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Let $E$ be part of a POVM $M = \{E,I-E\}$. Suppose that I know that $E = f(\rho_1, \rho_2)$. Suppose also that those two states are provided but we only know their type (dimension) and we also know $f$. For instance, $f$ might be a nonlinear functional.

Is it known when is possible to construct $E$?

I'll give an example for clarity; suppose that $E = \rho^M/\text{Tr}(\rho^M)$ and that I want to learn $\text{Tr}(E\sigma)$. Without knowing $\rho$, and if I understood correctly this paper https://journals.aps.org/prx/pdf/10.1103/PhysRevX.11.041036 it is possible to obtain this estimate by using a symmetric operator (see Eq.8 and Fig.1).

Are there other similar constructions but for other forms $f$ and other input states $\rho$'s instead of a single one?

Another concrete example that highlights the difficulty: Let $C(\rho_1,\rho_2) := -(i/2)[\rho_1,\rho_2]$. The operator $C$ is Hermitian, but can it be constructed the associated measurement to the observable $C$ without performing full tomography of $\rho_1$ and $\rho_2$, or better, without knowing if $C$ is zero or not?

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  • $\begingroup$ I'm not sure I understand. What are $\rho_1$ and $\rho_2$ here? Also, if I understand, you are given a function $f$, and want to "construct $E$". But if you know $f$, and presumably you know $\rho_1,\rho_2$, can't you just apply $f$ to those to get $E$? Also, how is "those two states are provided" compatible with "we only know their type"? $\endgroup$
    – glS
    Jan 19, 2023 at 15:26
  • $\begingroup$ It is in the spirit that we do now have full knowledge of the two rho1 and rho2. They are states. For instance, I am assuming that full tomography was not performed but nevertheless you still want to do an operation on them: the SWAP test is an example of an operation that extract information of two given states that have a known 'type' but that one does not need to have complete knowledge of. $\endgroup$
    – R.W
    Jan 19, 2023 at 15:32
  • $\begingroup$ right. So the context is essentially that you have $\rho_1\otimes \rho_2$ as input to some circuit/channel and you want to estimate some (probably not tomographically complete) property. But you want to do it with a POVM that is dependent on the states themselves it seems. So this seems a Fisher-like kind of scenario, where you have a POVM which is built to be optimal with respect to a specific input state. But I still don't understand $f$. Is it like an estimator, that gives an estimate of a property for each measurement output, or is it just a function you use to build the POVM $\endgroup$
    – glS
    Jan 19, 2023 at 15:41
  • $\begingroup$ I mean, if you don't know $\rho_i$ a priori, you clearly can't compute $f(\rho_1,\rho_2)$. So then you're looking for a way to estimate $f(\rho_1,\rho_2)$ from incomplete knowledge of the states? As in, you perform some initial measurement to get an estimate of the states, and then adaptively change the POVM to this $E$? I don't really understand. This doesn't seem to me like what's discussed in the paper. They're saying they measure $\rho^{\otimes M}$ to estimate ${\rm tr}(O\rho^M)$. The corresponding POVM is not $\rho^M$ here. It should be something like the eigenbasis of the symmetric op $\endgroup$
    – glS
    Jan 19, 2023 at 15:42
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    $\begingroup$ crucially, in such schemes, the POVM $E$ does not depend on the states. This seems like the direct generalization of estimating ${\rm tr}(\rho^2)$ by estimating the expectation value of the SWAP on the state $\rho\otimes\rho$, that is, ${\rm tr}({\rm SWAP}(\rho\otimes\rho))$. Practically, this amounts to doing a projective measurement on $\rho\otimes\rho$ in the eigenbasis of the SWAP plus some post-processing $\endgroup$
    – glS
    Jan 19, 2023 at 15:51

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