Proof of The Barycentric Decomposition of the Noncommutative Wasserstein Distance

Each result cited below is universally quantified over the data in its own statement. We work in the setting of Noncommutative Laws, Couplings and the Wasserstein Distance: Standing Notation, with $\mu,\nu\in\Sigma_{d}$ as in the statement, and use the following elementary facts throughout. Every variable satisfies $x_{i}^{*}=x_{i}$ by Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §adjoint, so the variables lie in the self-adjoint part by Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §self-adjoint. The inner product is linear in its second and conjugate-linear in its first argument by Complex Hilbert Spaces and Bounded Linear Maps: Standing Notation §spaces; for a bounded linear map $T$ with adjoint $T^{*}$ we have $\langle T^{*}\xi,\xi'\rangle=\langle\xi,T\xi'\rangle$, hence also $\langle T\xi',\xi\rangle=\langle\xi',T^{*}\xi\rangle$ by conjugate symmetry. For $k\in\mathbb{N}$, a real $r>0$, $\lambda\in\Sigma_{k,r}$ and $p,q\in\mathcal{P}_{k}$, 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

$$L_{p}\Omega_{\lambda}=\widehat{p}^{\,\lambda},\qquad\langle\widehat{p}^{\,\lambda},\widehat{q}^{\,\lambda}\rangle=\lambda(p^{*}q),\qquad\langle\Omega_{\lambda},L_{p}\Omega_{\lambda}\rangle=\lambda(p);$$

for $T\in\mathcal{M}_{\lambda}$, $\tau_{\lambda}(T)=\langle\Omega_{\lambda},T\Omega_{\lambda}\rangle$ by The Tracial Algebra of a Noncommutative Law and Its Trace §trace; $\tau_{\lambda}$ is linear with $\tau_{\lambda}(L_{p})=\lambda(p)$ by The Tracial Algebra of a Noncommutative Law: a Norm-Closed Unital *-Algebra with a Faithful Positive Trace, Determined by Vacuum Vectors, Closed under Square Roots §trace; and $\mathcal{M}_{\lambda}$ contains $I$ and every $L_{p}$ and is closed under sums, complex multiples, composition and adjoints by The Tracial Algebra of a Noncommutative Law: a Norm-Closed Unital *-Algebra with a Faithful Positive Trace, Determined by Vacuum Vectors, Closed under Square Roots §star-algebra. For a tuple $T$ of bounded operators, evaluation $p\mapsto p(T)$ is linear by Evaluation of Noncommutative Polynomials at a Tuple of Bounded Operators §evaluation, multiplicative by Evaluation of Noncommutative Polynomials at Bounded Operators: a Unital Homomorphism Compatible with Adjoints, Substitution, Representations and the GNS Multiplication Operators §homomorphism, and $1(T)=I$, $x_{i}(T)=T_{i}$ by Evaluation of Noncommutative Polynomials at Bounded Operators: a Unital Homomorphism Compatible with Adjoints, Substitution, Representations and the GNS Multiplication Operators §values.

Step 0 (Norm bounds, the marginal isometry and the barycentric vectors). By Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §law there are reals $R_{1},R_{2}>0$ with $\mu\in\Sigma_{d,R_{1}}$ and $\nu\in\Sigma_{d,R_{2}}$; with $R=\max\{R_{1},R_{2}\}$, Basic Properties of Noncommutative Laws: Adjoints, the Self-Adjoint Pairing, Cauchy-Schwarz, Monotonicity in the Bound, and Laws of Constant Tuples §monotone gives $\mu,\nu\in\Sigma_{d,R}$. We fix this $R$.

Fix $\gamma\in\Pi(\mu,\nu)$ and abbreviate $z_{j}=z^{\gamma}_{j}$, $z=(z_{1},\dots,z_{d})$, $\zeta=\zeta_{\gamma}$, $\beta=\beta_{\gamma}$ and $E=E_{\gamma}$; the operators $L_{p}$ act on $\mathcal{H}_{\gamma}$ for $p\in\mathcal{P}_{2d}$ and on $\mathcal{H}_{\mu}$ for $p\in\mathcal{P}_{d}$, as in the statement. 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 $\gamma\in\Sigma_{2d}$ by Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §law. Let $a=(x_{1},\dots,x_{d})$ and $a'=(x_{d+1},\dots,x_{2d})$, $d$-tuples in $\mathcal{P}_{2d,\mathrm{sa}}$. By Couplings of Two Noncommutative Laws and Their Quadratic Cost §marginals, $\sigma_{a}=\iota^{1}$ and $\sigma_{a'}=\iota^{2}$, so Couplings of Two Noncommutative Laws and Their Quadratic Cost §coupling gives $\gamma\circ\sigma_{a}=\mu$ and $\gamma\circ\sigma_{a'}=\nu$; and by Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §values, $\sigma_{a}(x_{j})=x_{j}$ and $\sigma_{a'}(x_{j})=x_{d+j}$ for $j\in[d]$.

