The Law of a Tuple of Bounded Self-Adjoint Operators in a Tracial Vector State

In the setting of Noncommutative Laws, Couplings and the Wasserstein Distance: Standing Notation and Complex Hilbert Spaces and Bounded Linear Maps: Standing Notation, let $n\in\mathbb{N}$, let $R>0$ be real, let $H$ be a complex Hilbert space, let $\Omega\in H$ with $\lVert\Omega\rVert=1$, and let $T=(T_{1},\dots,T_{n})$ be an $n$-tuple of self-adjoint elements of $\mathcal{L}(H)$ with $\lVert T_{j}\rVert_{\mathrm{op}}\le R$ for every $j\in[n]$. For $w\in W_{n}$ and $p\in\mathcal{P}_{n}$ let $T_{w}$ and $p(T)$ be the product along $w$ and the value of $p$ at $T$, and let $\lambda_{T}:\mathcal{P}_{n}\to\mathbb{C}$ be the map $\lambda_{T}(p)=\langle\Omega,p(T)\Omega\rangle$.

1. (State)¶ $\lambda_{T}$ is linear and $\lambda_{T}(1)=1$; for every $p\in\mathcal{P}_{n}$ the number $\lambda_{T}(p^{*}p)$ equals $\lVert p(T)\Omega\rVert^{2}$, so it is real and nonnegative; and $|\lambda_{T}(x_{w})|\le R^{k}$ for every $k\in\mathbb{N}$ and every word $w\in W_{n}$ of length $k$.

2. (Law)¶ If $\langle\Omega,T_{u}T_{v}\Omega\rangle=\langle\Omega,T_{v}T_{u}\Omega\rangle$ for all $u,v\in W_{n}$, then $\lambda_{T}\in\Sigma_{n,R}$.