1,721,027 research outputs found
Quantum optimal transport: an invitation
Full text linkThe optimal mass transport problem was formulated centuries ago, but only recently there has been a surge in its applications, particularly in functional inequalities, geometry, stochastic analysis, and numerical solutions for partial differential equations. Quantum optimal transport aims to extend this success story to non-commutative systems, where density operators replace probability measures. This brief review paper aims to describe the latest approaches, highlighting their advantages, disadvantages, and open mathematical problems relevant to applications
The Wasserstein Distance of Order 1 for Quantum Spin Systems on Infinite Lattices
Full text linkWe propose a generalization of the Wasserstein distance of order 1 to quantum spin systems on the lattice Z(d), which we call specific quantum W-1 distance. The proposal is based on the W-1 distance for qudits of De Palma et al. (IEEE Trans Inf Theory 67(10):6627-6643, 2021) and recovers Ornstein's (d) over bar -distance for the quantum states whose marginal states on any finite number of spins are diagonal in the canonical basis. We also propose a generalization of the Lipschitz constant to quantum interactions on Z(d) and prove that such quantum Lipschitz constant and the specific quantum W-1 distance are mutually dual. We prove a new continuity bound for the von Neumann entropy for a finite set of quantum spins in terms of the quantum W-1 distance, and we apply it to prove a continuity bound for the specific von Neumann entropy in terms of the specific quantum W-1 distance for quantum spin systems on Z(d). Finally, we prove that local quantum commuting interactions above a critical temperature satisfy a transportation-cost inequality, which implies the uniqueness of their Gibbs states
Towards Geometric Integration of Rough Differential Forms
No full textWe provide a draft of a theory of geometric integration of “rough differential forms” which are generalizations of classical (smooth) differential forms to similar objects with very low regularity, for instance, involving Hölder continuous functions that may be nowhere differentiable. Borrowing ideas from the theory of rough paths, we show that such a geometric integration can be constructed substituting appropriately differentials with more general asymptotic expansions. This can be seen as the basis of geometric integration similar to that used in geometric measure theory, but without any underlying differentiable structure, thus allowing Lipschitz functions and rectifiable sets to be substituted by far less regular objects (e.g. Hölder functions and their images which may be purely unrectifiable). Our construction includes both the one-dimensional Young integral and multidimensional integrals introduced recently by Züst, and provides also an alternative (and more geometric) view on the standard construction of rough paths. To simplify the exposition, we limit ourselves to integration of rough k-forms with k≤ 2
A Benamou-Brenier formulation of martingale optimal transport
No full textWe introduce a Benamou-Brenier formulation for the continuous-time martingale optimal transport problem as a weak length relaxation of its discrete-time counterpart. By the correspondence between classical martingale problems and Fokker-Planck equations, we obtain an equivalent PDE formulation for which basic properties such as existence, duality and geodesic equations can be analytically studied, yielding corresponding results for the stochastic formulation. In the one dimensional case, sufficient conditions for finiteness of the cost are also given and a link between geodesics and porous medium equations is partially investigated
Tra decoro e mercato.
No full textCome sta evolvendo il variegato mondo delle professional service firms? Quale posto occupano le professional service firms nell’ambito del più ampio settore dei knowledge intensive business services? Quali sono le forme organizzative emergenti e i sistemi operativi sui quali intervenire per migliorare le performance delle professional service firms? È possibile gestire in modo innovativo la comunicazione e il marketing (anche) negli studi professionali? Gli autori forniscono un percorso di lettura per rispondere a queste domande, prendendo spunto da alcuni volumi stranieri e italiani di recente pubblicazione
Optimal transport methods for combinatorial optimization over two random point sets
Full text linkWe investigate the minimum cost of a wide class of combinatorial optimization problems over random bipartite geometric graphs in Rd where the edge cost between two points is given by a p-th power of their Euclidean distance. This includes e.g.\ the travelling salesperson problem and the bounded degree minimum spanning tree. We establish in particular almost sure convergence, as n grows, of a suitable renormalization of the random minimum cost, if the points are uniformly distributed and d≥3, 1≤
The generalized strong subadditivity of the von Neumann entropy for bosonic quantum systems
No full textWe prove a generalization of the strong subadditivity of the von Neumann entropy for bosonic quantum Gaussian systems. Such generalization determines the minimum values of linear combinations of the entropies of subsystems associated to arbitrary linear functions of the quadratures, and holds for arbitrary quantum states including the scenario where the entropies are conditioned on a memory quantum system. We apply our result to prove new entropic uncertainty relations with quantum memory, a generalization of the quantum Entropy Power Inequality, and the linear time scaling of the entanglement entropy produced by quadratic Hamiltonians
Quantum Optimal Transport with Quantum Channels
No full textWe propose a new generalization to quantum states of the Wasserstein distance, which is a fundamental distance between probability distributions given by the minimization of a transport cost. Our proposal is the first where the transport plans between quantum states are in natural correspondence with quantum channels, such that the transport can be interpreted as a physical operation on the system. Our main result is the proof of a modified triangle inequality for our transport distance. We also prove that the distance between a quantum state and itself is intimately connected with the Wigner-Yanase metric on the manifold of quantum states. We then specialize to quantum Gaussian systems, which provide the mathematical model for the electromagnetic radiation in the quantum regime. We prove that the noiseless quantum Gaussian attenuators and amplifiers are the optimal transport plans between thermal quantum Gaussian states, and that our distance recovers the classical Wasserstein distance in the semiclassical limit
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