In the algebraic formulation of quantum physics/information, states $\omega: \mathcal{A}\rightarrow \mathbb{C}$ are defined as linear functionals on a $C^*$-algebra $\mathcal{A}$ (algebra of observables, representable as $\mathcal{B}(H)$ for some Hilbert space $H$ via the GNS construction) that are positive ($\omega(A^*A)\geq 0\,\forall A\in \mathcal{A}$) and normalized ($\omega(I)=1$ for $\mathcal{A}$ with unit element $I$ or an equivalent condition for non-unital $\mathcal{A}$). These quantum states are then usually represented as density operators defined via $\omega(A)=:\text{Tr}(\omega A)$, but it is well-known that in infinite dimensions there are so-called non-normal states that are not representable this way. Is this due to the fact that for infinite dimensions $\mathcal{B}(\mathcal{H})\simeq H\otimes H^*$ does not hold?
from Hot Weekly Questions - Mathematics Stack Exchange
Post a Comment