Let $V\in\mathcal{L}(\mathcal{H}_{\mu},\mathcal{H}_{\gamma})$ and $\pi:\mathcal{M}_{\mu}\to\mathcal{M}_{\gamma}$ be the isometry and the embedding of Marginals of a Noncommutative Law: the Isometry of GNS Spaces, the Trace-Preserving Embedding of Tracial Algebras and the Conditional Expectation for $\gamma$ and $a$. By Marginals of a Noncommutative Law: the Isometry of GNS Spaces, the Trace-Preserving Embedding of Tracial Algebras and the Conditional Expectation §isometry, $V^{*}V=I$, $V\Omega_{\mu}=\Omega_{\gamma}$ and $V\widehat{p}^{\,\mu}=\widehat{\iota^{1}(p)}^{\,\gamma}$ for $p\in\mathcal{P}_{d}$; by Marginals of a Noncommutative Law: the Isometry of GNS Spaces, the Trace-Preserving Embedding of Tracial Algebras and the Conditional Expectation §embedding, $\pi(T)\in\mathcal{M}_{\gamma}$, $\pi(T)\Omega_{\gamma}=VT\Omega_{\mu}$ and $\lVert\pi(T)\rVert_{\mathrm{op}}\le\lVert T\rVert_{\mathrm{op}}$ for $T\in\mathcal{M}_{\mu}$; by Marginals of a Noncommutative Law: the Isometry of GNS Spaces, the Trace-Preserving Embedding of Tracial Algebras and the Conditional Expectation §homomorphism, $\pi$ is linear, $\pi(I)=I$, $\pi(ST)=\pi(S)\pi(T)$, $\pi(T^{*})=\pi(T)^{*}$ and $\tau_{\gamma}(\pi(T))=\tau_{\mu}(T)$; and by Marginals of a Noncommutative Law: the Isometry of GNS Spaces, the Trace-Preserving Embedding of Tracial Algebras and the Conditional Expectation §expectation, $E(b)=V^{*}bV$ and $\lVert E(b)\rVert_{\mathrm{op}}\le\lVert b\rVert_{\mathrm{op}}$ for $b\in\mathcal{M}_{\gamma}$. For $j\in[d]$ put $y_{j}=\widehat{x_{d+j}}^{\,\gamma}=L_{x_{d+j}}\Omega_{\gamma}\in\mathcal{H}_{\gamma}$. Then

$$z_{j}\Omega_{\mu}=V^{*}L_{x_{d+j}}V\Omega_{\mu}=V^{*}L_{x_{d+j}}\Omega_{\gamma}=V^{*}y_{j}.\qquad(0.1)$$

Moreover, for every $y\in\mathcal{H}_{\gamma}$ we have $\lVert V^{*}y\rVert\le\lVert y\rVert$: indeed $\lVert VV^{*}y\rVert^{2}=\langle V^{*}y,V^{*}VV^{*}y\rangle=\lVert V^{*}y\rVert^{2}$ because $V^{*}V=I$, and by Cauchy-Schwarz Inequality in a Complex Inner Product Space

$$\lVert V^{*}y\rVert^{2}=\langle V^{*}y,V^{*}y\rangle=\bigl|\langle VV^{*}y,y\rangle\bigr|\le\lVert VV^{*}y\rVert\,\lVert y\rVert=\lVert V^{*}y\rVert\,\lVert y\rVert,\qquad(0.2)$$

so $\lVert V^{*}y\rVert\le\lVert y\rVert$ (trivially if $V^{*}y=0$, and after division by $\lVert V^{*}y\rVert$ otherwise).

Step 1 (Claim 1, coupling). Let $S>0$ be real with $\nu\in\Sigma_{d,S}$. Let $\pi':\mathcal{M}_{\nu}\to\mathcal{M}_{\gamma}$ be the embedding of Marginals of a Noncommutative Law: the Isometry of GNS Spaces, the Trace-Preserving Embedding of Tracial Algebras and the Conditional Expectation for $\gamma$ and $a'$ (that lemma's $\mu$ being our $\nu=\gamma\circ\sigma_{a'}$), and write $L^{\nu}_{p}$ for the multiplication operators on $\mathcal{H}_{\nu}$. By Marginals of a Noncommutative Law: the Isometry of GNS Spaces, the Trace-Preserving Embedding of Tracial Algebras and the Conditional Expectation §homomorphism, $\pi'(L^{\nu}_{x_{j}})=L_{\sigma_{a'}(x_{j})}=L_{x_{d+j}}$; by Marginals of a Noncommutative Law: the Isometry of GNS Spaces, the Trace-Preserving Embedding of Tracial Algebras and the Conditional Expectation §embedding, $\pi'$ does not increase operator norms; and 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 §multiplication for $\nu\in\Sigma_{d,S}$, $\lVert L^{\nu}_{x_{j}}\rVert_{\mathrm{op}}\le S$. With the contraction property of $E$ from Step 0,

$$\lVert z_{j}\rVert_{\mathrm{op}}=\lVert E(L_{x_{d+j}})\rVert_{\mathrm{op}}\le\lVert L_{x_{d+j}}\rVert_{\mathrm{op}}=\lVert\pi'(L^{\nu}_{x_{j}})\rVert_{\mathrm{op}}\le\lVert L^{\nu}_{x_{j}}\rVert_{\mathrm{op}}\le S\qquad(j\in[d]).\qquad(1.1)$$

By the statement, $z$ is a self-adjoint $d$-tuple in $\mathcal{M}_{\mu}$, and $(\mathcal{H}_{\mu},\mathcal{M}_{\mu},\Omega_{\mu})$ is a tracial W*-probability space by The Tracial Algebra of a Noncommutative Law is a Tracial W*-Probability Space: the W*-Closure of the Left Multiplications §w-star. Hence Laws of Self-Adjoint Tuples in a Tracial W*-Probability Space: Moments, Affine Images, Couplings, Embeddings and L^2 Approximation §law and (1.1) give $\zeta=\mathrm{law}(z)\in\Sigma_{d,S}$. This is the last sentence of claim 1; with $S=R$ it gives $\zeta\in\Sigma_{d,R}$, and so $\zeta\in\Sigma_{d}$ by Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §law.

Let $L_{x}=(L_{x_{1}},\dots,L_{x_{d}})$ on $\mathcal{H}_{\mu}$, a self-adjoint $d$-tuple in $\mathcal{M}_{\mu}$ (statement). By The Tracial Algebra of a Noncommutative Law is a Tracial W*-Probability Space: the W*-Closure of the Left Multiplications §law, $\langle\Omega_{\mu},p(L_{x})\Omega_{\mu}\rangle=\mu(p)$ for $p\in\mathcal{P}_{d}$, so by Self-Adjoint Tuples in a Tracial W*-Probability Space and Their Laws §law the law of $L_{x}$ is $\mu$. By Laws of Self-Adjoint Tuples in a Tracial W*-Probability Space: Moments, Affine Images, Couplings, Embeddings and L^2 Approximation §coupling, applied to the tuples $L_{x}$ and $z$, the law $\beta$ of $s=(L_{x_{1}},\dots,L_{x_{d}},z_{1},\dots,z_{d})$ belongs to $\Pi(\mu,\zeta)$. This proves claim 1.

