@article{12911, abstract = {This paper establishes new connections between many-body quantum systems, One-body Reduced Density Matrices Functional Theory (1RDMFT) and Optimal Transport (OT), by interpreting the problem of computing the ground-state energy of a finite-dimensional composite quantum system at positive temperature as a non-commutative entropy regularized Optimal Transport problem. We develop a new approach to fully characterize the dual-primal solutions in such non-commutative setting. The mathematical formalism is particularly relevant in quantum chemistry: numerical realizations of the many-electron ground-state energy can be computed via a non-commutative version of Sinkhorn algorithm. Our approach allows to prove convergence and robustness of this algorithm, which, to our best knowledge, were unknown even in the two marginal case. Our methods are based on a priori estimates in the dual problem, which we believe to be of independent interest. Finally, the above results are extended in 1RDMFT setting, where bosonic or fermionic symmetry conditions are enforced on the problem.}, author = {Feliciangeli, Dario and Gerolin, Augusto and Portinale, Lorenzo}, issn = {1096-0783}, journal = {Journal of Functional Analysis}, number = {4}, publisher = {Elsevier}, title = {{A non-commutative entropic optimal transport approach to quantum composite systems at positive temperature}}, doi = {10.1016/j.jfa.2023.109963}, volume = {285}, year = {2023}, } @article{13177, abstract = {In this note we study the eigenvalue growth of infinite graphs with discrete spectrum. We assume that the corresponding Dirichlet forms satisfy certain Sobolev-type inequalities and that the total measure is finite. In this sense, the associated operators on these graphs display similarities to elliptic operators on bounded domains in the continuum. Specifically, we prove lower bounds on the eigenvalue growth and show by examples that corresponding upper bounds cannot be established.}, author = {Hua, Bobo and Keller, Matthias and Schwarz, Michael and Wirth, Melchior}, issn = {1088-6826}, journal = {Proceedings of the American Mathematical Society}, number = {8}, pages = {3401--3414}, publisher = {American Mathematical Society}, title = {{Sobolev-type inequalities and eigenvalue growth on graphs with finite measure}}, doi = {10.1090/proc/14361}, volume = {151}, year = {2023}, } @article{13145, abstract = {We prove a characterization of the Dirichlet–Ferguson measure over an arbitrary finite diffuse measure space. We provide an interpretation of this characterization in analogy with the Mecke identity for Poisson point processes.}, author = {Dello Schiavo, Lorenzo and Lytvynov, Eugene}, issn = {1083-589X}, journal = {Electronic Communications in Probability}, pages = {1--12}, publisher = {Institute of Mathematical Statistics}, title = {{A Mecke-type characterization of the Dirichlet–Ferguson measure}}, doi = {10.1214/23-ECP528}, volume = {28}, year = {2023}, } @article{13318, abstract = {Bohnenblust–Hille inequalities for Boolean cubes have been proven with dimension-free constants that grow subexponentially in the degree (Defant et al. in Math Ann 374(1):653–680, 2019). Such inequalities have found great applications in learning low-degree Boolean functions (Eskenazis and Ivanisvili in Proceedings of the 54th annual ACM SIGACT symposium on theory of computing, pp 203–207, 2022). Motivated by learning quantum observables, a qubit analogue of Bohnenblust–Hille inequality for Boolean cubes was recently conjectured in Rouzé et al. (Quantum Talagrand, KKL and Friedgut’s theorems and the learnability of quantum Boolean functions, 2022. arXiv preprint arXiv:2209.07279). The conjecture was resolved in Huang et al. (Learning to predict arbitrary quantum processes, 2022. arXiv preprint arXiv:2210.14894). In this paper, we give a new proof of these Bohnenblust–Hille inequalities for qubit system with constants that are dimension-free and of exponential growth in the degree. As a consequence, we obtain a junta theorem for low-degree polynomials. Using similar ideas, we also study learning problems of low degree quantum observables and Bohr’s radius phenomenon on quantum Boolean cubes.}, author = {Volberg, Alexander and Zhang, Haonan}, issn = {1432-1807}, journal = {Mathematische Annalen}, publisher = {Springer Nature}, title = {{Noncommutative Bohnenblust–Hille inequalities}}, doi = {10.1007/s00208-023-02680-0}, year = {2023}, } @article{13271, abstract = {In this paper, we prove the convexity of trace functionals (A,B,C)↦Tr|BpACq|s, for parameters (p, q, s) that are best possible, where B and C are any n-by-n positive-definite matrices, and A is any n-by-n matrix. We also obtain the monotonicity versions of trace functionals of this type. As applications, we extend some results in Carlen et al. (Linear Algebra Appl 490:174–185, 2016), Hiai and Petz (Publ Res Inst Math Sci 48(3):525-542, 2012) and resolve a conjecture in Al-Rashed and Zegarliński (Infin Dimens Anal Quantum Probab Relat Top 17(4):1450029, 2014) in the matrix setting. Other conjectures in Al-Rashed and Zegarliński (Infin Dimens Anal Quantum Probab Relat Top 17(4):1450029, 2014) will also be discussed. We also show that some related trace functionals are not concave in general. Such concavity results were expected to hold in different problems.}, author = {Zhang, Haonan}, issn = {1424-0637}, journal = {Annales Henri Poincare}, publisher = {Springer Nature}, title = {{Some convexity and monotonicity results of trace functionals}}, doi = {10.1007/s00023-023-01345-7}, year = {2023}, }