Folded optimal transport

and its application to

separable quantum optimal transport

Thomas Borsoni*

under the supervision of Virginie Ehrlacher & Geneviève Dusson

Workshop Geometry, duality and convexity in new OT problems

November 19, 2025

*post-doc with Virginie Ehrlacher and Tony Lelièvre at CERMICS, ENPC (Champs-sur-Marne)

funded by the ERC starting grant HighLEAP  (Virginie Ehrlacher)

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 \(E\)

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

d
D
d
D

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

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

Folded Wasserstein distance

Simplex and set of probability measures

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

probability measure

 

convex combination

b
c
\cong
\mathcal{P}(E_0)
\bm{\delta}_x
\bm{\delta}_y
E = \{\bm{\delta}_z\}_{z \in E_0} \cong E_0

Optimal transport answers

for the simplex

\mathcal{P}(E_0)

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

  • Let \((E_0,d)\) a compact Polish* space 

* Polish = complete metric space with countable dense subset

  • For \(p\geqslant 1\), the Wasserstein-\(p\) distance \(W_p\) is a distance on \(\mathcal{P}(E_0)\) such that

\(\forall x,y \in E_0\),        \(W_p(\bm{\delta}_x, \bm{\delta}_y) = d(x,y)\)

\bm{\delta}_x
\bm{\delta}_y
E = \{\bm{\delta}_z\}_{z \in E_0} \cong E_0

\(W_p\) extends \(d\) from \(E \cong E_0\) to \(\mathcal{P}(E_0)\)

And for other convex sets?

Choquet theory

Representing every \(x \in C\) with convex combinations of points of \(E\)

C
E
\times
x
\bullet
\bullet
e_1
e_1'
  • \(\mu_1 = \frac34 \bm{\delta}_{e_1} + \frac14 \bm{\delta}_{e'_1} \) represents \(x\)

(\(x\) is the barycenter associated with \(\mu_1\))

Representing every \(x \in C\) with probability measures \(\mu \in \mathcal{P}(E)\)

Choquet theory

Representing every \(x \in C\) with convex combinations of points of \(E\)

Representing every \(x \in C\) with probability measures \(\mu \in \mathcal{P}(E)\)

\times
x
\bullet
\bullet
e_2
e_2'
  • \(\mu_1 = \frac34 \bm{\delta}_{e_1} + \frac14 \bm{\delta}_{e'_1} \) represents \(x\)
  • \(\mu_2 = \frac12 \bm{\delta}_{e_2} + \frac12 \bm{\delta}_{e'_2} \) represents \(x\)

(\(x\) is the barycenter associated with \(\mu_1\))

(\(x\) is the barycenter associated with \(\mu_2\))

C
E

Choquet theory

Representing every \(x \in C\) with convex combinations of points of \(E\)

Representing every \(x \in C\) with probability measures \(\mu \in \mathcal{P}(E)\)

\times
x
\bullet
\bullet
e_3
e_3'
\bullet
e''_3
  • \(\mu_3 = \frac14 \bm{\delta}_{e_3} + \frac14 \bm{\delta}_{e'_3} + \frac12 \bm{\delta}_{e''_3} \) represents \(x\)
  • \(\mu_1 = \frac34 \bm{\delta}_{e_1} + \frac14 \bm{\delta}_{e'_1} \) represents \(x\)
  • \(\mu_2 = \frac12 \bm{\delta}_{e_2} + \frac12 \bm{\delta}_{e'_2} \) represents \(x\)

(\(x\) is the barycenter associated with \(\mu_1\))

(\(x\) is the barycenter associated with \(\mu_2\))

(\(x\) is the barycenter associated with \(\mu_3\))

C
E

Choquet theory

Representing every \(x \in C\) with convex combinations of points of \(E\)

Representing every \(x \in C\) with probability measures \(\mu \in \mathcal{P}(E)\)

\times
x
  • \(\mu_1 = \frac34 \bm{\delta}_{e_1} + \frac14 \bm{\delta}_{e'_1} \) represents \(x\)
  • \(\mu_2 = \frac12 \bm{\delta}_{e_2} + \frac12 \bm{\delta}_{e'_2} \) represents \(x\)
  • \(\mu_3 = \frac14 \bm{\delta}_{e_3} + \frac14 \bm{\delta}_{e'_3} + \frac12 \bm{\delta}_{e''_3} \) represents \(x\)
\dots
\bullet
\bullet
\bullet
\bullet

many \(\mu \in \mathcal{P}(E)\) represent \(x\)  !

(\(x\) is the barycenter associated with many \(\mu \in \mathcal{P}(E)\))

C
E
\bullet
\bullet
\bullet
\times
x
  • \(\mu_1 = \frac34 \bm{\delta}_{e_1} + \frac14 \bm{\delta}_{e'_1} \) represents \(x\)
  • \(\mu_2 = \frac12 \bm{\delta}_{e_2} + \frac12 \bm{\delta}_{e'_2} \) represents \(x\)
\bullet
\bullet
\bullet
\bullet
  • \(\nu = \frac23 \bm{\delta}_{g} + \frac13 \bm{\delta}_{g'}\) represents \(y \neq x\)
\times
y
\bullet
\bullet
e_1
e_1'
e_2
e_2'
g
g'

\(\mu_1\) \(\sim \) \(\mu_2\)

\(\mu_1\), \(\mu_2\) \(\nsim \) \(\nu\)

then

but

  • Let \(\sim\) be the equivalence relation on \(\mathcal{P}(E)\):
\mu \sim \nu \qquad \iff \qquad

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

Choquet theory

Choquet-Bishop-DeLeeuw Theorem:

If \(C\) is convex and compact *, then

* and subset to a locally convex Hausdorff space

\forall x \in C,

(\(x\) is the barycenter associated with at least one \(\mu \in \mathcal{P}(E)\))

Since

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

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

Choquet-Bishop-DeLeeuw

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

 there exists at least one \(\mu \in \mathcal{P}(E)\) which represents \(x\) 

C
E

Representing every \(x \in C\) with probability measures \(\mu \in \mathcal{P}(E)\)

Choquet theory

Folded Wasserstein

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

unfold

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

extend

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

fold back

\cong

(optimal transport)

Folded Wasserstein distance on \(C\):   \(D_p := W_p / \sim\)

(Choquet)

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

?