Step 2 (Claim 2, orthogonal splitting). Let $j\in[d]$. Since $x_{j}^{*}=x_{j}$ and $\widehat{x_{j}}^{\,\gamma}=\widehat{\iota^{1}(x_{j})}^{\,\gamma}=V\widehat{x_{j}}^{\,\mu}$ (Step 0), the vacuum identities for $\gamma\in\Sigma_{2d,R}$ and $\mu\in\Sigma_{d,R}$ and (0.1) give

$$\gamma(x_{j}x_{d+j})=\langle\widehat{x_{j}}^{\,\gamma},\widehat{x_{d+j}}^{\,\gamma}\rangle=\langle V\widehat{x_{j}}^{\,\mu},y_{j}\rangle=\langle\widehat{x_{j}}^{\,\mu},V^{*}y_{j}\rangle=\langle L_{x_{j}}\Omega_{\mu},z_{j}\Omega_{\mu}\rangle.$$

By Laws of Self-Adjoint Tuples in a Tracial W*-Probability Space: Moments, Affine Images, Couplings, Embeddings and L^2 Approximation §moments for the $2d$-tuple $s$ of Step 1 (with indices $j$ and $d+j$), $\beta(x_{j}x_{d+j})=\langle L_{x_{j}}\Omega_{\mu},z_{j}\Omega_{\mu}\rangle$. Hence

$$\gamma(x_{j}x_{d+j})=\beta(x_{j}x_{d+j})\qquad(j\in[d]).\qquad(2.1)$$

Next, $\iota^{2}(x_{j}x_{j})=x_{d+j}x_{d+j}$ by Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §homomorphism and Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §values, so with $\gamma\circ\iota^{2}=\nu$ and the vacuum identities, $\nu(x_{j}^{2})=\gamma(x_{d+j}x_{d+j})=\lVert y_{j}\rVert^{2}$. By Laws of Self-Adjoint Tuples in a Tracial W*-Probability Space: Moments, Affine Images, Couplings, Embeddings and L^2 Approximation §moments for the tuple $z$ and (0.1), $\zeta(x_{j}^{2})=\lVert z_{j}\Omega_{\mu}\rVert^{2}=\lVert V^{*}y_{j}\rVert^{2}$. Summing over $j$, with $M$ as in Noncommutative Laws, Couplings and the Wasserstein Distance: Standing Notation §laws,

$$M(\nu)=\sum_{j=1}^{d}\lVert y_{j}\rVert^{2},\qquad M(\zeta)=\sum_{j=1}^{d}\lVert V^{*}y_{j}\rVert^{2},\qquad(2.2)$$

and (0.2) gives $M(\zeta)\le M(\nu)$. Finally, Couplings of Noncommutative Laws: the Norm Bound, the Cost Identity, the Tensor, Diagonal and Swapped Couplings, Weak-Star Closedness, and Displacement Interpolants §cost, applied to $\gamma\in\Pi(\mu,\nu)$ and to $\beta\in\Pi(\mu,\zeta)$ (with $\mu,\zeta\in\Sigma_{d}$ by Step 1), gives

$$I(\gamma)=M(\mu)+M(\nu)-2\sum_{j=1}^{d}\gamma(x_{j}x_{d+j}),\qquad I(\beta)=M(\mu)+M(\zeta)-2\sum_{j=1}^{d}\beta(x_{j}x_{d+j}),$$

and subtracting, by (2.1), $I(\gamma)=I(\beta)+M(\nu)-M(\zeta)$. This proves claim 2.

Step 3 (A second coupling of $\zeta$ and $\nu$). For $j\in[d]$ let $g_{j}=\pi(z_{j})$, and let $s'=(g_{1},\dots,g_{d},L_{x_{d+1}},\dots,L_{x_{2d}})$, a $2d$-tuple of operators on $\mathcal{H}_{\gamma}$. By Step 0, $g_{j}\in\mathcal{M}_{\gamma}$, $g_{j}^{*}=\pi(z_{j}^{*})=\pi(z_{j})=g_{j}$ and $\lVert g_{j}\rVert_{\mathrm{op}}\le\lVert z_{j}\rVert_{\mathrm{op}}\le R$ by (1.1) with $S=R$. The operators $L_{x_{d+j}}$ are self-adjoint elements of $\mathcal{M}_{\gamma}$ (statement) with $\lVert L_{x_{d+j}}\rVert_{\mathrm{op}}\le R$ by (1.1). So $s'$ is a self-adjoint $2d$-tuple in the tracial W*-probability space $(\mathcal{H}_{\gamma},\mathcal{M}_{\gamma},\Omega_{\gamma})$ of The Tracial Algebra of a Noncommutative Law is a Tracial W*-Probability Space: the W*-Closure of the Left Multiplications §w-star; let $\eta$ be its law, $\eta(p)=\langle\Omega_{\gamma},p(s')\Omega_{\gamma}\rangle$. By Laws of Self-Adjoint Tuples in a Tracial W*-Probability Space: Moments, Affine Images, Couplings, Embeddings and L^2 Approximation §law, $\eta\in\Sigma_{2d,R}$.

