@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},
}

@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{12087,
  abstract     = {Following up on the recent work on lower Ricci curvature bounds for quantum systems, we introduce two noncommutative versions of curvature-dimension bounds for symmetric quantum Markov semigroups over matrix algebras. Under suitable such curvature-dimension conditions, we prove a family of dimension-dependent functional inequalities, a version of the Bonnet–Myers theorem and concavity of entropy power in the noncommutative setting. We also provide examples satisfying certain curvature-dimension conditions, including Schur multipliers over matrix algebras, Herz–Schur multipliers over group algebras and generalized depolarizing semigroups.},
  author       = {Wirth, Melchior and Zhang, Haonan},
  issn         = {1424-0637},
  journal      = {Annales Henri Poincare},
  pages        = {717--750},
  publisher    = {Springer Nature},
  title        = {{Curvature-dimension conditions for symmetric quantum Markov semigroups}},
  doi          = {10.1007/s00023-022-01220-x},
  volume       = {24},
  year         = {2023},
}

@article{12210,
  abstract     = {The aim of this paper is to find new estimates for the norms of functions of a (minus) distinguished Laplace operator L on the ‘ax+b’ groups. The central part is devoted to spectrally localized wave propagators, that is, functions of the type ψ(L−−√)exp(itL−−√), with ψ∈C0(R). We show that for t→+∞, the convolution kernel kt of this operator satisfies
∥kt∥1≍t,∥kt∥∞≍1,
so that the upper estimates of D. Müller and C. Thiele (Studia Math., 2007) are sharp. As a necessary component, we recall the Plancherel density of L and spend certain time presenting and comparing different approaches to its calculation. Using its explicit form, we estimate uniform norms of several functions of the shifted Laplace-Beltrami operator Δ~, closely related to L. The functions include in particular exp(−tΔ~γ), t>0,γ>0, and (Δ~−z)s, with complex z, s.},
  author       = {Akylzhanov, Rauan and Kuznetsova, Yulia and Ruzhansky, Michael and Zhang, Haonan},
  issn         = {1432-1823},
  journal      = {Mathematische Zeitschrift},
  keywords     = {General Mathematics},
  number       = {4},
  pages        = {2327--2352},
  publisher    = {Springer Nature},
  title        = {{Norms of certain functions of a distinguished Laplacian on the ax + b groups}},
  doi          = {10.1007/s00209-022-03143-z},
  volume       = {302},
  year         = {2022},
}

@article{12216,
  abstract     = {Many trace inequalities can be expressed either as concavity/convexity theorems or as monotonicity theorems. A classic example is the joint convexity of the quantum relative entropy which is equivalent to the Data Processing Inequality. The latter says that quantum operations can never increase the relative entropy. The monotonicity versions often have many advantages, and often have direct physical application, as in the example just mentioned. Moreover, the monotonicity results are often valid for a larger class of maps than, say, quantum operations (which are completely positive). In this paper we prove several new monotonicity results, the first of which is a monotonicity theorem that has as a simple corollary a celebrated concavity theorem of Epstein. Our starting points are the monotonicity versions of the Lieb Concavity and the Lieb Convexity Theorems. We also give two new proofs of these in their general forms using interpolation. We then prove our new monotonicity theorems by several duality arguments.},
  author       = {Carlen, Eric A. and Zhang, Haonan},
  issn         = {0024-3795},
  journal      = {Linear Algebra and its Applications},
  keywords     = {Discrete Mathematics and Combinatorics, Geometry and Topology, Numerical Analysis, Algebra and Number Theory},
  pages        = {289--310},
  publisher    = {Elsevier},
  title        = {{Monotonicity versions of Epstein's concavity theorem and related inequalities}},
  doi          = {10.1016/j.laa.2022.09.001},
  volume       = {654},
  year         = {2022},
}