Folded optimal transport

and its application to

separable quantum optimal transport

Thomas Borsoni*

ArpiLYSM: Simplex vs Non-simplex

November 11, 2025

CERMICS, École Nationale des Ponts et Chaussées, France

*post-doc with Virginie Ehrlacher and Tony Lelièvre 

funded by the ERC starting grant HighLEAP

pure states

pure states

mixed states

mixed states

classical

quantum

\(\bullet\) convex set \(C\)

\(\bullet\) extreme boundary \(E\)

How to extend a distance \(d\) from \(E\) to \(C\)?

d
D
d
D

\(\bullet\) convex set \(C\)

\(\bullet\) extreme boundary \(E\)

Simplex and set of probability measures

a
\mathcal{P}(\{a,b,c\})

probability measure on \(E\)

 

convex combination of points of \(E\)

b
c
\mathcal{P}(E)
E

Optimal transport answers

for simplices

C=\mathcal{P}(E)

How to extend a distance from \(E\) to \(C\)?

  • Start with a Polish* space \((E,d)\)

* Polish = complete metric space with countable dense subset

For \(p\geq 1\), we can construct the Wasserstein-\(p\)  distance \(W_p\) on \(\mathcal{P}(E)\)

  • \(W_p(\bm{\delta}_x, \bm{\delta}_y) = d(x,y)\)
\bm{\delta}_x
\bm{\delta}_y
E \cong \{\bm{\delta}_z\}_{z \in E}

Let's do this for non-simplices!

Choquet theory

Representing every \(x \in C\) as a convex combination of points of \(E\)

Choquet-Bishop-DeLeeuw Theorem:

If \(C\) compact convex*, then

* and subset to a locally convex Hausdorff space

\forall x \in C, \; \; \exists \mu \in \mathcal{P}(E),

\(x\) is the barycenter associated with \(\mu\)

Let \(\sim\) be the equivalence relation on \(\mathcal{P}(E)\):

\mu \sim \nu \qquad \iff \qquad

\(\mu\) and \(\nu\) have the same barycenter \(x \in C\),

Choquet-Bishop-DeLeeuw

C \cong \mathcal{P}(E)/\sim
C
E

Folded optimal transport

C
(\mathcal{P}(E)/\sim, \; W_p / \sim)
(E,d)

unfold

\mathcal{P}(E)
(E,d)

extend

(\mathcal{P}(E),W_p)

fold back

\cong

(optimal transport)

the folded Wasserstein distance on \(C\) is   \(D_p := W_p / \sim\)

(Choquet)

How to extend \(d\) from \(E\) to \(C\)?

?

?

Folded Wasserstein as a quotient (pseudo-)distance

*pseudo-distance: would be a distance if it separated points

\widehat{D}_p(x,y) := \inf_{\mu \in \mathcal{P}(E) \text{ represents } x} \; \inf_{\nu \in \mathcal{P}(E) \text{ represents } y} \; W_p(\mu, \nu)
(\mathcal{P}(E),W_p)
y
z
x
\widehat{D}_p(x,y)

a priori, fails the triangle inequality!

  • Folded Wasserstein pseudo-distance*
D_p(x,y) := \inf_{z_1, \dots, \, z_N} \{ \widehat{D}_p(x,z_1) + \widehat{D}_p(z_1,z_2) + \dots + \widehat{D}_p(z_N,y) \}
W_p / \sim
x,y,z \in C
D_p(x,y)

The folded Wasserstein metric space

Theorem

(C,D_p)
(E,d)
  • \(C\) compact convex subset of \((X,\|\cdot\|)\) Banach
  • \((E,d)\) (compact) Polish and \(d\) continuous w.r.t. \(\|\cdot\|\)
  • For all \(x,y \in E\),    \(d(x,y) \geqslant \|x-y\| \)

Assume:

Then:

  • \(D_p\) is a distance on \(C\), and if \(Ri(C) \neq \emptyset\), continuous w.r.t. \(\|\cdot\|\)
  • For all \(x,y \in E\),    \(D_p(x,y) \leqslant d(x,y) \)       and      if \(d(x,y) = \|x-y\|\), then     \(D_p(x,y) = d(x,y) \)
  • For all \(x,y \in C\), \(D_p(x,y) \geq \|x-y\|\)
  • If \((E,d)\) geodesic and \(p>1\), then \((C,D_p)\) geodesic

Application to separable quantum optimal transport

C = S^+_1
E = \mathbf{P}_\mathcal{H}
C \subseteq (\mathcal{B}(\mathcal{H}), \|\cdot\|)

\(\mathcal{H}\) complex Hilbert of finite dimension

rank-one projectors on \(\mathcal{H}\)

pure states

self-ajoint semi-definite operators on \(\mathcal{H}\), with trace 1

mixed states

Quantum optimal transport: to define a distance on \(S^+_1\) from one on \(\mathbf{P}_{\mathcal{H}}\)

Families of existing formulations

  • Dynamic (Carlen-Maas)
  • Nonseparable* static (Biane-Voilescu, Golse-Mouhot-Paul, DePalma-Trévisan,...)
  • Separable static (Tóth-Pitrik, Beatty-Stilck França)

*Includes entanglement

quantum folded Wasserstein

classical OT

separable quantum OT

Conclusion

folded optimal transport

also includes semiclassical cost (Golse-Paul)

classical OT

TH

NK

Y

U

for your attention!