Proof of The Noncommutative Wasserstein Distance: Existence of Optimal Couplings, Symmetry, Separation, a Moment Bound, Weak-Star Lower Semicontinuity, and Displacement Interpolation

We use the following items: Couplings of Two Noncommutative Laws and Their Quadratic Cost §marginals, Couplings of Two Noncommutative Laws and Their Quadratic Cost §coupling and Couplings of Two Noncommutative Laws and Their Quadratic Cost §cost; Couplings of Noncommutative Laws: the Norm Bound, the Cost Identity, the Tensor, Diagonal and Swapped Couplings, Weak-Star Closedness, and Displacement Interpolants §cost, Couplings of Noncommutative Laws: the Norm Bound, the Cost Identity, the Tensor, Diagonal and Swapped Couplings, Weak-Star Closedness, and Displacement Interpolants §tensor, Couplings of Noncommutative Laws: the Norm Bound, the Cost Identity, the Tensor, Diagonal and Swapped Couplings, Weak-Star Closedness, and Displacement Interpolants §diagonal, Couplings of Noncommutative Laws: the Norm Bound, the Cost Identity, the Tensor, Diagonal and Swapped Couplings, Weak-Star Closedness, and Displacement Interpolants §swap, Couplings of Noncommutative Laws: the Norm Bound, the Cost Identity, the Tensor, Diagonal and Swapped Couplings, Weak-Star Closedness, and Displacement Interpolants §closed and Couplings of Noncommutative Laws: the Norm Bound, the Cost Identity, the Tensor, Diagonal and Swapped Couplings, Weak-Star Closedness, and Displacement Interpolants §interpolation; The Noncommutative Quadratic Wasserstein Distance and Optimal Couplings §distance and The Noncommutative Quadratic Wasserstein Distance and Optimal Couplings §optimal; Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §tracial-state and Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §law; Basic Properties of Noncommutative Laws: Adjoints, the Self-Adjoint Pairing, Cauchy-Schwarz, Monotonicity in the Bound, and Laws of Constant Tuples §cauchy-schwarz; The Algebra of Noncommutative Polynomials in Finitely Many Self-Adjoint Variables §monomials; Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §linear-extension, Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §monomials, Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §algebra, Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §adjoint and Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §self-adjoint; Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §values and Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §homomorphism; Basic Properties of Words: Associativity, Reversal, Finitely Many Factorisations, and Countability §last-letter; Weak-Star Convergence of Noncommutative Laws §weak-star; Limit of a Sequence of Real Numbers; claim 1 of Order Properties of Limits of Real Sequences; Strictly Increasing Sequences of Natural Numbers Dominate Their Index; claim 4 of Approximation Property of the Supremum and the Infimum in $\mathbb{R}$; Axiom of Countable Choice; Comparison of Real Numbers with Arbitrary Positive Slack §slack-above; Existence and Uniqueness of the Nonnegative Square Root; claims 3 and 5 of Elementary Arithmetic in an Ordered Field; claims 1, 3, 5, 6, 7, 8 and 10 of Elementary Order Arithmetic in an Ordered Field; claims 2 and 3 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field; claim 4 of Properties of Finite Sums of Vectors; Finite Sum Notation in a Field; claim 5 of Properties of Finite Sums; condition 1 of The Complex Numbers; Real and Imaginary Parts of a Complex Number; claims 1, 3 and 6 of Properties of Complex Conjugation and Modulus; claims 1 and 3 of Properties of the Order on the Natural Numbers; the setting of Real Powers Through the Exponential, and Elementary Asymptotic Tools: Monotonicity, Null Sequences of Negative Powers, Exponential Domination, Integer Rounding, and Square-Root and Exponential Inequalities (natural numbers as real numbers $\ge1$) and its claim 4; and Principle of Induction for the Natural Numbers.

