen in Eq. (7), the relative phases added to the periodic state | ψ q r,l 〉 follow precisely the same pattern, with p = l + r ( A − 1) / 2 . Therefore, they can be ignored for all entanglement considerations. We are left with two changes the QFT induces in periodic states: removing the shift and changing the period. Interestingly, although the importance of the QFT in Shor’s algorithm is in its canceling of the shift (which enables one to extract the period), this change is irrelevant to the state’s entanglement, as the shift does not appear at all in the approximated formula. Therefore we are left only with the change of period. Looking again at the approximated formula for the entanglement of periodic states given in Eq. (28), we see that the values of the two branches, 1 /r and r/Q , are swapped by the operation r → Q/r . It is now clear why 

