# A Equation in a paper about Dicke states

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1. Apr 30, 2018

### Danny Boy

Can anyone with basic knowledge of Dicke States assist with explaining how we arrive at equation (4) in the paper 'Entanglement detection in the vicinity of arbitrary Dicke states': <Moderator's note: link fixed>

$$\langle J^2_{x} \rangle_{\mu} = \sum_{i_1,i_2} \langle J_{xi_{_1}} \rangle_{\mu} \langle J_{xi_{_2}} \rangle_{\mu} + \sum_{i}\langle ( \Delta J_{xi})^2 \rangle_{\mu}~~~~~~~~~~~~~(4)$$

Any assistance is appreciated.

Last edited by a moderator: May 1, 2018
2. May 1, 2018

### Staff: Mentor

I am not very knowledgeable about Dicke states, but isn't that equation simply a rewriting of
$$\sigma^2 = \langle (\Delta x)^2 \rangle = \langle x^2 \rangle - \langle x \rangle^2$$

3. May 1, 2018

### Danny Boy

@DrClaude Yes I think you are correct. Since $J_{x}= \sum_{i=1}^{k}J_{xi}$ it follows that $$\langle J_{x} \rangle^{2}_{\mu} = \langle \sum_{i=1}^{k}J_{xi} \rangle_{\mu}^{2} = \sum_{i_1, i_2} \langle J_{xi_1}\rangle_{\mu}\langle J_{xi_2} \rangle_{\mu}$$ and $$\langle ( \Delta J_x )^2 \rangle = \sum_{i} \langle ( \Delta J_{xi})^2 \rangle_{\mu}$$ hence $$\sum_{i}\langle ( \Delta J_{xi})^2 \rangle_{\mu} = \langle J^2_{x} \rangle_{\mu} - \sum_{i_1,i_2}\langle J_{xi_{1}}\rangle_{\mu}\langle J_{xi_2}\rangle_{\mu}$$