Step 0. (Notation and two elementary facts.) By Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §law, $\Sigma_{d,R}\subseteq\Sigma_{d}$, so $W_{2}$ is defined on pairs from $\Sigma_{d,R}$. For $\lambda,\lambda'\in\Sigma_{d}$ let $C(\lambda,\lambda')=\{I(\gamma):\ \gamma\in\Pi(\lambda,\lambda')\}$ and $V(\lambda,\lambda')=\inf C(\lambda,\lambda')$. By The Noncommutative Quadratic Wasserstein Distance and Optimal Couplings §distance and the remarks preceding it, $C(\lambda,\lambda')$ is a nonempty set of nonnegative reals, $0\le V(\lambda,\lambda')$, and $W_{2}(\lambda,\lambda')$ is the nonnegative real number whose square is $V(\lambda,\lambda')$ (Existence and Uniqueness of the Nonnegative Square Root); thus $0\le W_{2}(\lambda,\lambda')$ and $W_{2}(\lambda,\lambda')^{2}=V(\lambda,\lambda')$. (0a) For every $\gamma\in\Pi(\lambda,\lambda')$ we have $W_{2}(\lambda,\lambda')^{2}\le I(\gamma)$, as the infimum is a lower bound of $C(\lambda,\lambda')$. (0b) If $a$ is a real number with $0\le a$ and $W_{2}(\lambda,\lambda')^{2}\le a^{2}$, then $W_{2}(\lambda,\lambda')\le a$, by claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field, as $0\le W_{2}(\lambda,\lambda')$.

Step 1. (Claim 1, optimal couplings exist.) Write $V=V(\mu,\nu)$. Let $m\in\mathbb{N}$. Regarded as a real number, $m\ge1>0$ (setting of Real Powers Through the Exponential, and Elementary Asymptotic Tools: Monotonicity, Null Sequences of Negative Powers, Exponential Domination, Integer Rounding, and Square-Root and Exponential Inequalities and claims 6 and 2 of Elementary Order Arithmetic in an Ordered Field), so $0<m^{-1}$ by claim 7 of Elementary Order Arithmetic in an Ordered Field. As $C(\mu,\nu)$ is nonempty and bounded below by $0$, claim 4 of Approximation Property of the Supremum and the Infimum in $\mathbb{R}$ with $\varepsilon=m^{-1}$ shows that the set

is nonempty. By Axiom of Countable Choice (this is the use of countable choice in this step) there is a sequence $(\gamma_{m})_{m\in\mathbb{N}}$ with $\gamma_{m}\in A_{m}$ for every $m$.

The constant sequences $\mu_{m}=\mu$ and $\nu_{m}=\nu$ lie in $\Sigma_{d,R}$ and converge weak-star to $\mu$ and $\nu$: for every $p\in\mathcal{P}_{d}$ the real sequences $(\operatorname{Re}\mu_{m}(p))_{m}$ and $(\operatorname{Im}\mu_{m}(p))_{m}$ are constant, so $|\operatorname{Re}\mu_{m}(p)-\operatorname{Re}\mu(p)|=0<\varepsilon$ for every $m$ and every $\varepsilon>0$, and likewise for the imaginary parts and for $\nu$; this is Weak-Star Convergence of Noncommutative Laws §weak-star with Limit of a Sequence of Real Numbers. Since $\gamma_{m}\in\Pi(\mu_{m},\nu_{m})$, Couplings of Noncommutative Laws: the Norm Bound, the Cost Identity, the Tensor, Diagonal and Swapped Couplings, Weak-Star Closedness, and Displacement Interpolants §closed gives a strictly increasing sequence $(m_{i})_{i\in\mathbb{N}}$ in $\mathbb{N}$ and $\gamma\in\Pi(\mu,\nu)$ such that $(I(\gamma_{m_{i}}))_{i\in\mathbb{N}}$ converges to $I(\gamma)$.

