Proof of Moments of Noncommutative Laws are Lipschitz in the Wasserstein Distance

Each result cited below is universally quantified over the data in its own statement.

Throughout, $d\in\mathbb{N}$ and the real $R>0$ are those of the statement. For a tracial state $\gamma$ on $\mathcal{P}_{2d}$ and $p\in\mathcal{P}_{2d}$ we use the notation $\|p\|_{\gamma}$ of Noncommutative Laws, Couplings and the Wasserstein Distance: Standing Notation §laws: the nonnegative square root of the real number $\gamma(p^{*}p)\ge0$. For a word $w\in W_{d}$ put

with $\iota^{1},\iota^{2}$ the marginal substitutions of Couplings of Two Noncommutative Laws and Their Quadratic Cost §marginals, and for $j\in[d]$ put $z_{j}=x_{j}-x_{d+j}\in\mathcal{P}_{2d}$. By the recursion theorem on $\mathbb{N}$ define reals $C_{k}$ $(k\in\mathbb{N})$ by $C_{1}=1$ and $C_{k+1}=R^{k}+R\,C_{k}$; since $R^{k}\ge0$ by claim 5 of Properties of Natural Number Powers in a Field, induction on $k$ gives $C_{k}\ge0$ for every $k$. For $w\in W_{d}$ let $c(w)=0$ if $w=\varnothing$ and $c(w)=C_{k}$ if $w$ has length $k\in\mathbb{N}$; every nonempty word has exactly one length by Words over a Finite Alphabet: the Empty Word, Concatenation and Reversal §words. The numbers $c(w)$ depend only on $w$ and $R$.

Step 1 (Four facts about $\|\cdot\|_{\gamma}$). Let $\gamma\in\Sigma_{2d,R}$ and $p,q\in\mathcal{P}_{2d}$.

(i) $|\gamma(q^{*}p)|\le\|p\|_{\gamma}\|q\|_{\gamma}$ and $|\gamma(p)|\le\|p\|_{\gamma}$. Indeed, Basic Properties of Noncommutative Laws: Adjoints, the Self-Adjoint Pairing, Cauchy-Schwarz, Monotonicity in the Bound, and Laws of Constant Tuples §cauchy-schwarz gives $|\gamma(q^{*}p)|^{2}\le\gamma(p^{*}p)\,\gamma(q^{*}q)=(\|p\|_{\gamma}\|q\|_{\gamma})^{2}$, and both $|\gamma(q^{*}p)|$ and $\|p\|_{\gamma}\|q\|_{\gamma}$ are nonnegative, so the first inequality follows from claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field. For the second take $q=1$: $1^{*}=1$ by Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §adjoint, $1p=p$ and $1\cdot1=1$ by Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §monomials, and $\gamma(1)=1$ by condition (a) of Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §tracial-state, so $\|1\|_{\gamma}=1$ by the uniqueness in Existence and Uniqueness of the Nonnegative Square Root.

(ii) $\|p+q\|_{\gamma}\le\|p\|_{\gamma}+\|q\|_{\gamma}$. By Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §adjoint and the distributive laws of Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §algebra, $(p+q)^{*}(p+q)=p^{*}p+p^{*}q+q^{*}p+q^{*}q$, so by linearity of $\gamma$

Put $s=\gamma(q^{*}p)$. By Basic Properties of Noncommutative Laws: Adjoints, the Self-Adjoint Pairing, Cauchy-Schwarz, Monotonicity in the Bound, and Laws of Constant Tuples §adjoint, $\gamma(p^{*}q)=\overline{s}$, and $s+\overline{s}=2\operatorname{Re}s\le2|s|$ by claims 2 and 6 of Properties of Complex Conjugation and Modulus; with (i) this gives $\|p+q\|_{\gamma}^{2}\le(\|p\|_{\gamma}+\|q\|_{\gamma})^{2}$, and claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field gives (ii).

(iii) $\|p^{*}\|_{\gamma}=\|p\|_{\gamma}$. Condition (c) of Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §tracial-state, applied to $p$ and $p^{*}$, gives $\gamma(pp^{*})=\gamma(p^{*}p)$; since $(p^{*})^{*}=p$ by Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §adjoint, this says $\|p^{*}\|_{\gamma}^{2}=\|p\|_{\gamma}^{2}$, and claim 3 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field gives (iii).

