A geometric interpretation of

quantum optimal transport

Thomas Borsoni\(^{1,2}\)

under the supervision of       Virginie Ehrlacher\(^{1,2}\) & Geneviève Dusson\(^{3}\)                    

96\(^{\text{th}}\) annual GAMM meeting, Stuttgart

March 17\(^{\text{th}}\), 2026

\(^{1}\) CERMICS, ENPC, Champs-sur-Marne, France

\(^{2}\) MATHERIALS team, INRIA, France

\(^{3}\) Université Bourgogne-Franche-Comté, Besançon, France

Unified geometric interpretation of Kantorovich formulations

of classical and quantum optimal transport

Aim of the presentation

Unified geometric interpretation of Kantorovich formulations

of classical and quantum optimal transport

Aim of the presentation

1. Classical and quantum optimal transport

Outline

2. Geometric interpretation of Kantorovich formulations

(with Choquet theory)

3. Folded Wasserstein distances

        unifying classical and quantum without entanglement

1. Classical and quantum optimal transport

Classical optimal transport

Aim of OT :

  • to compare two probability measures (compute a cost or a distance)
  • to compute interpolations between probability measures
\mu
\nu

Formulations of OT :            cost

 

1. (Monge-)Kantorovich formulation

c : \mathcal{X} \times \mathcal{Y} \to \R
K_c : (\mu, \nu) \in \mathcal{X} \times \mathcal{Y} \mapsto \inf_{\pi \in \Pi(\mu,\nu)} \iint_{\mathcal{X} \times \mathcal{Y}} c(x,y) \, \mathrm{d} \pi (x,y)

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!