Narrow Results

The momentum and position observables in an $n$-mode boson Fock space $\mathrm{\Gamma }({ℂ}^n)$ have the whole real line $ℝ$ as their spectrum. But the total number operator $N$ has a discrete spectrum ${\mathbb{Z}}_{+}=\{0,1,2,\dots \}$. An $n$-mode Gaussian state in $\mathrm{\Gamma }({ℂ}^n)$ is completely determined by the mean values of momentum and position observables and their covariance matrix which together constitute a family of $n(2n+3)$ real parameters. Starting with $N$ and its unitary conjugates by the Weyl displacement operators and operators from a representation of the symplectic group $\mathrm{\text{Sp}}(2n)$ in $\mathrm{\Gamma }({ℂ}^n)$, we construct $n(2n+3)$ observables with spectrum ${\mathbb{Z}}_{+}$ but whose expectation values in a Gaussian state determine all its mean and covariance parameters. Thus measurements of discrete-valued observables enable the tomography of the underlying Gaussian state and it can be done by using five one-mode and four two-mode Gaussian symplectic gates in single and pair mode wires of $\mathrm{\Gamma }({ℂ}^n)=\mathrm{\Gamma }{(ℂ)}^{\otimes n}$. Thus the tomography protocol admits a simple description in a language similar to circuits in quantum computation theory. Such a Gaussian tomography applied to outputs of a Gaussian channel with coherent input states permit a tomography of the channel parameters. However, in our procedure the number of counting measurements exceeds the number of channel parameters slightly. Presently, it is not clear whether a more efficient method exists for reducing this tomographic complexity.

As a byproduct of our approach an elementary derivation of the probability generating function of $N$ in a Gaussian state is given. In many cases the distribution turns out to be infinitely divisible and its underlying Lévy measure can be obtained. However, we are unable to derive the exact distribution in all cases. Whether this property of infinite divisibility holds in general is left as an open problem.