Trace-state cyclicity and spectral probability calculus

Since $\rho$ is the trace-one positive density of [F1], [F2] gives $\rho ={\sum }_j{p}_j\mathrm{\mid }{u}_j⟩⟨u_j\mathrm{\mid }$, $p_j>0$, with ${\sum }_j{p}_j=\mathrm{Tr}\rho =1$, and trace-norm convergence. For bounded $B$, [F3] makes $\rho B$ trace class. Applying the basis-independent trace to finite partial sums and passing to the trace-norm limit gives $\mathrm{Tr}(\rho B)={\sum }_j{p}_j⟨B{u}_j,u_j⟩$; its absolute sum is at most $\mathrm{\parallel }B\mathrm{\parallel }{\sum }_j{p}_j=\mathrm{\parallel }B\mathrm{\parallel }$. The bounded cyclicity statement in [F6] gives $\mathrm{Tr}(B\rho )=\mathrm{Tr}(\rho B)$.

The scalar mean is defined on the first absolute-moment domain stated above. If the second moment is finite, then $\mathrm{\mid }\lambda \mathrm{\mid }\le (1+{\lambda }^2)\mathrm{/}2$ and ${\mu }_{\rho }$ is a probability measure, so the first absolute moment is finite as well. Expanding $(\lambda -m_{\rho }(A){)}^2$ and using $\int 1d{\mu }_{\rho }=1$ gives the displayed variance identity. For a vector $u$, the PVM domain formula [F4] is the operator domain, while the first absolute moment defines the larger form domain $D(\mathrm{\mid }A{\mathrm{\mid }}^{1\mathrm{/}2})=\{u:\int \mathrm{\mid }\lambda \mathrm{\mid }d{E}_u(\lambda )<\mathrm{\infty }\}$. These definitions use scalar integrals and make no unbounded cyclic permutation.

Since $0<p_j\le 1$, every term $-p_j\log p_j$ is nonnegative, so the entropy sum is well defined in $[0,\mathrm{\infty }]$; multiplying by the given $k_B>0$ preserves that range. It can diverge: let $c^{-1}={\sum }_{n\ge 2}(n(\log n{)}^2{)}^{-1}$ and $p_n=c\mathrm{/}(n(\log n{)}^2)$. The normalizing series converges because the block $2^k\le n<2^{k+1}$ contributes at most $1\mathrm{/}(k\log 2{)}^2$, and ${\sum }_k{k}^{-2}$ converges by grouping dyadic blocks. For sufficiently large $n$, $-p_n\log p_n\ge c\mathrm{/}(2n\log n)$. Each dyadic block contributes at least $1\mathrm{/}(4(k+1)\log 2)$ to ${\sum }_{n}1\mathrm{/}(n\log n)$, so that entropy series diverges.

For each Borel set $C\subseteq \mathbb{R}$, [F5] and step 1.1 give ${\mu }_{\rho }(C)={\sum }_j{p}_j\mathrm{\parallel }{E}_A(C)u_j{\mathrm{\parallel }}^2\ge 0$ and ${\mu }_{\rho }(\mathbb{R})={\sum }_j{p}_j=1$. For pairwise disjoint $C_k$, the PVM property gives $\mathrm{\parallel }{E}_A({\bigcup }_k{C}_k)u_j{\mathrm{\parallel }}^2={\sum }_k\mathrm{\parallel }{E}_A(C_k)u_j{\mathrm{\parallel }}^2$, since the projected vectors are orthogonal. The nonnegative double series may be summed in either order; therefore ${\mu }_{\rho }({\bigcup }_k{C}_k)={\sum }_k{\mu }_{\rho }(C_k)$. This proves that ${\mu }_{\rho }$ is a probability measure and agrees with the canonical scalar-law interface [F7].

For densities ${\rho }_k$ and weights $t_k\ge 0$, ${\sum }_k{t}_k=1$, the partial sums ${\sum }_{k\le N}{t}_k{\rho }_k$ are Cauchy in trace norm because each $\mathrm{\parallel }{\rho }_k{\mathrm{\parallel }}_1=\mathrm{Tr}{\rho }_k=1$ and the tail norm is at most ${\sum }_{k>N}{t}_k$. Completeness in [F3] gives a trace-class limit. Its trace is one by [F2] and trace continuity; its quadratic forms are nonnegative as limits of nonnegative quadratic forms. Thus the countable mixture is again a normal density state and the convergence is in trace norm.