By (0a), $V\le I(\gamma)$. For the reverse inequality let $\varepsilon>0$ and $\eta=\varepsilon\cdot2^{-1}$, so $0<\eta$ and $\eta+\eta=\varepsilon$ by claim 8 of Elementary Order Arithmetic in an Ordered Field. By claim 4 of Real Powers Through the Exponential, and Elementary Asymptotic Tools: Monotonicity, Null Sequences of Negative Powers, Exponential Domination, Integer Rounding, and Square-Root and Exponential Inequalities there is $N_{0}\in\mathbb{N}$ with $\eta^{-1}<N$ for every natural number $N\ge N_{0}$; for such $N$, multiplying $\eta^{-1}<N$ by the positive number $N^{-1}\eta$ (claims 7 and 5 of Elementary Order Arithmetic in an Ordered Field) gives $N^{-1}<\eta$ by claim 10 of Elementary Order Arithmetic in an Ordered Field. By Limit of a Sequence of Real Numbers there is $N_{1}\in\mathbb{N}$ with $|I(\gamma_{m_{i}})-I(\gamma)|<\eta$ for every $i\ge N_{1}$. Let $i$ be the larger of $N_{0}$ and $N_{1}$ (claim 3 of Properties of the Order on the Natural Numbers). By Strictly Increasing Sequences of Natural Numbers Dominate Their Index and claim 1 of Properties of the Order on the Natural Numbers, $N_{0}\le i\le m_{i}$, so $m_{i}^{-1}<\eta$. For a real number $y$ we have $y=y+0i$, so $\operatorname{Re}y=y$ (Real and Imaginary Parts of a Complex Number) and $-y\le|y|$ by claim 6 of Properties of Complex Conjugation and Modulus; with $y=I(\gamma_{m_{i}})-I(\gamma)$ this gives $I(\gamma)-I(\gamma_{m_{i}})<\eta$, hence $I(\gamma)<I(\gamma_{m_{i}})+\eta$ by claim 1 of Elementary Order Arithmetic in an Ordered Field. Since $\gamma_{m_{i}}\in A_{m_{i}}$, $I(\gamma_{m_{i}})<V+m_{i}^{-1}<V+\eta$, and by claim 3 of Elementary Order Arithmetic in an Ordered Field

As $\varepsilon>0$ was arbitrary, Comparison of Real Numbers with Arbitrary Positive Slack §slack-above gives $I(\gamma)\le V$. By antisymmetry of the order, $I(\gamma)=V=W_{2}(\mu,\nu)^{2}$, so $\gamma$ is optimal in the sense of The Noncommutative Quadratic Wasserstein Distance and Optimal Couplings §optimal. The argument used only $\mu,\nu\in\Sigma_{d,R}$, so it applies to every pair in $\Sigma_{d,R}$.

Step 2. (Claim 2, symmetry.) If $\gamma\in\Pi(\mu,\nu)$, then $\gamma\circ s\in\Pi(\nu,\mu)$ and $I(\gamma\circ s)=I(\gamma)$ by Couplings of Noncommutative Laws: the Norm Bound, the Cost Identity, the Tensor, Diagonal and Swapped Couplings, Weak-Star Closedness, and Displacement Interpolants §swap; hence $C(\mu,\nu)\subseteq C(\nu,\mu)$. The same clause applied to the pair $(\nu,\mu)$ gives $C(\nu,\mu)\subseteq C(\mu,\nu)$. So the two sets are equal, their infima are equal, and so are the nonnegative square roots of the infima: $W_{2}(\mu,\nu)=W_{2}(\nu,\mu)$.

Step 3. (First half of claim 3, separation.) By Couplings of Noncommutative Laws: the Norm Bound, the Cost Identity, the Tensor, Diagonal and Swapped Couplings, Weak-Star Closedness, and Displacement Interpolants §diagonal, $\mu\circ\delta\in\Pi(\mu,\mu)$ and $I(\mu\circ\delta)=0$, so $V(\mu,\mu)\le0$; with $0\le V(\mu,\mu)$ (Step 0) antisymmetry gives $V(\mu,\mu)=0$. Since $0\le0$ and $0\cdot0=0$, uniqueness in Existence and Uniqueness of the Nonnegative Square Root gives $W_{2}(\mu,\mu)=0$.

Step 4. (Second half of claim 3, separation.) Suppose $W_{2}(\mu,\nu)=0$. By Step 1 there is an optimal $\gamma\in\Pi(\mu,\nu)$, so $I(\gamma)=W_{2}(\mu,\nu)^{2}=0$. For $j\in[d]$ put $q_{j}=x_{j}-x_{d+j}=x_{j}+(-1)x_{d+j}\in\mathcal{P}_{2d}$. By Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §self-adjoint (variables are self-adjoint, and the self-adjoint part is closed under sums and real multiples), $q_{j}^{*}=q_{j}$, so $q_{j}^{2}=q_{j}q_{j}=q_{j}^{*}q_{j}$, and $\gamma(q_{j}^{2})$ is real and nonnegative by (b) of Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §tracial-state.

