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
- goal of classical optimal transport: to compare probability measures
- probability measures are classical objects : it is a classical way of describing the state of a system
- the state of a quantum system is better described by density matrices
- goal of quantum optimal transport: to compare density matrices
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 :
- (Monge-)Kantorovich formulation
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!
presentation-stuttgart
By Thomas Borsoni
