Couplings of Two Noncommutative Laws and Their Quadratic Cost

Let $d\in\mathbb{N}$, where $\mathbb{N}$ is the set of natural numbers, and for $n\in\mathbb{N}$ let $\mathcal{P}_{n}=\mathbb{C}\langle x_{1},\dots,x_{n}\rangle$ be the noncommutative polynomials in $n$ variables, a complex vector space by Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §vector-space, with its product. Let $\Sigma_{d}$ be the set of noncommutative laws of $d$ variables, and for a tuple $a$ of polynomials let $\sigma_{a}$ be its substitution.

1. (Marginal substitutions)¶ The first and second marginal substitutions are $\iota^{1}=\sigma_{(x_{1},\dots,x_{d})}$ and $\iota^{2}=\sigma_{(x_{d+1},\dots,x_{2d})}$, maps from $\mathcal{P}_{d}$ to $\mathcal{P}_{2d}$ substituting the first $d$, respectively the last $d$, variables of $\mathcal{P}_{2d}$ for $x_{1},\dots,x_{d}$.

2. (Couplings)¶ Let $\mu,\nu\in\Sigma_{d}$. A coupling of $\mu$ and $\nu$ is a tracial state $\gamma$ on $\mathcal{P}_{2d}$ such that $\gamma\circ\iota^{1}=\mu$ and $\gamma\circ\iota^{2}=\nu$. The set of couplings of $\mu$ and $\nu$ is written $\Pi(\mu,\nu)$.

3. (Cost)¶ The cost polynomial is

the finite sum in $\mathcal{P}_{2d}$, where $q^{2}=qq$. The cost of a coupling $\gamma\in\Pi(\mu,\nu)$ is the number $I(\gamma)=\gamma(\Delta_{d})$.