Connes distance and optimal transport

We give a brief overview on the relation between Connes spectral distance in noncommutative geometry and the Wasserstein distance of order 1 in optimal transport. We first recall how these two distances coincide on the space of probability measures on a Riemannian manifold. Then we work out a simple example on a discrete space, showing that the spectral distance between arbitrary states does not coincide with the Wasserstein distance with cost the spectral distance between pure states.


The metric aspect of noncommutative geometry
Topology is the minimal structure required for a set to be called space, and for its elements to become points. Topology gives sense to the notion of neighbourhood (and more generally to the notion of open sets). The algebraic dual notion is that of continuity (a continuous function being, by definition, such that the inverse image of any open set is an open set). Actually this duality goes quite far, since all the information of a topological space is contained within the algebra of continuous functions defined on it. More precisely, Gelfand's duality states that any complex commutative C * -algebra A is isomorphic to the algebra of continuous functions vanishing at infinity on some locally compact topological space -the space P(A) of pure states of A -and conversely any locally compact topological space X is homeomorphic to the space of pure states of the commutative algebra C 0 (X ) of continuous functions on X vanishing at infinity: A ≃ C 0 (P(A)), X ≃ P(C 0 (X )). (1) Recall that the pure states of an involutive algebra A are the extremal points of the set of states, the latter being the linear maps ϕ on A wich are positive -ϕ(a * a) ∈ R + -and of norm 1 (where ||ϕ|| = sup a∈A |ϕ(a)| ||a|| ). In particular, a state of the commutative algebra C 0 (X ) is the integration with respect to a probability measure µ, with pure states given by Dirac δ measures (i.e. evaluation at a point): Connes' noncommutative geometry extends Gelfand's duality beyond topology, so that to encompass all the aspects of Riemannian geometry, in particular the metric. To do so, one needs more than an algebra: a spectral triple [2] consists in an involutive algebra A acting faithfully on an Hilbert space H, with D a selfadjoint operator on H such that the commutator [D, a] is bounded and a[D − λI] −1 is compact for any a ∈ A and λ / ∈ Sp D. When a set of conditions (dimension, regularity, finitude, first order, orientability) is satisfied, then one is able to characterize a Riemannian manifold by purely spectral data [5]: • For M a compact Riemannian manifold, then is a spectral triple, where Ω • (M) is the Hilbert space of square integrable differential forms on M and d+d † is the Hodge-Dirac operator (d the exterior derivative, d † its Hodge-adjoint). By adding two extra-conditions (real structure and Poincaré duality), the result is extended to spin manifolds.
Why is such a spectral characterization of manifolds interesting ? Because the properties defining a spectral triple still make sense for a noncommutative A [2]. A noncommutative geometry is thus a spectral triple where the algebra A is noncommutative. At the light of Gelfand duality, this is the geometrical object whose algebra of functions defined on it is non commutative. As such it cannot be a usual topological space (otherwise its algebra of continuous functions would be commutative), but it rather appears as a "space without points". commutative spectral triple → noncommutative spectral triple ↓ Riemannian geometry non-commutative geometry However, always at the light of Gelfand duality, it is tempting to consider the pure states of the noncommutative algebra as the equivalent of points in the noncommutative context. This is all the more appealing that the same formula that allows to retrieve the Riemannian geodesic distance in Connes reconstruction theorem, also provides the space of states with a distance.
Explicitly, given a spectral triple (A, H, D), with A commutative or not, one defines on its state space S(A) the spectral distance [3] It is not difficult to check that d D has all the properties of a distance (zero if and only if ϕ = ψ, d D (ϕ, ψ) = d D (ψ, ϕ), triangle inequality), except that it may be infinite. By a slight abuse of language, we still call it distance. For an overview of explicit computation of this distance in various examples of commutative and noncommutative spectral triples, see [11].
2. Rieffel's remark and Wasserstein distance of order 1 Rieffel noticed in [12] that formula (3), applied to the spectral triple of a Riemannian manifold (2), was nothing but the Wasserstein distance of order 1 in the theory of optimal transport, or more exactly a reformulation of Kantorovich dual of the Wasserstein distance. To see it, let us first remind what the Wasserstein (or Monge Kantorovich distance) is. Let X be a locally compact Polish space, c(x, y) a positive real function, the "cost". The minimal work W required to transport the probability measure µ 1 to µ 2 is where the infimum is over all transportation plans, i.e. measures π on X × X with marginals µ 1 , µ 2 . When the cost function c is a distance d, then is a distance (possibly infinite) on the space of probability measures on X , called the Monge-Kantorovich (or Wasserstein) distance of order 1.