?

C
(E,d)
(C,D_p)

(represent)

(quotient)

The quotient (pseudo-)distance

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

\widehat{D}_p([\mu],[\nu]) := \inf_{\mu' \sim \mu} \; \inf_{\nu' \sim \nu} \; W_p(\mu', \nu')
(\mathcal{P}(E),W_p)
[\mu]
\widehat{D}_p([\mu],[\nu])

a priori, fails the triangle inequality!

D_p([\mu],[\nu]) := \inf_{\gamma_1, \dots, \, \gamma_N} \{ \widehat{D}_p([\mu],[\gamma_1]) + \widehat{D}_p([\gamma_1],[\gamma_2]) + \dots + \widehat{D}_p([\gamma_N],[\nu]) \}
W_p / \sim
D_p([\mu],[\nu])
  • Candidate quotient distance:
  • The actual quotient pseudo*-distance:
[\nu]
[\gamma]
W_p / \sim

?

\mathcal{P}(E) / \sim

on

The folded Wasserstein (pseudo-)distance

\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),
D_p(x,y) := \inf_{z_1, \dots, \, z_N \in C} \left\{ \widehat{D}_p(x,z_1) + \widehat{D}_p(z_1,z_2) + \dots + \widehat{D}_p(z_N,y) \right\}
d
D_p
C
E

An answer to: how to extend \(d\) from \(E\) to \(C\)?

For \(p \geqslant 1\), the folded Wasserstein-\(p\) (pseudo-)distance associated with \(d\) is

with

and

\(W_p\) is the standard Wasserstein distance on \(\mathcal{P}(E)\) associated with \(d\).

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

unfold

fold back

extend

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

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* \(\mathrm{Ri}(C) \neq \emptyset\), is continuous w.r.t. \(\|\cdot\|\)
  • For all \(x,y \in C\), \(D_p(x,y) \geqslant \|x-y\|\)
  • \(D_p\) sub-extends \(d\),                        and if \(d = \|\cdot - \cdot\|\), \(D_p\) extends \(d\)

An answer to: how to extend \(d\) from \(E\) to \(C\)?

*\(\mathrm{Ri}(C)\) is the relative interior of \(C\). In finite dimension, \(C \neq \emptyset \implies \mathrm{Ri}(C) \neq \emptyset\).

\(\forall x,y \in E\),    \(D_p(x,y) \leqslant d(x,y) \)                                                          \(\forall x,y \in E\),   \(D_p(x,y) = d(x,y) \)

 

  • If \((E,d)\) is geodesic and \(p>1\), then \((C,D_p)\) is geodesic
(X,\|\cdot\|)

(TB 2025)

Folded general optimal transport

C_2
E_1
C_1
E_2
c : E_1 \times E_2 \longrightarrow \R

How to extend \(c\) from \(E_1 \times E_2\) to \(C_1 \times C_2\)?

Optimal transport answers

for simplicies

\mathcal{P}(E_1)

How to extend \(c\) from \(E_1 \times E_2\) to \(C_1 \times C_2\)?

E_1
\mathcal{P}(E_2)
E_2
c : E_1 \times E_2 \longrightarrow \R
K_c : (\mu,\nu) \in \mathcal{P}(E_1) \times \mathcal{P}(E_2) \; \; \mapsto \; \; \inf_{\pi \in \mathcal{C}(\mu,\nu)} \iint_{E_1 \times E_2} c(e_1,e_2) \, \mathrm{d} \pi(e_1,e_2)

Kantorovitch cost associated with \(c\)

E_1
C_1
C_2
E_2
c : E_1 \times E_2 \longrightarrow \R

unfold

(represent)

E_1
E_2
\mathcal{P}(E_1)
\mathcal{P}(E_2)
c : E_1 \times E_2 \longrightarrow \R

extend

(Kantorovitch cost)

E_1
E_2
\mathcal{P}(E_1)
\mathcal{P}(E_2)
K_c : \mathcal{P}(E_1) \times \mathcal{P}(E_2) \longrightarrow \R

fold back

(quotient)

E_1
E_2
\mathcal{P}(E_1) / \sim
\mathcal{P}(E_2) / \sim
\overline{K}_c \equiv K_c / \sim : C_1 \times C_2 \longrightarrow \R
\cong

(Choquet)

folded Kantorovitch cost:

K_c / \sim

Folded Kantorovitch cost

Application to quantum optimal transport

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 \(\mathbf{P}_{\mathcal{H}}\)

Some existing formulations

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

*Includes entanglement

quantum folded optimal transport

classical OT

separable quantum OT

Conclusion (quantum)

folded optimal transport

classical OT

semiclassical OT

nonseparable quantum OT

To sum up

  • Standard optimal transport answers

How to extend a cost from extreme boundaries to the whole convexes ?

in the case of the simplex

  • Folded optimal transport answers (?)

How to extend a cost from extreme boundaries to the whole convexes ?

in the general case

  • Quantum OT without entanglement is folded OT
  • Folded OT is constructed from standard OT

is quantum without entanlement just classical?

TH

NK

Y

U

for your attention!