# How to bound the dimension of infinite dimensional Hilbert space?

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?

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.
• 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 $ε<1/N$ – Frédéric Grosshans Jan 10 at 11:50