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Lets say I have a density matrix $\rho$ encoded into a physical system. Lets say I can have access to as many copy of $\rho$ as I want. I perform hmodyne detection on the copies in phase and out of phase and based on the measurement result always lies in a circle nemely $$P^2+X^2\leq r$$ where $P$ and $X$ are real values and correspond to measurement results out of phase and in phase respectively. Based on this measurment results what can I say about the dim of hilbert space corresponding to $\rho$? Is it possible to bound the dimention?

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Basically, the average value of $P^2+X^2$ is the average value of the energy, hence of $4n+2$ (with the suitable normalization). Therefore, what you describe is known ans an energy test in the literature. Basically, if your state is essentially restricted to a superposition of the Fock states of $O$ to $d-1$ photons, we cans say that it is restricted into a subspace of dimension $d$.

The details of the energy test vary, and there is still ongoing research work to find the most efficient one. One recent example is in PRL 118 200501 / arXiv:1701.03393, by Anthony Leverrier which does not assume a product state and use heterodyne detection. Basically, if your state has a non-negligible support of Fock states of $d$ photons or more, it will have a non-negligible probability to have a value of $P^2+X^2$ to be higher than $r$ and to be detected by the energy test.

One way to compute this probability would be to use the explicit expression of the Fock states using Hilbert polynomials, but I don’t know a nice and simple expression from the top of my head.

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  • $\begingroup$ Thanks for your answer! I am a little confused about this quantum de finneti approach. Each time that we make a measurement the quantum state collapse and basically we lose some information about the quantum state. How can we justify that by performing some measurement the fidelity between resultant state and the original state is reasonably high? For instance, look at this paper: arXiv:0809.2243. Basically the idea of the paper is to perform measurements on some part of the system and bound the dim of Hilbert space but how can we justify the collapse of quantum density operator? $\endgroup$ – Heisenberg Jan 8 at 19:21
  • $\begingroup$ And what if we have two modes? how things work in that case? $\endgroup$ – Heisenberg Jan 8 at 20:07
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    $\begingroup$ For two or more modes, things work essentially the same way, but with a worse scaling (exponential in the number of modes) $\endgroup$ – Frédéric Grosshans Jan 10 at 11:42
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    $\begingroup$ About the de Finnetti approach: as in all verification problem, we can never guarantee the we weren’t especially unlucky. All we can do is gurantee that a “bad” state has a low probability to pass our test. In our case, it would be something like a state with a support higher than $η$ out a given subspace of dimension $d$ has a probability lower than $ε=f(r, η, N)$ to pass the test. In De Finetti test, this is ensured by symmetrizing the state (or perform a symmetric test) such that the optimal way to cheat is to send a symmetric state anyway. But we can never have $ε<1/N$ $\endgroup$ – Frédéric Grosshans Jan 10 at 11:50

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