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I came across an expression for Entanglement of formation for a mixed state $$E_F(\rho_{AB}) = \text{min}\sum_i p_i S(\rho^i_B)\leq S(\rho_B)$$ where minimum is taken over all the decompositions of the mixed state. What does this mean, 'all the decompositions of the mixed state'?

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A mixed state can be decomposed into a convex combination of pure states. This is always possible, however we do not have any form of uniqueness guarantee, in fact there may be even infinitely many decompositions of a mixed state into pure states. This is the reason we talk about ALL the decompositions of rho. The explicit computation of this minimum is in general hard. – ckrk Dec 29 '13 at 23:44
@ckrk: Thank you. Just to be clear, in the above expression, $\rho^i_B$ is a particular pure state component in the mixed state $\rho_B$? – Cain Dec 30 '13 at 4:37
Yes, that is correct! – ckrk Dec 30 '13 at 15:32
up vote 2 down vote accepted

Every mixed state $\rho$ can decomposed into convex combination of pure states $\rho=\sum_{i}p_i|\psi_i\rangle\langle\psi_i|$, where $p_i$ form a probability vector.

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