Let $g=(g_{1},\dots,g_{d})$ and $h=(L_{x_{d+1}},\dots,L_{x_{2d}})$. For $q\in\mathcal{P}_{d}$, Evaluation of Noncommutative Polynomials at Bounded Operators: a Unital Homomorphism Compatible with Adjoints, Substitution, Representations and the GNS Multiplication Operators §transport, applied with $\mathcal{H}_{\mu}$, $\mathcal{H}_{\gamma}$, $d$, $z$, $\mathcal{M}_{\mu}$, $\pi$ in place of its $H$, $K$, $n$, $T$, $\mathcal{A}$, $\Phi$ (the set $\mathcal{M}_{\mu}$ contains $I$ and every $z_{j}$ and is closed under sums, complex multiples and composition, and $\pi$ is linear, multiplicative and unital, by Step 0), gives

$$q(z)\in\mathcal{M}_{\mu}\qquad\text{and}\qquad q(g)=\pi\bigl(q(z)\bigr).\qquad(3.1)$$

Hence $\langle\Omega_{\gamma},q(g)\Omega_{\gamma}\rangle=\tau_{\gamma}(\pi(q(z)))=\tau_{\mu}(q(z))=\langle\Omega_{\mu},q(z)\Omega_{\mu}\rangle=\zeta(q)$, so the law of $g$ is $\zeta$. For the second block, write $L^{\gamma}_{x}=(L_{x_{1}},\dots,L_{x_{2d}})$; then $a'(L^{\gamma}_{x})=h$ by Evaluation of Noncommutative Polynomials at Bounded Operators: a Unital Homomorphism Compatible with Adjoints, Substitution, Representations and the GNS Multiplication Operators §values, so Evaluation of Noncommutative Polynomials at Bounded Operators: a Unital Homomorphism Compatible with Adjoints, Substitution, Representations and the GNS Multiplication Operators §substitution and Evaluation of Noncommutative Polynomials at Bounded Operators: a Unital Homomorphism Compatible with Adjoints, Substitution, Representations and the GNS Multiplication Operators §gns (for $\gamma\in\Sigma_{2d}$) give $q(h)=(\iota^{2}(q))(L^{\gamma}_{x})=L_{\iota^{2}(q)}$, and $\langle\Omega_{\gamma},q(h)\Omega_{\gamma}\rangle=\gamma(\iota^{2}(q))=\nu(q)$. So the law of $h$ is $\nu$, and Laws of Self-Adjoint Tuples in a Tracial W*-Probability Space: Moments, Affine Images, Couplings, Embeddings and L^2 Approximation §coupling, applied to the tuples $g$ and $h$, gives $\eta\in\Pi(\zeta,\nu)$.

Step 4 (A martingale identity for $\eta$). We show

$$\eta\bigl(\iota^{1}(q)\,x_{d+j}\bigr)=\eta\bigl(\iota^{1}(q)\,x_{j}\bigr)\qquad(q\in\mathcal{P}_{d},\ j\in[d]).\qquad(4.1)$$

Let $k\in[2d]$. By the homomorphism and value rules of evaluation, $(\iota^{1}(q)x_{k})(s')=(\iota^{1}(q))(s')\,s'_{k}$; and $a(s')=g$ by Evaluation of Noncommutative Polynomials at Bounded Operators: a Unital Homomorphism Compatible with Adjoints, Substitution, Representations and the GNS Multiplication Operators §values, so Evaluation of Noncommutative Polynomials at Bounded Operators: a Unital Homomorphism Compatible with Adjoints, Substitution, Representations and the GNS Multiplication Operators §substitution and (3.1) give $(\iota^{1}(q))(s')=q(g)=\pi(q(z))$. With $\pi(q(z))^{*}=\pi(q(z)^{*})$ and $\pi(q(z)^{*})\Omega_{\gamma}=Vq(z)^{*}\Omega_{\mu}$ (Step 0; $q(z)^{*}\in\mathcal{M}_{\mu}$), we get

$$\eta\bigl(\iota^{1}(q)x_{k}\bigr)=\langle\Omega_{\gamma},\pi(q(z))\,s'_{k}\Omega_{\gamma}\rangle=\langle Vq(z)^{*}\Omega_{\mu},s'_{k}\Omega_{\gamma}\rangle.$$

For $k=d+j$, $s'_{k}\Omega_{\gamma}=y_{j}$ and, by (0.1), $\langle Vq(z)^{*}\Omega_{\mu},y_{j}\rangle=\langle q(z)^{*}\Omega_{\mu},V^{*}y_{j}\rangle=\langle q(z)^{*}\Omega_{\mu},z_{j}\Omega_{\mu}\rangle$. For $k=j$, $s'_{j}\Omega_{\gamma}=\pi(z_{j})\Omega_{\gamma}=Vz_{j}\Omega_{\mu}$ and $\langle Vq(z)^{*}\Omega_{\mu},Vz_{j}\Omega_{\mu}\rangle=\langle q(z)^{*}\Omega_{\mu},V^{*}Vz_{j}\Omega_{\mu}\rangle=\langle q(z)^{*}\Omega_{\mu},z_{j}\Omega_{\mu}\rangle$. The two values agree, proving (4.1).

Step 5 (Gluing freely over $\zeta$: the data). By Step 1, $\zeta,\mu\in\Sigma_{d,R}$, so The Noncommutative Wasserstein Distance: Existence of Optimal Couplings, Symmetry, Separation, a Moment Bound, Weak-Star Lower Semicontinuity, and Displacement Interpolation §attained gives a coupling $\alpha\in\Pi(\zeta,\mu)$ that is optimal, i.e. by The Noncommutative Quadratic Wasserstein Distance and Optimal Couplings §optimal and The Noncommutative Wasserstein Distance: Existence of Optimal Couplings, Symmetry, Separation, a Moment Bound, Weak-Star Lower Semicontinuity, and Displacement Interpolation §symmetry,

