# Quantum Transport Equations and Applications (18w5059)

Arriving in Oaxaca, Mexico Sunday, September 2 and departing Friday September 7, 2018

## Organizers

George Androulakis (University of South Carolina)

Franco Fagnola (Politecnico di Milano)

Eric Carlen (Rutgers University)

Roberto Quezada (Universidad Autonoma Metropolitana, Iztapalapa Campus)

## Objectives

Transport equations naturally arise in several models of evolution phenomena in classical and quantum systems. In classical systems these are partial differential equations on a space of probability densities, while in the quantum case they are dissipative evolution equations for density matrices. In the second case, they are often proposed in a phenomenological way. Meaningful models can be deduced from fundamental physical laws through stochastic limit methods. We propose to study a class of such models as a paradigm for quantum transport phenomena in equilibrium and local equilibrium conditions. \\

In a more precise way we expect that the main topics of the workshop will include: \\

• Transport equations. \\

Talks of experts in the field will present the state of the art in classical transport equations highlighting the main current streams, open problems and future perspectives. Special attention will be given to those methods potentially applicable in the quantum case, as for instance, gradient flows and entropy methods for the study of the speed of convergence to steady states. \\ • Stochastic limit type semigroups.\\

Evolution equations of open quantum systems are suitably formulated via quantum Markov semigroups. The powerful technique of the stochastic limit, allows one to deduce, from fundamental physical laws, appropriate evolution equations for modeling quantum phenomena, in particular quantum transport. Problems of convergence to steady states and evaluation of the speed of convergence for quantum Markov semigroups with equilibrium (detailed balance) or local equilibrium steady states will be investigated. Extensions of most fruitful methods in partial differential equations to the quantum case will be analyzed. The identification of good classes of local equilibrium states will also be discussed. \\

• Quantum kinetic equations.\\

As in the classical case of the Boltzmann equation, kinetic evolution equations arise when one seeks a closed evolution equation for the single particle marginal of a many particle system with pair interactions. This is possible when chaos is propagated meaning that as the number of particles tends to infinity, the 2-particle reduced density matrix remains the tensor product of the one-particle density matrix with itself. This has been proved in some cases, but many open problems will be investigated. Physically and mathematically interesting problems involving quantum master equations (a class of the linear quantum Markov semigroups discussed above) and the non-linear equations one gets for single particle marginals (such as the Hartree equation for Fermi systems) will be discussed. This subject ties well into the next topic; kinetic theory was the origin of the “entropy method”. \\

• Hypercontractivity, Hypocoercivity and Entropy Methods.\\

Hypercontractivity, hypocoercivity and entropy methods play a fundamental role in the study of the qualitative behavior of solutions to partial differential equations and often allow one to derive explicit or even optimal constants in functional inequalities giving quantitative evaluation of convergence rates to steady states. Among these methods only a few have been studied in the noncommutative case. We expect to broaden the class of those which are fruitfully extendable and also understand intrinsic obstrutions due to noncommutativity. Hypercontractivity for QMSs arising from second quantizations in interacting Fock spaces will also be discussed.\\

• Applications to Physics, Information and Biology.\\

Decoherence is usually considered as the loss of quantum features of a quantum state. We believe that the above methods for studying convergence to invariant states will provide new explicit evaluations for the speed of decoherence. If time allows we will also discuss the symmetric point of view, of exploiting decoherence for the goals of quantum control.\\

The recent attempts, to use quantum master equations to verify the conjecture that quantum effects may enhance energy transfer processes in photosynthesis, will be discussed side by side with more traditional applications of classical transport equations in Biology.\\

Other application to problems in quantum information theory and quantum control will be discussed as well as non-Markovian transfer equations. \\

References:\\

[1] Accardi L., Fagnola F. and Quezada R., On three new principles in non-equibrium statistical mechanics and Markov semigroups of weak coupling limit type, Infinite Dimensional Analysis, Quantum Probability and Related Topics Vol. 19, No. 2 (2016) 1650009 (37 pages) DOI: 10.1142/S0219025716500090\\

[2] Accardi L., Kuo H.H. and Stan A:, An interacting Fock space characterization of probability measures, Commun. Stoch. Anal. 3 (2009), 85—99.\\

[3] Carbone R., Sasso E. and Umanitá V., Decoherence for positive semigroups on M2, J. Math. Phys. 52, (2011) 032202, 17 pp. \\

[4] Carbone R., Sasso E. and Umanitá V., Decoherence for quantum Markov semigroups on matrix algebras, Ann. Henri Poincaré 14, (2013), 681—697. \\

