Proof of The Noncommutative Wasserstein Distance Satisfies the Triangle Inequality and is a Metric on Noncommutative Laws

Each result cited below is universally quantified over the data in its own statement. We adopt the setting of Noncommutative Laws, Couplings and the Wasserstein Distance: Standing Notation, with $d$ and $W_{2}$ as in the statement.

Step 1 (a common norm bound). Let $\mu,\nu,\rho\in\Sigma_{d}$. By Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §law there are reals $R_{1},R_{2},R_{3}>0$ with $\mu\in\Sigma_{d,R_{1}}$, $\nu\in\Sigma_{d,R_{2}}$ and $\rho\in\Sigma_{d,R_{3}}$. Let $R=\max\{\max\{R_{1},R_{2}\},R_{3}\}$, maxima in the totally ordered $\mathbb{R}$. Claim 1 of Elementary Properties of the Maximum of Two Elements, applied twice, gives $R_{1}\le R$, $R_{2}\le R$ and $R_{3}\le R$; in particular $R\ge R_{1}>0$. By Basic Properties of Noncommutative Laws: Adjoints, the Self-Adjoint Pairing, Cauchy-Schwarz, Monotonicity in the Bound, and Laws of Constant Tuples §monotone, $\mu,\nu,\rho\in\Sigma_{d,R}$. The same argument with two laws shows that any two elements of $\Sigma_{d}$ lie in a common $\Sigma_{d,R}$ with $R>0$.

Step 2 (optimal couplings and gluing). By The Noncommutative Wasserstein Distance: Existence of Optimal Couplings, Symmetry, Separation, a Moment Bound, Weak-Star Lower Semicontinuity, and Displacement Interpolation §attained, applied to $\mu,\nu\in\Sigma_{d,R}$ and to $\nu,\rho\in\Sigma_{d,R}$, there are optimal couplings $\gamma_{1}\in\Pi(\mu,\nu)$ and $\gamma_{2}\in\Pi(\nu,\rho)$; by The Noncommutative Quadratic Wasserstein Distance and Optimal Couplings §optimal,

Let $\sigma^{12},\sigma^{23},\sigma^{13}:\mathcal{P}_{2d}\to\mathcal{P}_{3d}$ be the substitutions of the $2d$-tuples $(x_{1},\dots,x_{2d})$, $(x_{d+1},\dots,x_{3d})$ and $(x_{1},\dots,x_{d},x_{2d+1},\dots,x_{3d})$ in $\mathcal{P}_{3d}$. By Gluing Two Noncommutative Couplings along a Common Marginal there is $\psi\in\Sigma_{3d,R}$ with $\psi\circ\sigma^{12}=\gamma_{1}$ (Gluing Two Noncommutative Couplings along a Common Marginal §first), $\psi\circ\sigma^{23}=\gamma_{2}$ (Gluing Two Noncommutative Couplings along a Common Marginal §second) and $\gamma_{3}=\psi\circ\sigma^{13}\in\Pi(\mu,\rho)$ (Gluing Two Noncommutative Couplings along a Common Marginal §composite). Let $m$ be the infimum of $\{I(\gamma):\gamma\in\Pi(\mu,\rho)\}$. By The Noncommutative Quadratic Wasserstein Distance and Optimal Couplings §distance, $W_{2}(\mu,\rho)$ is the nonnegative real number whose square is $m$; and $m$, the greatest lower bound of a set containing $I(\gamma_{3})$ (Existence of the Infimum of a Nonempty Subset of $\mathbb{R}$ Bounded Below), satisfies $m\le I(\gamma_{3})$. Hence

Step 3 (the three costs). For $j\in[d]$ let

where $p-q=p+(-1)q$. The variables are self-adjoint and $\mathcal{P}_{3d,\mathrm{sa}}$ is 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 $u_{j}$, $v_{j}$ and $u_{j}+v_{j}$ lie in $\mathcal{P}_{3d,\mathrm{sa}}$. The three substitutions are linear by Substitution of Noncommutative Polynomials into the Variables §substitution, and by Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §values each sends $x_{k}$ to the $k$-th entry of its tuple; the $j$-th and $(d+j)$-th entries are $x_{j},x_{d+j}$ for $\sigma^{12}$, $x_{d+j},x_{2d+j}$ for $\sigma^{23}$, and $x_{j},x_{2d+j}$ for $\sigma^{13}$. Hence, using the vector space rules of Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §vector-space for the last equality,

Let $\sigma$ be one of the three substitutions and $w_{j}$ the corresponding right-hand side. The cost polynomial $\Delta_{d}$ of Couplings of Two Noncommutative Laws and Their Quadratic Cost §cost is a finite sum in $\mathcal{P}_{2d}$, so claim 4 of Properties of Finite Sums of Vectors (for the linear map $\sigma$) and Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §homomorphism give $\sigma(\Delta_{d})=\sum_{j=1}^{d}\sigma(x_{j}-x_{d+j})^{2}=\sum_{j=1}^{d}w_{j}^{2}$, and claim 4 of Properties of Finite Sums of Vectors again (for the linear map $\psi$, Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §tracial-state) gives $\psi(\sigma(\Delta_{d}))=\sum_{j=1}^{d}\psi(w_{j}^{2})$. By Couplings of Two Noncommutative Laws and Their Quadratic Cost §cost and Step 2 therefore