$$I(\alpha)=W_{2}(\zeta,\mu)^{2}=W_{2}(\mu,\zeta)^{2}.\qquad(5.1)$$

By Couplings of Noncommutative Laws: the Norm Bound, the Cost Identity, the Tensor, Diagonal and Swapped Couplings, Weak-Star Closedness, and Displacement Interpolants §bound, $\alpha\in\Sigma_{2d,R}$; also $\eta\in\Sigma_{2d,R}$ (Step 3). Put $m_{1}=m_{2}=2d$, $n=d$, $\gamma_{1}=\alpha$, $\gamma_{2}=\eta$ and $a^{1}=a^{2}=a$. Since $\sigma_{a}=\iota^{1}$, Couplings of Two Noncommutative Laws and Their Quadratic Cost §coupling gives $\alpha\circ\sigma_{a}=\zeta=\eta\circ\sigma_{a}$, so the data of Two Noncommutative Laws with a Common Marginal: Standing Notation for Their Amalgamated Free Product §data hold. The letters clash: the common marginal, which that setting (and every result stated in it) calls $\mu$, is here $\zeta$; our $\mu$ plays no role in the amalgamation. With this renaming we use the notation of its algebras and its embeddings and expectations: $N=\mathcal{M}_{\zeta}$ on $\mathcal{H}_{\zeta}$, $A_{1}=\mathcal{M}_{\alpha}$ on $\mathcal{H}_{\alpha}$, $A_{2}=\mathcal{M}_{\eta}$ on $\mathcal{H}_{\eta}$, $\pi_{\varepsilon}:N\to A_{\varepsilon}$ and $E_{\varepsilon}:A_{\varepsilon}\to N$; let $V_{\varepsilon}$ be the isometry of Marginals of a Noncommutative Law: the Isometry of GNS Spaces, the Trace-Preserving Embedding of Tracial Algebras and the Conditional Expectation §isometry for $\gamma_{\varepsilon}$ and $a$. Let $\mathcal{H}$, $\Omega$, $\varphi$ and $\Lambda_{\varepsilon}$ be the amalgamated free product space, its vacuum vector, its vacuum state and the actions, as in The Amalgamated Free Product of Two Noncommutative Laws over a Common Marginal: Representations, Amalgamation, Freeness and the Tracial Vacuum State; by The Amalgamated Free Product Space of Two Noncommutative Laws over a Common Marginal, Its Vacuum Vector and Vacuum State §state, $\varphi(X)=\langle\Omega,X\Omega\rangle_{\mathcal{H}}$, so $\varphi$ is linear. Write $L^{1}_{q}$, $L^{2}_{q}$ for the multiplication operators on $\mathcal{H}_{\alpha}$, $\mathcal{H}_{\eta}$ ($q\in\mathcal{P}_{2d}$), and $L^{\zeta}_{q}$ for those on $\mathcal{H}_{\zeta}$ ($q\in\mathcal{P}_{d}$). They lie in $A_{1}$, $A_{2}$, $N$ respectively (preliminaries); 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 §multiplication 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 §adjoint for $\alpha,\eta\in\Sigma_{2d,R}$, $\lVert L^{\varepsilon}_{x_{i}}\rVert_{\mathrm{op}}\le R$ and $(L^{\varepsilon}_{x_{i}})^{*}=L^{\varepsilon}_{x_{i}}$ for $i\in[2d]$; and by Marginals of a Noncommutative Law: the Isometry of GNS Spaces, the Trace-Preserving Embedding of Tracial Algebras and the Conditional Expectation §homomorphism and $\sigma_{a}(x_{j})=x_{j}$,

$$\pi_{\varepsilon}(L^{\zeta}_{x_{j}})=L^{\varepsilon}_{x_{j}}\qquad(\varepsilon\in\{1,2\},\ j\in[d]).\qquad(5.2)$$

We use repeatedly, for $\varepsilon\in\{1,2\}$, $b\in A_{\varepsilon}$ and $X,Y\in N$: $\pi_{\varepsilon}$ is linear and multiplicative with $\pi_{\varepsilon}(I)=I$ and $\tau_{\gamma_{\varepsilon}}(\pi_{\varepsilon}(X))=\tau_{\zeta}(X)$ (Marginals of a Noncommutative Law: the Isometry of GNS Spaces, the Trace-Preserving Embedding of Tracial Algebras and the Conditional Expectation §homomorphism); $E_{\varepsilon}(\pi_{\varepsilon}(X)\,b\,\pi_{\varepsilon}(Y))=X\,E_{\varepsilon}(b)\,Y$, $\tau_{\zeta}(E_{\varepsilon}(b))=\tau_{\gamma_{\varepsilon}}(b)$ and $\langle\xi,E_{\varepsilon}(b)\xi'\rangle=\langle V_{\varepsilon}\xi,bV_{\varepsilon}\xi'\rangle$ for $\xi,\xi'\in\mathcal{H}_{\zeta}$ (Marginals of a Noncommutative Law: the Isometry of GNS Spaces, the Trace-Preserving Embedding of Tracial Algebras and the Conditional Expectation §expectation); $\Lambda_{\varepsilon}$ is linear and multiplicative with $\Lambda_{\varepsilon}(b^{*})=\Lambda_{\varepsilon}(b)^{*}$ and $\lVert\Lambda_{\varepsilon}(b)\rVert_{\mathrm{op}}\le\lVert b\rVert_{\mathrm{op}}$ (The Amalgamated Free Product of Two Noncommutative Laws over a Common Marginal: Representations, Amalgamation, Freeness and the Tracial Vacuum State §representation); $\Lambda_{1}(\pi_{1}(X))=\Lambda_{2}(\pi_{2}(X))$ (The Amalgamated Free Product of Two Noncommutative Laws over a Common Marginal: Representations, Amalgamation, Freeness and the Tracial Vacuum State §amalgamation); and $\varphi(\Lambda_{\varepsilon}(b))=\tau_{\gamma_{\varepsilon}}(b)$, $\lVert\Omega\rVert_{\mathcal{H}}=1$ (The Amalgamated Free Product of Two Noncommutative Laws over a Common Marginal: Representations, Amalgamation, Freeness and the Tracial Vacuum State §marginals).

