I'm learning about different possibilities to measure entanglement right now, and I'm struggling with the geometric measure. My question is probably quite trivial, but I can just not comprehend how the following example was calculated.
So, what I know is that if I have an $n$-partice pure state, in general: $$\lvert{\psi}\rangle = \sum_{p_1,p_2,\dots p_n} C_{p_1 \dots p_n}\lvert{p_n}^{(1)}\rangle \otimes \ \lvert{p_2}^{(2)}\rangle \otimes \dots \lvert{p_n}^{(n)}\rangle $$ Then I have to find the closest seperable pure state $$\lvert\phi\rangle.$$ The measure is then $$ \Lambda = \max_{\phi} \lvert\langle\phi|\psi\rangle\rvert$$
In the given example, there is the GHZ-State: $$ \lvert \text{GHZ}\rangle= \frac{1}{\sqrt{3}} ( \lvert001\rangle + \lvert010\rangle + \lvert100\rangle) $$ And the solution should be $\Lambda=2/3$. But I do not know, how this is calculated. The result implies (so I think) that the relevant $\lvert\phi\rangle$, which creates the maximum value, is a linear combination of 3 states of which 2 are also in the GHZ-state. But a state like: $$ \lvert\phi\rangle = \frac{1}{\sqrt{3}} ( \lvert001\rangle + \lvert010\rangle + \lvert000\rangle ) $$ Is not separable in factors of the 3 separate Hilbert-spaces. What do I miss? Or which thing did I misunderstood?
Edit: relevant link to slides : http://insti.physics.sunysb.edu/conf/simons-qcomputation/talks/wei.pdf Definition on Slide 10, Example on Slide 14)
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for dirac noation (see the help page on notation) as this will make it much easier to read your question. – Sebastian Riese May 31 '15 at 21:10