# Is the set of density operators invariant under the induced action of the unitary group?

Show that the set of density operators is invariant under the induced action of $$U(H)$$ on $$End(H)$$.

I know that a density operator must be positive and have a trace equal to one. But I don't know how to prove the invariance under the unitary group.

I assume I need to show that the set of density operators consists of the same elements even after applying $$U(H)$$ to $$End(H)$$, or am I misunderstanding the question entirely?

I'm not sure what $$End(H)$$ is but here is a proof that unitaries take density operators to density operators. Let $$X$$ be a positive semidefinite matrix with unit trace. We have to show that for a unitary $$U$$, $$U^\dagger XU$$ is also positive semidefinite with unit trace. Using the cyclicity of trace and $$UU^\dagger = I$$
$$Tr(U^\dagger X U) = Tr(XUU^\dagger) = Tr(XI) = Tr(X) = 1$$
As for positive semidefiniteness, this means that for any $$\vert \psi\rangle$$, you have $$\langle \psi\vert X\vert\psi\rangle \geq 0$$. Now we have
$$\langle \psi\vert U^\dagger XU\vert\psi\rangle = \langle \phi\vert X\vert\phi\rangle \geq 0,$$
where I have set $$\vert\phi\rangle = U\vert\psi\rangle$$ and the inequality holds since $$X$$ is positive semidefinite.