(4a) Each $\gamma(q_{j}^{2})$ vanishes. Since $\gamma$ is linear, claim 4 of Properties of Finite Sums of Vectors and Couplings of Two Noncommutative Laws and Their Quadratic Cost §cost give $0=I(\gamma)=\gamma(\Delta_{d})=\sum_{j=1}^{d}\gamma(q_{j}^{2})$, a finite sum in $\mathbb{C}$ of real numbers. The partial-sum map of Finite Sum Notation in a Field formed in $\mathbb{R}$ satisfies the defining recursion also when the additions are read in $\mathbb{C}$ (condition 1 of The Complex Numbers), so by the uniqueness of that map for $\mathbb{C}$ this complex sum is the real sum of the same terms. By claim 5 of Properties of Finite Sums (nonnegative summands with sum $0$), $\gamma(q_{j}^{2})=0$ for every $j\in[d]$.

(4b) For every $j\in[d]$ and $r\in\mathcal{P}_{2d}$, $\gamma(q_{j}r)=0$. Apply Basic Properties of Noncommutative Laws: Adjoints, the Self-Adjoint Pairing, Cauchy-Schwarz, Monotonicity in the Bound, and Laws of Constant Tuples §cauchy-schwarz to the tracial state $\gamma$ on $\mathcal{P}_{2d}$ with $p=r$ and $q=q_{j}$: since $q_{j}^{*}=q_{j}$,

Also $0\le|\gamma(q_{j}r)|^{2}$ by claim 5 of Elementary Arithmetic in an Ordered Field, so $|\gamma(q_{j}r)|^{2}=0=0^{2}$; claim 3 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field gives $|\gamma(q_{j}r)|=0$, and claim 3 of Properties of Complex Conjugation and Modulus gives $\gamma(q_{j}r)=0$.

(4c) For every $j\in[d]$ and $r\in\mathcal{P}_{2d}$, $\gamma(x_{j}r)=\gamma(x_{d+j}r)$. Indeed, by Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §algebra (distributivity and $(cp)q=c(pq)$) and linearity of $\gamma$, $\gamma(x_{j}r)-\gamma(x_{d+j}r)=\gamma\bigl((x_{j}+(-1)x_{d+j})r\bigr)=\gamma(q_{j}r)=0$ by (4b).

(4d) Induction over the length of words. By Couplings of Two Noncommutative Laws and Their Quadratic Cost §marginals, $\iota^{1}=\sigma_{a}$ and $\iota^{2}=\sigma_{a'}$ with $a=(x_{1},\dots,x_{d})$ and $a'=(x_{d+1},\dots,x_{2d})$, so $\iota^{1}(x_{j})=x_{j}$ and $\iota^{2}(x_{j})=x_{d+j}$ for $j\in[d]$ by Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §values. Let $A$ be the set of those $k\in\mathbb{N}$ such that, for every word $w\in W_{d}$ of length $k$ and every $r\in\mathcal{P}_{2d}$, $\gamma(\iota^{1}(x_{w})\,r)=\gamma(\iota^{2}(x_{w})\,r)$. We verify the hypotheses of Principle of Induction for the Natural Numbers for $A$.

$1\in A$: a word $w$ of length $1$ is a letter $(j)$ by Basic Properties of Words: Associativity, Reversal, Finitely Many Factorisations, and Countability §last-letter, so $x_{w}=x_{j}$ (The Algebra of Noncommutative Polynomials in Finitely Many Self-Adjoint Variables §monomials) and $\gamma(\iota^{1}(x_{w})r)=\gamma(x_{j}r)=\gamma(x_{d+j}r)=\gamma(\iota^{2}(x_{w})r)$ by (4c).

$k\in A$ implies $S(k)\in A$: let $w$ have length $S(k)$ and let $r\in\mathcal{P}_{2d}$. 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 $j\in[d]$, so $x_{w}=x_{w'}x_{j}$ by Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §monomials, and by Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §homomorphism, $\iota^{1}(x_{w})=P\,x_{j}$ and $\iota^{2}(x_{w})=P'\,x_{d+j}$ with $P=\iota^{1}(x_{w'})$ and $P'=\iota^{2}(x_{w'})$. Using associativity (Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §algebra), the hypothesis $k\in A$ (with $x_{j}r$ in place of $r$), traciality (c) of Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §tracial-state, and (4c) (with $rP'$ in place of $r$),

So $S(k)\in A$, and $A=\mathbb{N}$ by Principle of Induction for the Natural Numbers.