[5] Carbone R. and Sasso E:, Hypercontractivity for a quantum Ornstein Uhlenbeck semigroup, Prob. Theory and Related Fields 140, \\

[6] Carlen E. and Maas J., Gradient flow and entropy inequalities for quantum Markov semigroups with detailed balance, arXiv:1609.01254v1 [math-OA] 5 Sep 2016. \\

[ 7] Carlen E., Carvalho M.C., Degond P. and Wennberg B., A Boltzmann model for rod alignment and schooling fish, Nonlinearity 28, (2015), 1783--1803.\\

[8 ] Carlen E., Lieb E.H. and Reuver R., Entropy and entanglement bounds for reduced density matrices of fermionic states, Commun. Math. Phys. 344 (2016), 655 - 671. \\

[9] Deschamps J., Fagnola F, Sasso E. and Umanit\'a V., Structure of uniformly continuous quantum Markov semigroups. Rev. Math. Phys. 28 (2016), no. 1, 1650003, 32 pp\\

[10] Bolanos J.R. and Fagnola F., On the range of the generator of a quantum Markov semigroup, Infin. Dimens. Anal. Quantum Probab. Rel. Top., 18 (2015), 1550027, 10 pp.\\

[ 11] Lindsay J.M. and Wills S.J., Existence, positivity and contractivity for quantum stochastic flows with infinite dimensional noise, Probab. Theory and related Fields 116, (200), 505-543.\\

[12 ] Arnold A., Carlen E. and Qiangchang J., Large-time behavior of non-symmetric Fokker-Planck type equations, Commun. Stoch. Anal. 2, (2008), 153 - 175.\\

[13 ] Hermida R. and Quezada R., On the spectral gap of the n-photon absorption-emission process.Quantum probability and related topics, 143 - 159,QP-PQ: Quantum Probab. White Noise Anal. 29, World Scientific Pub. (2013)\\

[14] Rebolledo R., Decoherence of quantum Markov semigroups, Ann. Inst. H.Poincar\'e Prob. Statist. 41 (2005), 349 - 373.\\

[15] Da Pelo P., Lanconelli A. and Stan A., A sharp intepolation between the Holder and the Gaussian Young inequalities, Infin. Dimens. Anal. Quantum Probab, Relat. Top. 19 (2016), 165001 37 pp.\\

[16] Da Pelo P., Lanconelli A. and Stan A., An extensi\'on of Beckner's type Poincar\'e inequality to convolution measures in abstract Wiener spaces, Stoch. Anal. Appl. 34 (2016), 47 - 64.\\

[17] Villani C., Hypocoercivity, arXiv:math/0609050v1 [math.AP] 1 Sep. 2006. \\

[18] Temme K., Non-commutative Nash inequalities, J. Math. Phys. 57 (2016), 015217, 11 pp.\\

[19] Cubbit T., Kastoryano M., Montanaro A. and Temme K., Quantum reverse hypercontractivity, J. Math. Phys, 56 (2015), 102204, 11 pp.

In a more precise way we expect that the main topics of the workshop will include: \\

• Transport equations. \\

Talks of experts in the field will present the state of the art in classical transport equations highlighting the main current streams, open problems and future perspectives. Special attention will be given to those methods potentially applicable in the quantum case, as for instance, gradient flows and entropy methods for the study of the speed of convergence to steady states. \\ • Stochastic limit type semigroups.\\

Evolution equations of open quantum systems are suitably formulated via quantum Markov semigroups. The powerful technique of the stochastic limit, allows one to deduce, from fundamental physical laws, appropriate evolution equations for modeling quantum phenomena, in particular quantum transport. Problems of convergence to steady states and evaluation of the speed of convergence for quantum Markov semigroups with equilibrium (detailed balance) or local equilibrium steady states will be investigated. Extensions of most fruitful methods in partial differential equations to the quantum case will be analyzed. The identification of good classes of local equilibrium states will also be discussed. \\

• Quantum kinetic equations.\\

As in the classical case of the Boltzmann equation, kinetic evolution equations arise when one seeks a closed evolution equation for the single particle marginal of a many particle system with pair interactions. This is possible when chaos is propagated meaning that as the number of particles tends to infinity, the 2-particle reduced density matrix remains the tensor product of the one-particle density matrix with itself. This has been proved in some cases, but many open problems will be investigated. Physically and mathematically interesting problems involving quantum master equations (a class of the linear quantum Markov semigroups discussed above) and the non-linear equations one gets for single particle marginals (such as the Hartree equation for Fermi systems) will be discussed. This subject ties well into the next topic; kinetic theory was the origin of the “entropy method”. \\