Step 4 (costs as squared norms). Since $\psi\in\Sigma_{3d,R}$, Left and Right Multiplication Operators on the Complex GNS Space of a Noncommutative Law: Boundedness, Algebra Rules, Adjoints, Commutation, the Vacuum and the Conjugation applies with $3d$, $R$, $\psi$ in place of that lemma's $d$, $r$, $\lambda$; let $\mathcal{H}_{\psi}$ be the complex GNS space of $\psi$, $\Omega_{\psi}$ its vacuum vector, $\widehat{p}$ the class of $p\in\mathcal{P}_{3d}$ and $L_{p}$ the left multiplication operators. For $a\in\mathcal{P}_{3d,\mathrm{sa}}$, Left and Right Multiplication Operators on the Complex GNS Space of a Noncommutative Law: Boundedness, Algebra Rules, Adjoints, Commutation, the Vacuum and the Conjugation §vacuum gives $\lVert\widehat{a}\rVert^{2}=\langle\widehat{a},\widehat{a}\rangle=\psi(a^{*}a)=\psi(a^{2})$. Moreover, by Left and Right Multiplication Operators on the Complex GNS Space of a Noncommutative Law: Boundedness, Algebra Rules, Adjoints, Commutation, the Vacuum and the Conjugation §vacuum and Left and Right Multiplication Operators on the Complex GNS Space of a Noncommutative Law: Boundedness, Algebra Rules, Adjoints, Commutation, the Vacuum and the Conjugation §algebra,

Let $\mathcal{H}_{\psi}^{d}$ be the finite direct sum of Finite Direct Sums of a Complex Hilbert Space: the Hilbert Structure, Coordinate Inclusions, Block Entries of Bounded Operators and Commutation with Diagonal Operators, with $\mathcal{H}_{\psi}$ and $d$ in place of that lemma's $H$ and $m$, and let $\xi=(\widehat{u_{1}},\dots,\widehat{u_{d}})$ and $\eta=(\widehat{v_{1}},\dots,\widehat{v_{d}})$, so that $\xi+\eta=(\widehat{u_{1}+v_{1}},\dots,\widehat{u_{d}+v_{d}})$ by the componentwise addition. By Finite Direct Sums of a Complex Hilbert Space: the Hilbert Structure, Coordinate Inclusions, Block Entries of Bounded Operators and Commutation with Diagonal Operators §hilbert and (2),

Step 5 (the triangle inequality). $\mathcal{H}_{\psi}^{d}$ is a complex Hilbert space by Finite Direct Sums of a Complex Hilbert Space: the Hilbert Structure, Coordinate Inclusions, Block Entries of Bounded Operators and Commutation with Diagonal Operators §hilbert, so by claim 2 of The Induced Norm is a Norm, and Induces a Metric its induced norm is a norm: it is nonnegative and $\lVert\xi+\eta\rVert\le\lVert\xi\rVert+\lVert\eta\rVert$. The numbers $\lVert\xi\rVert$ and $W_{2}(\mu,\nu)$ are nonnegative with the same square $I(\gamma_{1})$ (Steps 2 and 4), so they are equal by the uniqueness in Existence and Uniqueness of the Nonnegative Square Root; likewise $\lVert\eta\rVert=W_{2}(\nu,\rho)$. Claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field, in the ordered field $\mathbb{R}$ with the nonnegative numbers $\lVert\xi+\eta\rVert$ and $\lVert\xi\rVert+\lVert\eta\rVert$, turns the triangle inequality into $\lVert\xi+\eta\rVert^{2}\le(\lVert\xi\rVert+\lVert\eta\rVert)^{2}$. With (1),

Both $W_{2}(\mu,\rho)$ and $W_{2}(\mu,\nu)+W_{2}(\nu,\rho)$ are nonnegative by The Noncommutative Quadratic Wasserstein Distance and Optimal Couplings §distance, so claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field gives $W_{2}(\mu,\rho)\le W_{2}(\mu,\nu)+W_{2}(\nu,\rho)$. This proves claim 1.

Step 6 (the metric). By The Noncommutative Quadratic Wasserstein Distance and Optimal Couplings §distance, $W_{2}$ is a map $\Sigma_{d}\times\Sigma_{d}\to\mathbb{R}$. Let $\mu,\nu,\rho\in\Sigma_{d}$, and by Step 1 choose a real $R>0$ with $\mu,\nu\in\Sigma_{d,R}$. We check the four conditions of Metric Space. Condition 1: $W_{2}(\mu,\nu)\ge0$ by The Noncommutative Quadratic Wasserstein Distance and Optimal Couplings §distance. Condition 2: if $\mu=\nu$ then $W_{2}(\mu,\nu)=W_{2}(\mu,\mu)=0$, and if $W_{2}(\mu,\nu)=0$ then $\mu=\nu$, both by The Noncommutative Wasserstein Distance: Existence of Optimal Couplings, Symmetry, Separation, a Moment Bound, Weak-Star Lower Semicontinuity, and Displacement Interpolation §separation. Condition 3: $W_{2}(\mu,\nu)=W_{2}(\nu,\mu)$ by The Noncommutative Wasserstein Distance: Existence of Optimal Couplings, Symmetry, Separation, a Moment Bound, Weak-Star Lower Semicontinuity, and Displacement Interpolation §symmetry. Condition 4: $W_{2}(\mu,\rho)\le W_{2}(\mu,\nu)+W_{2}(\nu,\rho)$ by claim 1. Hence $W_{2}$ is a metric on $\Sigma_{d}$. For real $R>0$, every element of $\Sigma_{d,R}$ is a noncommutative law by Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §law, so $\Sigma_{d,R}\subseteq\Sigma_{d}$; the restriction of $W_{2}$ is therefore a map $\Sigma_{d,R}\times\Sigma_{d,R}\to\mathbb{R}$, and the four conditions for points of $\Sigma_{d,R}$ are instances of those just proved. This proves claim 2.