Step 6 (The expectation of the $\nu$-variables of $\eta$). We show $E_{2}(L^{2}_{x_{d+j}})=L^{\zeta}_{x_{j}}$ for $j\in[d]$. Let $q\in\mathcal{P}_{d}$. By Marginals of a Noncommutative Law: the Isometry of GNS Spaces, the Trace-Preserving Embedding of Tracial Algebras and the Conditional Expectation §isometry, $V_{2}\widehat{q}^{\,\zeta}=\widehat{\iota^{1}(q)}^{\,\eta}$ and $V_{2}\Omega_{\zeta}=\Omega_{\eta}$; by Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §adjoint, Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §homomorphism and Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §values, $\iota^{1}(q)^{*}=\iota^{1}(q^{*})$ and $\iota^{1}(q^{*})x_{j}=\iota^{1}(q^{*}x_{j})$. With the inner-product rule for $E_{2}$ of Step 5, the vacuum identities for $\eta\in\Sigma_{2d,R}$ and $\zeta\in\Sigma_{d,R}$, (4.1) for $q^{*}$, and $\eta\circ\iota^{1}=\zeta$ (Step 3),

$$\langle\widehat{q}^{\,\zeta},E_{2}(L^{2}_{x_{d+j}})\Omega_{\zeta}\rangle=\langle\widehat{\iota^{1}(q)}^{\,\eta},\widehat{x_{d+j}}^{\,\eta}\rangle=\eta\bigl(\iota^{1}(q^{*})x_{d+j}\bigr)=\eta\bigl(\iota^{1}(q^{*})x_{j}\bigr)=\zeta(q^{*}x_{j})=\langle\widehat{q}^{\,\zeta},L^{\zeta}_{x_{j}}\Omega_{\zeta}\rangle.$$

So $D=E_{2}(L^{2}_{x_{d+j}})-L^{\zeta}_{x_{j}}\in N$ satisfies $\langle\widehat{q}^{\,\zeta},D\Omega_{\zeta}\rangle=0$ for every $q\in\mathcal{P}_{d}$. The classes are the image of $\mathcal{P}_{d}$ under the canonical map of the completion $\mathcal{H}_{\zeta}$ (The Complex GNS Space of a Tracial State on Noncommutative Polynomials §gns and The Complex GNS Space of a Tracial State on Noncommutative Polynomials §classes), which is dense by The Complex Hilbert Completion is a Complex Hilbert Space Containing a Dense Isometric Image, and Bounded Complex-Linear Maps Extend to It §dense; choosing classes $\widehat{q_{k}}^{\,\zeta}$ converging to $D\Omega_{\zeta}$, continuity of the inner product gives $\lVert D\Omega_{\zeta}\rVert^{2}=\lim_{k}\langle\widehat{q_{k}}^{\,\zeta},D\Omega_{\zeta}\rangle=0$. Thus $D\Omega_{\zeta}=0$, and $D=0$ by The Tracial Algebra of a Noncommutative Law: a Norm-Closed Unital *-Algebra with a Faithful Positive Trace, Determined by Vacuum Vectors, Closed under Square Roots §vacuum.

Step 7 (Mixed moments across the free product). We show $\varphi(\Lambda_{1}(b)\Lambda_{2}(c))=\tau_{\zeta}(E_{1}(b)E_{2}(c))$ for $b\in A_{1}$ and $c\in A_{2}$. Let $X=E_{1}(b)$, $Y=E_{2}(c)$, and let $b^{\circ}=b-\pi_{1}(X)$ and $c^{\circ}=c-\pi_{2}(Y)$ be the centred parts, so $b^{\circ}\in A_{1}^{\circ}$, $c^{\circ}\in A_{2}^{\circ}$, i.e. $E_{1}(b^{\circ})=0$ and $E_{2}(c^{\circ})=0$. As $\Lambda_{1},\Lambda_{2},\varphi$ are linear and composition is bilinear, $\varphi(\Lambda_{1}(b)\Lambda_{2}(c))$ is the sum of the four terms below, evaluated with the rules of Step 5. First,

$$\varphi\bigl(\Lambda_{1}(\pi_{1}(X))\Lambda_{2}(\pi_{2}(Y))\bigr)=\varphi\bigl(\Lambda_{2}(\pi_{2}(XY))\bigr)=\tau_{\eta}(\pi_{2}(XY))=\tau_{\zeta}(XY).$$

Second,

$$\varphi\bigl(\Lambda_{1}(\pi_{1}(X))\Lambda_{2}(c^{\circ})\bigr)=\varphi\bigl(\Lambda_{2}(\pi_{2}(X)c^{\circ})\bigr)=\tau_{\eta}(\pi_{2}(X)c^{\circ}\pi_{2}(I))=\tau_{\zeta}\bigl(X\,E_{2}(c^{\circ})\bigr)=0.$$

Third,

$$\varphi\bigl(\Lambda_{1}(b^{\circ})\Lambda_{2}(\pi_{2}(Y))\bigr)=\varphi\bigl(\Lambda_{1}(b^{\circ}\pi_{1}(Y))\bigr)=\tau_{\alpha}(\pi_{1}(I)b^{\circ}\pi_{1}(Y))=\tau_{\zeta}\bigl(E_{1}(b^{\circ})\,Y\bigr)=0.$$

Fourth, $((1,b^{\circ}),(2,c^{\circ}))$ is an alternating tuple of type $(1,2)$, so by The Amalgamated Free Product of Two Noncommutative Laws over a Common Marginal: Representations, Amalgamation, Freeness and the Tracial Vacuum State §freeness, $\Lambda_{1}(b^{\circ})\Lambda_{2}(c^{\circ})\Omega=\Xi(b^{\circ},c^{\circ})$ and $\varphi(\Lambda_{1}(b^{\circ})\Lambda_{2}(c^{\circ}))=\langle\Omega,\Xi(b^{\circ},c^{\circ})\rangle_{\mathcal{H}}=0$. Summing, $\varphi(\Lambda_{1}(b)\Lambda_{2}(c))=\tau_{\zeta}(XY)$.