(iv) For every $i\in[2d]$: $\|x_{i}p\|_{\gamma}\le R\|p\|_{\gamma}$ and $\|px_{i}\|_{\gamma}\le R\|p\|_{\gamma}$. The first inequality is the last sentence of The Norm Bound of a Noncommutative Law: Multiplication by a Variable is Bounded, the Bound is Detected by Even Moments, and Laws Pull Back under Self-Adjoint Substitutions §multiplication, applied with $2d$ variables to $\gamma\in\Sigma_{2d,R}$. For the second, $(px_{i})^{*}=x_{i}^{*}p^{*}=x_{i}p^{*}$ by Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §adjoint, so by (iii) and the first inequality $\|px_{i}\|_{\gamma}=\|x_{i}p^{*}\|_{\gamma}\le R\|p^{*}\|_{\gamma}=R\|p\|_{\gamma}$.

Step 2 (Products along words). By Couplings of Two Noncommutative Laws and Their Quadratic Cost §marginals and Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §values, $\iota^{1}(x_{j})=x_{j}$ and $\iota^{2}(x_{j})=x_{d+j}$ in $\mathcal{P}_{2d}$ for $j\in[d]$. Let $w\in W_{d}$ have length $k+1$ for some $k\in\mathbb{N}$. By Basic Properties of Words: Associativity, Reversal, Finitely Many Factorisations, and Countability §last-letter, $w=w'(j)$ with $w'$ a word of length $k$ and $j\in[d]$; then $x_{w}=x_{w'}x_{j}$ by Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §monomials, and Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §homomorphism gives

Also, a word of length $1$ is a letter $(j)$ by Basic Properties of Words: Associativity, Reversal, Finitely Many Factorisations, and Countability §last-letter, and then $X_{(j)}=x_{j}$, $Y_{(j)}=x_{d+j}$ and $E_{(j)}=z_{j}$.

Step 3 (Left multiplication by $X_{u}$). Let $\gamma\in\Sigma_{2d,R}$. We show by induction on $k\in\mathbb{N}$ that $\|X_{u}p\|_{\gamma}\le R^{k}\|p\|_{\gamma}$ for every word $u\in W_{d}$ of length $k$ and every $p\in\mathcal{P}_{2d}$. For $k=1$, $u=(j)$ and $X_{u}=x_{j}$ by Step 2, and Step 1(iv) applies, with $R^{1}=R$ by claim 1 of Properties of Natural Number Powers in a Field. If the claim holds for $k$ and $u$ has length $k+1$, write $u=u'(j)$ as in Step 2; by associativity (Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §algebra) $X_{u}p=X_{u'}(x_{j}p)$, so the induction hypothesis, Step 1(iv) and claims 1 and 5 of Properties of Natural Number Powers in a Field give $\|X_{u}p\|_{\gamma}\le R^{k}\|x_{j}p\|_{\gamma}\le R^{k}R\|p\|_{\gamma}=R^{k+1}\|p\|_{\gamma}$.

