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For the GHZ state, the nearest product state is any one of the two basis states that comprise it: 

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and for both states: P$_{max}$ (GHZ) = 1 / 2 . The second special case is the 2 n -qubit balanced generalized W state, denoted by | φ ( n, 2 n ) 〉 . It consists of all the basis states of 2 n qubits that have n zeros and n ones: 

In this case the nearest product state is the complete ES state 

and P$_{max}$ is given by 

## 3. The approximated formula 

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Given a general ES state | ψ q S 〉 , our aim is to find the nearest product states | ϕ 〉 . The results presented previously motivate us to examine two types of product states as candidate states: 

(i) | ϕ 〉 = | k 〉 for some k ∈ S , which means that each x$_{m}$ is either 0 or 1 . This yields P q S = 1 / | S | , where P q S is the estimated value of P . 

(ii) | ϕ 〉 = | η 〉 (the complete ES state), which means that x$_{m}$ = 1 / 2 for all m ’s. This yields P q S = | S | /Q . 

Clearly, the first guess is better for a small S , and the second guess is better for a large S . They become equally good for | S | $_{=}$√$_{Q}$$_{. Thus, we can combine them into one improved}$ guess: 