Step 8 (The glued coupling $\psi$). Define $f:[2d]\to\{1,2\}$ and $c_{i}\in A_{f(i)}$ by $f(i)=1$, $c_{i}=L^{1}_{x_{d+i}}$ and $f(d+i)=2$, $c_{d+i}=L^{2}_{x_{d+i}}$ for $i\in[d]$, and let $T=(T_{1},\dots,T_{2d})$ with $T_{i}=\Lambda_{f(i)}(c_{i})\in\mathcal{L}(\mathcal{H})$. By Step 5, $T_{i}^{*}=\Lambda_{f(i)}(c_{i}^{*})=T_{i}$ and $\lVert T_{i}\rVert_{\mathrm{op}}\le\lVert c_{i}\rVert_{\mathrm{op}}\le R$, and $\lVert\Omega\rVert_{\mathcal{H}}=1$; so The Law of a Tuple of Bounded Self-Adjoint Operators in a Tracial Vector State applies with $2d$, $\mathcal{H}$, $T$ in place of its $n$, $H$, $T$. Let $\psi=\lambda_{T}$, i.e. $\psi(p)=\langle\Omega,p(T)\Omega\rangle=\varphi(p(T))$ for $p\in\mathcal{P}_{2d}$.

To apply The Law of a Tuple of Bounded Self-Adjoint Operators in a Tracial Vector State §law we show $\varphi(T_{u}T_{v})=\varphi(T_{v}T_{u})$ for all words $u,v\in W_{2d}$. If $u=\varnothing$, then $T_{u}=I$ by Evaluation of Noncommutative Polynomials at a Tuple of Bounded Operators §word-products and both sides equal $\varphi(T_{v})$; likewise if $v=\varnothing$. For nonempty $u,v$, $T_{u}T_{v}=T_{uv}$ and $T_{v}T_{u}=T_{vu}$ by Evaluation of Noncommutative Polynomials at Bounded Operators: a Unital Homomorphism Compatible with Adjoints, Substitution, Representations and the GNS Multiplication Operators §values, so it suffices to show $\varphi(T_{uv})=\varphi(T_{vu})$ for all nonempty $u,v$. First, for a letter $(i)$ and a nonempty word $w$ of length $l$ with letters $w_{1},\dots,w_{l}$, we have $\varphi(T_{(i)w})=\varphi(T_{w(i)})$: indeed $T_{(i)w}=T_{i}T_{w}$, $T_{w(i)}=T_{w}T_{i}$ and $T_{w}=T_{w_{1}}\cdots T_{w_{l}}$ by Evaluation of Noncommutative Polynomials at Bounded Operators: a Unital Homomorphism Compatible with Adjoints, Substitution, Representations and the GNS Multiplication Operators §values and Evaluation of Noncommutative Polynomials at a Tuple of Bounded Operators §word-products, so by associativity of composition $T_{(i)w}$ is the product $\Lambda_{f(i)}(c_{i})\Lambda_{f(w_{1})}(c_{w_{1}})\cdots\Lambda_{f(w_{l})}(c_{w_{l}})$ of $l+1\ge2$ factors and $T_{w(i)}$ is the product of the same factors with the first one moved to the end, and the claim is The Amalgamated Free Product of Two Noncommutative Laws over a Common Marginal: Representations, Amalgamation, Freeness and the Tracial Vacuum State §trace with $r=l+1$. Now induct on the length $k$ of $u$, for all nonempty $v$ at once. If $k=1$, $u$ is a letter by Basic Properties of Words: Associativity, Reversal, Finitely Many Factorisations, and Countability §last-letter and the claim is the rotation just shown. If the claim holds for words of length $k$ and $u$ has length $k+1$, then $u=u'(i)$ with $u'$ of length $k$ by Basic Properties of Words: Associativity, Reversal, Finitely Many Factorisations, and Countability §last-letter; by Basic Properties of Words: Associativity, Reversal, Finitely Many Factorisations, and Countability §monoid, $uv=u'((i)v)$, $((i)v)u'=(i)(vu')$ with $(i)v$ and $vu'$ nonempty, and $(vu')(i)=v(u'(i))=vu$; so the induction hypothesis (for $u'$ and $(i)v$) and the rotation (for $i$ and $vu'$) give $\varphi(T_{uv})=\varphi(T_{(i)(vu')})=\varphi(T_{(vu')(i)})=\varphi(T_{vu})$. Hence $\psi\in\Sigma_{2d,R}$ by The Law of a Tuple of Bounded Self-Adjoint Operators in a Tracial Vector State §law; in particular $\psi$ is a tracial state by Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §norm-bound.

Marginals of $\psi$. Let $p\in\mathcal{P}_{d}$. As $a(T)=(T_{1},\dots,T_{d})$ by Evaluation of Noncommutative Polynomials at Bounded Operators: a Unital Homomorphism Compatible with Adjoints, Substitution, Representations and the GNS Multiplication Operators §values, Evaluation of Noncommutative Polynomials at Bounded Operators: a Unital Homomorphism Compatible with Adjoints, Substitution, Representations and the GNS Multiplication Operators §substitution gives $(\iota^{1}(p))(T)=p(T_{1},\dots,T_{d})$. Apply Evaluation of Noncommutative Polynomials at Bounded Operators: a Unital Homomorphism Compatible with Adjoints, Substitution, Representations and the GNS Multiplication Operators §transport with $\mathcal{H}_{\alpha}$, $\mathcal{H}$, $d$, $(L^{1}_{x_{d+1}},\dots,L^{1}_{x_{2d}})$, $A_{1}$, $\Lambda_{1}$ in place of its $H$, $K$, $n$, $T$, $\mathcal{A}$, $\Phi$ (the set $A_{1}$ has the required closure properties by the preliminaries, and $\Lambda_{1}$ is linear, multiplicative and unital by Step 5): $p(T_{1},\dots,T_{d})=\Lambda_{1}\bigl(p(L^{1}_{x_{d+1}},\dots,L^{1}_{x_{2d}})\bigr)$. With $L^{1}_{x}=(L^{1}_{x_{1}},\dots,L^{1}_{x_{2d}})$ we have $a'(L^{1}_{x})=(L^{1}_{x_{d+1}},\dots,L^{1}_{x_{2d}})$, so Evaluation of Noncommutative Polynomials at Bounded Operators: a Unital Homomorphism Compatible with Adjoints, Substitution, Representations and the GNS Multiplication Operators §substitution and Evaluation of Noncommutative Polynomials at Bounded Operators: a Unital Homomorphism Compatible with Adjoints, Substitution, Representations and the GNS Multiplication Operators §gns (for $\alpha\in\Sigma_{2d}$) give $p(L^{1}_{x_{d+1}},\dots,L^{1}_{x_{2d}})=(\iota^{2}(p))(L^{1}_{x})=L^{1}_{\iota^{2}(p)}$. Hence, as $\alpha\in\Pi(\zeta,\mu)$,

$$\psi(\iota^{1}(p))=\varphi\bigl(\Lambda_{1}(L^{1}_{\iota^{2}(p)})\bigr)=\tau_{\alpha}(L^{1}_{\iota^{2}(p)})=\alpha(\iota^{2}(p))=\mu(p).$$