Step 4 (The optimal coupling and the cost). Now fix $\mu,\nu\in\Sigma_{d,R}$, and write $W=W_{2}(\mu,\nu)\ge0$. By The Noncommutative Wasserstein Distance: Existence of Optimal Couplings, Symmetry, Separation, a Moment Bound, Weak-Star Lower Semicontinuity, and Displacement Interpolation §attained choose an optimal coupling $\gamma\in\Pi(\mu,\nu)$, so that $I(\gamma)=W^{2}$ by The Noncommutative Quadratic Wasserstein Distance and Optimal Couplings §optimal; by Couplings of Noncommutative Laws: the Norm Bound, the Cost Identity, the Tensor, Diagonal and Swapped Couplings, Weak-Star Closedness, and Displacement Interpolants §bound, $\gamma\in\Sigma_{2d,R}$, so Steps 1 and 3 apply to $\gamma$. By Couplings of Two Noncommutative Laws and Their Quadratic Cost §coupling, $\mu=\gamma\circ\iota^{1}$ and $\nu=\gamma\circ\iota^{2}$. We claim $\|z_{j}\|_{\gamma}\le W$ for every $j\in[d]$. The variables are self-adjoint and the self-adjoint polynomials are closed under sums and real multiples by Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §self-adjoint, so $z_{j}^{*}=z_{j}$ and $z_{j}^{*}z_{j}=z_{j}^{2}$, whence $\gamma(z_{j}^{2})=\|z_{j}\|_{\gamma}^{2}$ is a nonnegative real by condition (b) of Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §tracial-state. The cost polynomial of Couplings of Two Noncommutative Laws and Their Quadratic Cost §cost is the finite sum $\Delta_{d}=\sum_{j=1}^{d}z_{j}^{2}$, and a linear map carries a finite sum to the finite sum of its values (claim 4 of Properties of Finite Sums of Vectors, applied to the linear map $\gamma$ from $\mathcal{P}_{2d}$ to the complex vector space $\mathbb{C}$), so

The summands being nonnegative reals, claim 6 of Properties of Finite Sums gives $\|z_{j}\|_{\gamma}^{2}=\gamma(z_{j}^{2})\le W^{2}$, and claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field gives $\|z_{j}\|_{\gamma}\le W$.

Step 5 (The telescoped difference). We show by induction on $k\in\mathbb{N}$ that $\|E_{w}\|_{\gamma}\le C_{k}W$ for every word $w\in W_{d}$ of length $k$. For $k=1$, $E_{w}=z_{j}$ by Step 2, and Step 4 gives $\|E_{w}\|_{\gamma}\le W=C_{1}W$. Suppose the claim holds for $k$, and let $w=w'(j)$ have length $k+1$ as in Step 2. By Step 2 and the rules of Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §algebra,

Step 1(ii), Step 3 (with $u=w'$ and $p=z_{j}$), Step 1(iv) (with $i=d+j$), Step 4 and the induction hypothesis give

Step 6 (Monomials). We claim $|\mu(x_{w})-\nu(x_{w})|\le c(w)\,W$ for every $w\in W_{d}$. If $w=\varnothing$, then $x_{w}=1$ by The Algebra of Noncommutative Polynomials in Finitely Many Self-Adjoint Variables §monomials, and $\mu(1)=\nu(1)=1$ by condition (a) of Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §tracial-state, so both sides vanish. If $w$ has length $k$, then by Step 4 and linearity of $\gamma$, $\mu(x_{w})-\nu(x_{w})=\gamma(X_{w})-\gamma(Y_{w})=\gamma(E_{w})$, and Step 1(i) and Step 5 give $|\gamma(E_{w})|\le\|E_{w}\|_{\gamma}\le C_{k}W=c(w)W$.

Step 7 (Claim 1). Let $p\in\mathcal{P}_{d}$; the constant $C$ is chosen from $p$ alone, before $\mu$ and $\nu$. If $p=0$, take $C=0$: $\mu(0)=\nu(0)=0$ by linearity. Otherwise $F=\operatorname{supp}p$ is a nonempty finite set, and we take $C=\sum_{w\in F}|p(w)|\,c(w)$, the sum of Sum over a Finite Index Set; it is a nonnegative real by claim 5 of Properties of Finite Sums, applied through a bijection $\varphi:[n]\to F$ as in that definition. Now let $\mu,\nu\in\Sigma_{d,R}$. The map $\ell=\mu-\nu:\mathcal{P}_{d}\to\mathbb{C}$ is linear, so the uniqueness and the formula in Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §linear-extension (part (a), with $c'(w)=\mu(x_{w})-\nu(x_{w})$) give

By Sums over Finite Index Sets: Finite Unions, Disjoint Unions, Vanishing Terms, Dependent Pairs, Conjugation and the Modulus §modulus and claim 4 of Properties of Complex Conjugation and Modulus, $|\mu(p)-\nu(p)|\le\sum_{w\in F}|p(w)|\,|\mu(x_{w})-\nu(x_{w})|$. Each summand is at most $|p(w)|\,c(w)\,W$ by Step 6. Writing both sums over $F$ as finite sums over $[n]$ through the same bijection $\varphi$, claims 2, 3 and 5 of Properties of Finite Sums (applied to the nonnegative differences of the summands) show that the first sum is at most the second, and claim 3 of that lemma pulls out the factor $W$. Hence $|\mu(p)-\nu(p)|\le C\,W_{2}(\mu,\nu)$.

