# How is postselection used in quantum tomography?

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.