Research

My research activity mainly concerns the study of quantum Markov semigroups (QMSs), usually defined as a weak*continuous semigroup of operators acting on a von Neumann algebra with appropriate properties. These semigroups can also be naturally considered in discrete time, in which case the semigroup can be represented as the family of powers of a proper fixed operator Φ. This superoperator Φ is the quantum analogue of the transition matrix of a classical Markov chain and is usually called a quantum channel; in this case, it is sometimes referred to as a quantum Markov chain.

For QMSs, one can study a multitude of properties, some of which are entirely analogous to those studied in classical probability theory, while others are typical of the non-commutative context, such as, for example, decoherence.

Some of my research topics:

Probabilistic properties of quantum channels: accessibility, transience and recurrence, ergodic properties, period.
Environment-induced decoherence for QMSs on finite and infinite-dimensional algebras: description of the decoherence-free algebra, necessary and sufficient conditions for decoherence, relationships with related properties such as period.
Open quantum random walks: reducibility, period, structure of invariant states, central limit theorems and large deviations for the position process.
Fundamental properties for certain notable families of QMSs, such as the construction of the minimal semigroup from the Lindblad form of the generator, conservativity, existence of invariant states, Feller properties.
Functional inequalities and contraction properties. Spectral gap estimation and study of hypercontractivity and its related properties for some quantum evolutions, with particular attention to the Ornstein-Uhlenbeck quantum process and to semigroups on the algebra of 2×2 complex matrices.

Other topics in classical probability: - Algorithms for the evaluation of exotic options in the Black-Scholes model with binomial approximation; - Existence and uniqueness results for backward stochastic differential equations.

A list of my publications follows,

partially organized by topics

Properties of quantum channels

Carbone, Raffaella; Girotti, Federico. Absorption in invariant domains for semigroups of quantum channels. Ann. Henri Poincaré 22 (2021), no. 8, 2497–2530.

Carbone, Raffaella; Jenčová, Anna. On period, cycles and fixed points of a quantum channel. Ann. Henri Poincaré 21 (2020), no. 1, 155–188.

Carbone, Raffaella; Pautrat, Yan. Irreducible decompositions and stationary states of quantum channels. Rep. Math. Phys. 77 (2016), no. 3, 293–313.

Open quantum walks

Carbone, Raffaella; Girotti, Federico; Melchor Hernandez, Anderson. On a generalized central limit theorem and large deviations for homogeneous open quantum walks. J. Stat. Phys. 188 (2022), no. 1, Paper No. 8, 33 pp.

Carbone, Raffaella; Pautrat, Yan. Open quantum random walks: reducibility, period, ergodic properties. Ann. Henri Poincaré 17 (2016), no. 1, 99–135.

Carbone, Raffaella; Pautrat, Yan. Homogeneous open quantum random walks on a lattice. J. Stat. Phys. 160 (2015), no. 5, 1125–1153.

Decoherence

Carbone, R.; Sasso, E.; Umanità, V.. Structure of generic quantum Markov semigroup. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 20 (2017), no. 2, 1750012, 19 pp.

Carbone, Raffaella; Jenčová, Anna. On period, cycles and fixed points of a quantum channel. Ann. Henri Poincaré 21 (2020), no. 1, 155–188.

Carbone, Raffaella; Sasso, Emanuela; Umanità, Veronica. Environment induced decoherence for Markovian evolutions. J. Math. Phys. 56 (2015), no. 9, 092704, 22 pp.

Carbone, R.; Sasso, E.; Umanità, V.. Ergodic quantum Markov semigroups and decoherence. J. Operator Theory 72 (2014), no. 2, 293–312.

Carbone, R.; Sasso, E.; Umanità, V.. On the asymptotic behavior of generic quantum Markov semigroups. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 17 (2014), no. 1, 1450001, 18 pp.

Carbone, Raffaella; Sasso, Emanuela; Umanità, Veronica. Decoherence for quantum Markov semi-groups on matrix algebras. Ann. Henri Poincaré 14 (2013), no. 4, 681–697.

Carbone, Raffaella; Sasso, Emanuela; Umanità, Veronica. Decoherence for positive semigroups on M2(ℂ). J. Math. Phys. 52 (2011), no. 3, 032202, 17 pp.

Variational inequalities and hypercontractivity

Carbone, Raffaella; Martinelli, Andrea. Logarithmic Sobolev inequalities in non-commutative algebras. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 18 (2015), no. 2, 1550011, 30 pp.

Bolaños-Servin, Jorge R.; Carbone, Raffaella; Quezada, Roberto. On reducibility and spectral properties of circulant Markov processes. Statist. Probab. Lett. 123 (2017), 27–33. + Corrigendum Statist. Probab. Lett. 135 (2018), 140.

Carbone, R.; Fagnola, F.; García, J. C.; Quezada, R.. Spectral properties of the two-photon absorption and emission process. J. Math. Phys. 49 (2008), no. 3, 032106, 18 pp.

Carbone, Raffaella; Sasso, Emanuela. Hypercontractivity for a quantum Ornstein-Uhlenbeck semigroup. Probab. Theory Related Fields 140 (2008), no. 3-4, 505–522.

Carbone, Raffaella; Fagnola, Franco. Exponential ergodicity of classical and quantum Markov birth and death semigroups. Kluwer Academic Publishers, Dordrecht, 2004, 169–183. ISBN: 1-4020-2467-3

Carbone, Raffaella. Optimal log-Sobolev inequality and hypercontractivity for positive semigroups on M2(ℂ). Infin. Dimens. Anal. Quantum Probab. Relat. Top. 7 (2004), no. 3, 317–335.

Carbone, R.; Fanʹola, F.. Exponential convergence in L2 of quantum Markov semigroups on ℬ(h). Mat. Zametki 68 (2000), no. 4, 523–538. Math. Notes 68 (2000), no. 3-4, 452–463.

Carbone, Raffaella. Exponential L2-convergence of some quantum Markov semigroups related to birth-and-death processes. Trends Math.. Birkhäuser Boston, Inc., Boston, MA, 2000, 1–22.

Other

Girotti, Federico; van Horssen, Merlijn; Carbone, Raffaella; Guţă, Mădălin. Large deviations, central limit, and dynamical phase transitions in the atom maser. J. Math. Phys. 63 (2022), no. 6, Paper No. 062202, 24 pp.

Bolaños-Servin, Jorge R.; Carbone, Raffaella; Quezada, Roberto. Structure and block representation for circulant quantum processes. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 22 (2019), no. 3, 1950017, 16 pp.

Bolaños-Servin, Jorge R.; Carbone, Raffaella. Spectral properties of circulant quantum Markov semigroups. Open Syst. Inf. Dyn. 21 (2014), no. 4, 1450007, 18 pp.

Carbone, Raffaella; Fagnola, Franco. The Feller property of a class of quantum Markov semigroups. II. QP–PQ: Quantum Probab. White Noise Anal., 15. World Scientific Publishing Co., Inc., River Edge, NJ, 2003, 57–76. ISBN: 981-238-288-7

Carbone, Raffaella; Fagnola, Franco. The Feller property of a class of quantum Markov semigroups. Aportaciones Mat. Investig., 16. Sociedad Matemática Mexicana, México, 2001, 143–158. ISBN: 968-36-9572-8

Carbone, Raffaella. Binomial approximation of Brownian motion and its maximum. Statist. Probab. Lett. 69 (2004), no. 3, 271–285.

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