(4e) Conclusion. Let $w\in W_{d}$. If $w=\varnothing$, then $x_{w}=1$ and $\mu(1)=1=\nu(1)$ by (a) of Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §tracial-state. Otherwise $w$ has a length $k\in\mathbb{N}=A$, and with $r=1$ (so that $q1=q$ by Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §monomials) and Couplings of Two Noncommutative Laws and Their Quadratic Cost §coupling,

Thus the linear maps $\mu,\nu:\mathcal{P}_{d}\to\mathbb{C}$ agree on every monomial, and the uniqueness in (a) of Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §linear-extension, with $c(w)=\mu(x_{w})$, gives $\mu=\nu$.

Step 5. (Claim 4, moment bound.) By Couplings of Noncommutative Laws: the Norm Bound, the Cost Identity, the Tensor, Diagonal and Swapped Couplings, Weak-Star Closedness, and Displacement Interpolants §tensor, $\mu\otimes\nu\in\Pi(\mu,\nu)$, so (0a) and Couplings of Noncommutative Laws: the Norm Bound, the Cost Identity, the Tensor, Diagonal and Swapped Couplings, Weak-Star Closedness, and Displacement Interpolants §cost give $W_{2}(\mu,\nu)^{2}\le I(\mu\otimes\nu)\le2M(\mu)+2M(\nu)$.

Step 6. (Claim 5, weak-star lower semicontinuity.) For $m\in\mathbb{N}$ let $B_{m}$ be the set of optimal couplings of $\mu_{m}$ and $\nu_{m}$, a subset of the set of tracial states on $\mathcal{P}_{2d}$; it is nonempty by Step 1 applied to $\mu_{m},\nu_{m}\in\Sigma_{d,R}$. By Axiom of Countable Choice (the use of countable choice in this step) there is a sequence $(\gamma_{m})$ with $\gamma_{m}\in B_{m}$, so $\gamma_{m}\in\Pi(\mu_{m},\nu_{m})$ and $I(\gamma_{m})=W_{2}(\mu_{m},\nu_{m})^{2}$. By Step 0, $0\le W_{2}(\mu_{1},\nu_{1})\le c$, so $0\le c$; and for every $m$, $0\le W_{2}(\mu_{m},\nu_{m})\le c$ gives $I(\gamma_{m})\le c^{2}$ by claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field. By Couplings of Noncommutative Laws: the Norm Bound, the Cost Identity, the Tensor, Diagonal and Swapped Couplings, Weak-Star Closedness, and Displacement Interpolants §closed there are a strictly increasing sequence $(m_{i})$ and $\gamma\in\Pi(\mu,\nu)$ such that $(I(\gamma_{m_{i}}))_{i}$ converges to $I(\gamma)$. The constant sequence with value $c^{2}$ converges to $c^{2}$ (Limit of a Sequence of Real Numbers), and $I(\gamma_{m_{i}})\le c^{2}$ for every $i$, so claim 1 of Order Properties of Limits of Real Sequences gives $I(\gamma)\le c^{2}$. By (0a), $W_{2}(\mu,\nu)^{2}\le I(\gamma)\le c^{2}$, and by (0b), $W_{2}(\mu,\nu)\le c$.

Step 7. (Claim 6, displacement interpolation.) Let $\gamma\in\Pi(\mu,\nu)$ be optimal, so $I(\gamma)=W^{2}$ with $W=W_{2}(\mu,\nu)$, and let $0\le t\le1$. By Couplings of Noncommutative Laws: the Norm Bound, the Cost Identity, the Tensor, Diagonal and Swapped Couplings, Weak-Star Closedness, and Displacement Interpolants §interpolation, $\gamma_{t}\in\Sigma_{d,R}\subseteq\Sigma_{d}$, and $\Pi(\mu,\gamma_{t})$ contains a coupling of cost $t^{2}I(\gamma)=(tW)^{2}$, while $\Pi(\gamma_{t},\nu)$ contains a coupling of cost $(1-t)^{2}I(\gamma)=((1-t)W)^{2}$. By (0a), $W_{2}(\mu,\gamma_{t})^{2}\le(tW)^{2}$ and $W_{2}(\gamma_{t},\nu)^{2}\le((1-t)W)^{2}$. Now $0\le1-t$ by claim 3 of Elementary Arithmetic in an Ordered Field, and $0\le W$, so $0\le tW$ and $0\le(1-t)W$ by claim 5 of Elementary Arithmetic in an Ordered Field. Hence (0b) gives $W_{2}(\mu,\gamma_{t})\le tW$ and $W_{2}(\gamma_{t},\nu)\le(1-t)W$.