Proposition: Finite matrix separation certificate

Let $\mathfrak{A}\subseteq M_d$ be a unital $\ast$-subalgebra and $h=h^{\ast }\notin \mathfrak{A}$. Let $E_{\mathrm{HS}}$ be the Hilbert–Schmidt orthogonal projection onto $\mathfrak{A}$, and put

For sufficiently small $t>0$,

are states with equal restrictions to $\mathfrak{A}$, but different expectations of $h$.

Proof. The projection preserves adjoints because the subspace is adjoint closed and the Hilbert–Schmidt inner product is compatible with adjoints. Thus $H$ is Hermitian. Orthogonality to the unit gives $\mathrm{Tr}H=0$. For $t\Vert H\Vert <1/d$, both matrices are positive and normalized.

For every $a\in \mathfrak{A}$, orthogonality gives $\mathrm{Tr}(Ha)=0$, so the restrictions agree. Finally,