I refer to this paper but reproduce a simplified version of their argument. Apologies if I have misrepresented the argument of the paper!

Alice has a classical description of a quantum state $\rho$. Alice and Bob both agree on a two outcome observable $M$. Now, the goal for Bob is to come up with a classical description of a state $\sigma$ that gives the right measurement statistics i.e. $Tr(M\rho) \approx Tr(M\sigma)$.

The way this is done is that Bob has a guess state, say the maximally mixed state, $I$. Alice then tells him the value of $Tr(M\rho)$. Bob then measures the maximally mixed state repeatedly (or he runs a classical simulation of this with many copies of the maximally mixed state) and "postselects" the outcomes where he obtains $Tr(M\rho) \approx Tr(MI)$. In this way, he obtains a state that reproduces the measurement statistics of the true state.

What is the meaning of postselection in this context? How does Bob go from $I$ to $\sigma$ in this procedure?


In principle, Bob here just has to guess the $2\times 2$ matrix $\sigma$. If he starts with any parametric state $\sigma(\alpha,\beta)$ with $\alpha,\beta\in\mathbb{C}$ and measures the outcome Tr$(M\sigma)$ with the post measurement state $\sigma '=M\rho M^\dagger/\text{Tr}(M\rho)$, he receives a number and he has to tune $\alpha,\beta$ to come close to the value Tr$(M\rho)$. This tuning is done on the basis of postselection by which it is implied that he selects the state $\sigma$ close to $\rho$ on the constraint of minimising the relative entropy: \begin{equation} \rho\text{ln}\rho-\rho\text{ln}\sigma \end{equation} so as to move closer to the state $\rho$, which can be found out in terms of $\alpha,\beta$. In such problems usually one starts with a parametric state in one variable and optimises over it.


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