Step 8 (Claim 2). Let $(\mu_{m})$ and $\mu$ be as in claim 2, let $p\in\mathcal{P}_{d}$, and let $C$ be the constant of claim 1 for $p$. Let $\varepsilon>0$. Since $C+1>0$, the real sequence $(W_{2}(\mu_{m},\mu))_{m}$, which converges to $0$ and consists of nonnegative reals, gives by Limit of a Sequence of Real Numbers an $N\in\mathbb{N}$ with $W_{2}(\mu_{m},\mu)<\varepsilon/(C+1)$ for all $m\ge N$. For such $m$, write $s=\mu_{m}(p)-\mu(p)$. The uniqueness of real and imaginary parts in Real and Imaginary Parts of a Complex Number gives $\operatorname{Re}s=\operatorname{Re}\mu_{m}(p)-\operatorname{Re}\mu(p)$ and $\operatorname{Im}s=\operatorname{Im}\mu_{m}(p)-\operatorname{Im}\mu(p)$. By claim 6 of Properties of Complex Conjugation and Modulus and claim 1 of Properties of the Absolute Value in an Ordered Field, $|\operatorname{Re}s|\le|s|$ and $|\operatorname{Im}s|\le|s|$, and by claim 1 (applied to $\mu_{m}$ and $\mu$)

So $(\operatorname{Re}\mu_{m}(p))_{m}$ and $(\operatorname{Im}\mu_{m}(p))_{m}$ converge to $\operatorname{Re}\mu(p)$ and $\operatorname{Im}\mu(p)$, for every $p$; that is, $\mu_{m}\to\mu$ weak-star in the sense of Weak-Star Convergence of Noncommutative Laws §weak-star.

Step 9 (Claim 3). Let $K$ be as in claim 3, and let $\lambda\in\Sigma_{d,R}\setminus K$. By Open Subset of a Metric Space and Open Ball in a Metric Space we must find a real $r>0$ such that every $\kappa\in\Sigma_{d,R}$ with $W_{2}(\lambda,\kappa)<r$ lies outside $K$. Suppose no such $r$ exists. For each $m\in\mathbb{N}$ the real $1/m$ is positive by claim 3 of Properties of the Canonical Map from the Natural Numbers to an Ordered Field, so we may choose $\mu_{m}\in K$ with $W_{2}(\lambda,\mu_{m})<1/m$. Then $W_{2}(\mu_{m},\lambda)=W_{2}(\lambda,\mu_{m})$ by The Noncommutative Wasserstein Distance: Existence of Optimal Couplings, Symmetry, Separation, a Moment Bound, Weak-Star Lower Semicontinuity, and Displacement Interpolation §symmetry, and this sequence converges to $0$: given $\varepsilon>0$, claim 3 of The Archimedean Property of the Real Numbers gives $N\in\mathbb{N}$ with $1/N<\varepsilon$, and for $m\ge N$ we have $1/m\le1/N$ (equality if $m=N$; if $N<m$, then $N<m$ as reals by claim 6 of Properties of the Canonical Map from the Natural Numbers to an Ordered Field, and multiplying by the positive real $(1/N)(1/m)$, using claims 5 and 10 of Elementary Order Arithmetic in an Ordered Field, gives $1/m<1/N$), so $0\le W_{2}(\mu_{m},\lambda)<1/m\le1/N<\varepsilon$. By claim 2, $\mu_{m}\to\lambda$ weak-star; this is a sequence in $K$ converging weak-star to $\lambda\in\Sigma_{d,R}$, so $\lambda\in K$ by the hypothesis on $K$, a contradiction. Hence such an $r$ exists for every $\lambda\in\Sigma_{d,R}\setminus K$, and $\Sigma_{d,R}\setminus K$ is open in $(\Sigma_{d,R},W_{2})$. $\blacksquare$