In [9], Kantorovich showed that Monge problem of minimizing the cost (eq. (4)) had an equivalent dual formulation (interpreted as maximizing a profit). Namely, W (µ 1 , µ 2 ) is equal to where ϕ 1 , ϕ 2 are the states of C(X ) defined by the measure µ 1 , µ 2 : and the supremum in (6) is on all the functions 1-Lipschitz with respect to the cost, that is Let X = M be a complete, connected, without boundary, Riemannian manifold. For any ϕ,φ ∈ S(C 0 (M)), where W is the Wasserstein distance associated to the cost d geo , while d D is the spectral distance associated to C ∞ 0 (M), Ω • (M), D = d + d † . That Kantorovich dual (6) coincides with the spectral distance (3) then follows from the observation that the supremum in the latter can be searched equivalently on selfadjoint elements, for which one has As pointed out in [7], one has to be careful that the manifold is complete, otherwise there is no guarantee that the supremum on the 1-Lipschitz functions in (6) coincides with the supremum on 1-Lipschitz functions vanishing at infinity in (3).
3. Towards a theory of optimal transport in noncommutative geometry ? Connes spectral distance on a manifold coincides with Kantorovich dual formulation of the Wasserstein distance of order 1. It is quite natural to wonder if the same is true is the noncommutative setting. But there does not exist any "noncommutative Wasserstein distance" of whom the spectral distance would be the dual. Is it possible to build one ? More specifically, given a spectral triple (A, H, D) with noncommutative A, is there a Wasserstein distance W D on S(A) such that its Kantorovich dual is the spectral distance d D ?
Commutative case: Noncommutative case: Kantorovich duality ? ↓ ↓ Wassertein distance W W D for some noncommutative cost ? with d D (δ x , δ y ) as a cost function In the commutative case A = C ∞ (M), one retrieves the cost function as the Wasserstein distance between pure states: W (δ x , δ y ) = c(x, y).
So it is tempting to define W D on the whole space of states S(A) as the Wasserstein distance associated with the cost c defined by the spectral distance on the space of pure states P(A), that is We worked out this construction in [10], restricting to unital separable C * -algebras, for which it is known that a state is a probability measure on P(A) [1, p.144] 1 , namely to any ϕ ∈ S(A), there exists a (non-necessarily unique) probability measure µ ∈ Prob(P(A)) such that whereâ(ω) .
= ω(a) denotes the evaluation at ω ∈ P(A) of an element a of A, viewed as a function on P(A). The Wasserstein distance on S(A) associated with the cost (10) has Kantorovich-dual formulation where is the set of element of A that are 1-Lipschitz with respect to the cost (10). It is not difficult to show that the Wasserstein distance provides a upper bound to the spectral distance [10, Prop. III.1], The equality holds on any subset of S(A) given by a convex linear combination of two pure states: fixed ω 1 , ω 2 ∈ P(A), one denotes ϕ λ := λω 1 + (1 − λ)ω 2 . Then This shows in particular that W D = d D on the whole of S(A) if A = M 2 (C), since the pure state space of the algebra of 2 × 2 matrices is homeomorphic to the 2-sphere, so that the space of states is the 2-ball, and any two states ϕ 1 , ϕ 2 can always be decomposed as two convex linear combinations ϕ λ 1 , ϕ λ 2 of the same two pure states.
However the two distances are not equal in general, as can be seen in the following counterexample, taken from [13, §7]. Consider A = C 3 acting on H = C 3 as a diagonal matrix, and take as Dirac operator There are three pure states δ i for A, defined as So the space of states is the plain triangle with summit δ 1 , δ 2 , δ 3 . By (15), one has that W D coincides with d D on each edge of the triangle. But the two distances do not agree on the whole triangle.
Proof. The cost function (10) is obtained computing explicitly the spectral distance between pure states (see e.g. [8,Prop. 7]): The Lipschitz ball (13) is thus For any a = (z 1 , z 2 , z 3 ) ∈ C 3 , one has Therefore Since Λ 1 e Λ 2 have the same sign, this upper bound is attained by a 0 in Lip D (C 3 ) defined by To prove (21), one computes the commutator of a = (z 1 , z 2 , z 3 ) with D (see e.g. [8]) By subtracting z 1 I, one can always assume that z 3 = 0. The commutator condition ||[D, a]|| ≤ 1 becomes α 2 |z 1 | 2 + β 2 |z 2 | 2 ≤ 1, which is equivalent to For such a, one obtains from (24) reaches its maximum when f ′ vanishes, that is for This maximum, is an upper bound for the spectral distance, reached by the element a = (x 0 , f (x 0 ), 0).

Conclusion and outlook
On a manifold, Connes spectral distance between arbitrary states coincides with the Wasserstein distance of order 1 with cost the geodesic distance. On an arbitrary spectral triple, the spectral distance d D on the space of states S(A) (viewed as the convex hull of the pure states space P(A)) does not coincide with the Wasserstein distance W D with cost function d D (on P(A)). However, W D always provides an upper bound to the spectral distance, and the two distances coincides on any convex combination of two fixed pure states. This shows that the interpretation of the spectral distance as a Wasserstein distance is more involved that could be initially thought, although the intriguing example worked out in this paper (W D is the sum of the opposite and adjacent sides of a right a triangle, d D is the length of the hypothenuse) suggests that there might exist a simple relation between the two distances.
Let us also mention that there do exists a formulation of the spectral distance as an infimum rather than a supremum (an analogue to "dual of the dual" formula of the Wasserstein distance in optimal transport), whose possible interpretation as a noncommutative cost still has to be elucidated [7].