In the same way, with $a'(T)=(T_{d+1},\dots,T_{2d})=(\Lambda_{2}(L^{2}_{x_{d+1}}),\dots,\Lambda_{2}(L^{2}_{x_{2d}}))$ and $\mathcal{H}_{\eta}$, $A_{2}$, $\Lambda_{2}$, $\eta$ in place of $\mathcal{H}_{\alpha}$, $A_{1}$, $\Lambda_{1}$, $\alpha$, we get $\psi(\iota^{2}(p))=\tau_{\eta}(L^{2}_{\iota^{2}(p)})=\eta(\iota^{2}(p))=\nu(p)$, as $\eta\in\Pi(\zeta,\nu)$. By Couplings of Two Noncommutative Laws and Their Quadratic Cost §coupling, $\psi\in\Pi(\mu,\nu)$.

Step 9 (Claim 3, upper bound). Let $j\in[d]$. By the homomorphism and value rules of evaluation, Step 7, Step 6, the expectation and trace rules of Step 5 (with $\pi_{1}(I)=I$) and (5.2), then 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, the identity $\tau_{\alpha}(L^{1}_{p})=\alpha(p)$ of the preliminaries, and condition (c) of Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §tracial-state for $\alpha$,

$$\psi(x_{j}x_{d+j})=\varphi(T_{j}T_{d+j})=\tau_{\zeta}\bigl(E_{1}(L^{1}_{x_{d+j}})\,L^{\zeta}_{x_{j}}\bigr)=\tau_{\zeta}\bigl(E_{1}(L^{1}_{x_{d+j}}\pi_{1}(L^{\zeta}_{x_{j}}))\bigr)=\tau_{\alpha}(L^{1}_{x_{d+j}}L^{1}_{x_{j}})=\alpha(x_{d+j}x_{j})=\alpha(x_{j}x_{d+j}).$$

By Couplings of Noncommutative Laws: the Norm Bound, the Cost Identity, the Tensor, Diagonal and Swapped Couplings, Weak-Star Closedness, and Displacement Interpolants §cost, applied to $\psi\in\Pi(\mu,\nu)$ and to $\alpha\in\Pi(\zeta,\mu)$ (with $\zeta,\mu\in\Sigma_{d}$),

$$I(\psi)=M(\mu)+M(\nu)-2\sum_{j=1}^{d}\alpha(x_{j}x_{d+j}),\qquad I(\alpha)=M(\zeta)+M(\mu)-2\sum_{j=1}^{d}\alpha(x_{j}x_{d+j}),$$

so $I(\psi)=I(\alpha)+M(\nu)-M(\zeta)$. By The Noncommutative Quadratic Wasserstein Distance and Optimal Couplings §distance, $W_{2}(\mu,\nu)^{2}$ is the infimum of the costs of couplings of $\mu$ and $\nu$, so $W_{2}(\mu,\nu)^{2}\le I(\psi)$, and with (5.1)

$$W_{2}(\mu,\nu)^{2}\le W_{2}(\mu,\zeta)^{2}+M(\nu)-M(\zeta).\qquad(9.1)$$

As $\gamma\in\Pi(\mu,\nu)$ was arbitrary, this proves claim 3.

Step 10 (Claim 4, optimal couplings). Let $\gamma$ be optimal, so $I(\gamma)=W_{2}(\mu,\nu)^{2}$ by The Noncommutative Quadratic Wasserstein Distance and Optimal Couplings §optimal. By Laws of Self-Adjoint Tuples in a Tracial W*-Probability Space: Moments, Affine Images, Couplings, Embeddings and L^2 Approximation §coupling for the tuples $L_{x}$ and $z$ of Step 1 (whose laws are $\mu$ and $\zeta$), $I(\beta)\ge W_{2}(\mu,\zeta)^{2}$. With claim 2 and (9.1),

$$W_{2}(\mu,\nu)^{2}=I(\gamma)=I(\beta)+M(\nu)-M(\zeta)\ge W_{2}(\mu,\zeta)^{2}+M(\nu)-M(\zeta)\ge W_{2}(\mu,\nu)^{2}.$$

Hence equality holds throughout: $W_{2}(\mu,\nu)^{2}=W_{2}(\mu,\zeta)^{2}+M(\nu)-M(\zeta)$ and $I(\beta)=W_{2}(\mu,\zeta)^{2}$, so $\beta\in\Pi(\mu,\zeta)$ is optimal by The Noncommutative Quadratic Wasserstein Distance and Optimal Couplings §optimal. This proves claim 4.

Step 11 (Claim 5, variational formula). By claim 3, $W_{2}(\mu,\nu)^{2}$ is a lower bound of the set $\{W_{2}(\mu,\zeta_{\gamma})^{2}+M(\nu)-M(\zeta_{\gamma}):\ \gamma\in\Pi(\mu,\nu)\}$, and by claim 4 it is the value of this expression at every optimal $\gamma$. Since $\mu,\nu\in\Sigma_{d,R}$, The Noncommutative Wasserstein Distance: Existence of Optimal Couplings, Symmetry, Separation, a Moment Bound, Weak-Star Lower Semicontinuity, and Displacement Interpolation §attained provides an optimal $\gamma\in\Pi(\mu,\nu)$, so $W_{2}(\mu,\nu)^{2}$ belongs to the set and is its least element. This proves claim 5.