# Density matrix and wigner function from first and second moments

Let's say I know the first and second moments of position and momentum for all times.

$$\langle x\rangle$$, $$\langle p\rangle$$,$$\langle x^2\rangle$$, $$\langle p^2\rangle$$, $$\langle xp\rangle$$, $$\langle px\rangle$$.

I know the state is Gaussian, so the full state of the system should be fully characterized by above mentioned moments.

How do you reconstruct the density matrix and the Wigner function from them?

• But this is a probability distribution question: How do you specify a bivariate Gaussian given the first and second moments? Talking about Wigner functions and density matrices and states is merely confusing your (simple) question. What does WP say? – Cosmas Zachos Sep 15 '19 at 0:52
• – Cosmas Zachos Sep 15 '19 at 1:00
• Thanks, that help a lot! – Luke Sep 15 '19 at 4:23