# POVM construction with little input information

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?

• 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"?
– glS
Jan 19, 2023 at 15:26
• 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.
– R.W
Jan 19, 2023 at 15:32
• 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
– glS
Jan 19, 2023 at 15:41
• 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
– glS
Jan 19, 2023 at 15:42
• 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
– glS
Jan 19, 2023 at 15:51