Quantum Optimal Transport is Cheaper
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Quantum Optimal Transport is Cheaper E. Caglioti1
· F. Golse2 · T. Paul3
Received: 10 February 2020 / Accepted: 16 May 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract We compare bipartite (Euclidean) matching problems in classical and quantum mechanics. The quantum case is treated in terms of a quantum version of the Wasserstein distance. We show that the optimal quantum cost can be cheaper than the classical one. We treat in detail the case of two particles: the equal mass case leads to equal quantum and classical costs. Moreover, we show examples with different masses for which the quantum cost is strictly cheaper than the classical cost.
1 Introduction The paradigm of modern optimal transport theory uses extensively the 2-Wasserstein distance between two probability measures μ, ν on Rn , defined as W2 (μ, ν)2 := |x − y|2 (d x, dy). inf (1) coupling of μ and ν
We have called coupling (or transport plan) of the two probabilities μ and ν any probability measure (d x, dy) on Rn × Rn whose marginals on the first and the second factors are μ and ν resp., i.e. R
n ×Rn
Rn ×Rn
a(x)(d x, dy) =
a(x)μ(d x), R
b(y)(d x, dy) =
n
(2) b(y)ν(dy)
Rn
Communicated by Eric A. Carlen.
B
E. Caglioti [email protected] F. Golse [email protected] T. Paul [email protected]
1
Dipartimento di Matematica, Sapienza Università di Roma, Piazzale Aldo Moro 5, 00185 Rome, Italy
2
Centre de Mathématiques Laurent Schwartz, Ecole polytechnique, 91128 Palaiseau Cedex, France
3
Centre de Mathématiques Laurent Schwartz, CNRS, Ecole polytechnique, 91128 Palaiseau Cedex, France
123
E. Caglioti et al.
for all test (i.e. continuous and bounded) functions a and b. Restricting the definition of W2 to couplings of the form = δ(y − T (x))μ(d x)
(3)
where T is a transformation of Rn such that ν is the image T# μ of μ by T , one sees that: M(μ, ν)2 := inf (x − T (x))2 μ(d x) ≥ W2 (μ, ν)2 . (4) T# μ=ν Rn
The converse inequality is due to Knott, Smith and Brenier: under certain restrictions on the regularity of μ and ν, any optimal coupling for the minimization problem defined by (1) is of the form (3) for some transport map T , so that the inequality in (4) is an equality (see e.g. [3] Sect. 1 for some details and Theorem 2.12 in [14] for an extensive study). Associated to W2 is the bipartite matching problem which can be described as follows. Let us consider M material points on the real line {xi }i=1,...,M with xi < xi+1 , and with masses {m i }i=1,...,M , and on the other hand N points {yi }i=1,...,N with y j < y j+1 , and with masses {n i }i=1,...,N . We normalize the total mass as follows: M
mi =
i=1
N
n j = 1.
j=1
The bipartite problem consists in finding a coupling matrix ( pi, j )|i=1,...,N , j=1,...M satisfying N
M
pi, j = m i ,
j=1
pi, j = n j ,
pi, j ≥ 0 for each i, j
i=1
which minimizes the quantity
pi, j |xi − y j |2 .
i, j
That is to say, we define the optimal transport cost as inf pi, j |xi − y j |2 . Cc := N
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