= GNS dimension from density matrix rank
{c}
{title2=$\dim H_\rho=d\operatorname{rank}\rho$}
For the <state on a C-star algebra> $a\mapsto\operatorname{Tr}(\rho a)$ on $M_d(\mathbb C)$, the <GNS construction> has the displayed Hilbert-space dimension. Here $\rho$ is a <density matrix>. Its null space for the GNS seminorm consists of $b$ with $b\rho^{1/2}=0$, because $\operatorname{Tr}(\rho b^*b)=\|b\rho^{1/2}\|_{\mathrm{HS}}^2$. Thus the GNS space is the space of matrices $b\rho^{1/2}$ with the <Hilbert-Schmidt inner product>. If $\rho$ has <rank> $r$, diagonalization identifies these matrices with $r$ arbitrary columns, giving dimension $dr$. Left multiplication is the representation and $\rho^{1/2}$ is its <cyclic vector>. In particular a pure <rank>-one matrix state gives the defining $d$-dimensional representation, while a full-<rank> state gives a $d^2$-dimensional space.
Back to article page