• Hypercontractivity, Hypocoercivity and Entropy Methods.\\

Hypercontractivity, hypocoercivity and entropy methods play a fundamental role in the study of the qualitative behavior of solutions to partial differential equations and often allow one to derive explicit or even optimal constants in functional inequalities giving quantitative evaluation of convergence rates to steady states. Among these methods only a few have been studied in the noncommutative case. We expect to broaden the class of those which are fruitfully extendable and also understand intrinsic obstrutions due to noncommutativity. Hypercontractivity for QMSs arising from second quantizations in interacting Fock spaces will also be discussed.\\

• Applications to Physics, Information and Biology.\\

Decoherence is usually considered as the loss of quantum features of a quantum state. We believe that the above methods for studying convergence to invariant states will provide new explicit evaluations for the speed of decoherence. If time allows we will also discuss the symmetric point of view, of exploiting decoherence for the goals of quantum control.\\

The recent attempts, to use quantum master equations to verify the conjecture that quantum effects may enhance energy transfer processes in photosynthesis, will be discussed side by side with more traditional applications of classical transport equations in Biology.\\

Other application to problems in quantum information theory and quantum control will be discussed as well as non-Markovian transfer equations. \\

References:\\

[1] Accardi L., Fagnola F. and Quezada R., On three new principles in non-equibrium statistical mechanics and Markov semigroups of weak coupling limit type, Infinite Dimensional Analysis, Quantum Probability and Related Topics Vol. 19, No. 2 (2016) 1650009 (37 pages) DOI: 10.1142/S0219025716500090\\

[2] Accardi L., Kuo H.H. and Stan A:, An interacting Fock space characterization of probability measures, Commun. Stoch. Anal. 3 (2009), 85—99.\\

[3] Carbone R., Sasso E. and Umanitá V., Decoherence for positive semigroups on M2, J. Math. Phys. 52, (2011) 032202, 17 pp. \\

[4] Carbone R., Sasso E. and Umanitá V., Decoherence for quantum Markov semigroups on matrix algebras, Ann. Henri Poincaré 14, (2013), 681—697. \\

[5] Carbone R. and Sasso E:, Hypercontractivity for a quantum Ornstein Uhlenbeck semigroup, Prob. Theory and Related Fields 140, \\

[6] Carlen E. and Maas J., Gradient flow and entropy inequalities for quantum Markov semigroups with detailed balance, arXiv:1609.01254v1 [math-OA] 5 Sep 2016. \\

[ 7] Carlen E., Carvalho M.C., Degond P. and Wennberg B., A Boltzmann model for rod alignment and schooling fish, Nonlinearity 28, (2015), 1783--1803.\\

[8 ] Carlen E., Lieb E.H. and Reuver R., Entropy and entanglement bounds for reduced density matrices of fermionic states, Commun. Math. Phys. 344 (2016), 655 - 671. \\

[9] Deschamps J., Fagnola F, Sasso E. and Umanit\'a V., Structure of uniformly continuous quantum Markov semigroups. Rev. Math. Phys. 28 (2016), no. 1, 1650003, 32 pp\\

[10] Bolanos J.R. and Fagnola F., On the range of the generator of a quantum Markov semigroup, Infin. Dimens. Anal. Quantum Probab. Rel. Top., 18 (2015), 1550027, 10 pp.\\

[ 11] Lindsay J.M. and Wills S.J., Existence, positivity and contractivity for quantum stochastic flows with infinite dimensional noise, Probab. Theory and related Fields 116, (200), 505-543.\\

[12 ] Arnold A., Carlen E. and Qiangchang J., Large-time behavior of non-symmetric Fokker-Planck type equations, Commun. Stoch. Anal. 2, (2008), 153 - 175.\\

[13 ] Hermida R. and Quezada R., On the spectral gap of the n-photon absorption-emission process.Quantum probability and related topics, 143 - 159,QP-PQ: Quantum Probab. White Noise Anal. 29, World Scientific Pub. (2013)\\

[14] Rebolledo R., Decoherence of quantum Markov semigroups, Ann. Inst. H.Poincar\'e Prob. Statist. 41 (2005), 349 - 373.\\

[15] Da Pelo P., Lanconelli A. and Stan A., A sharp intepolation between the Holder and the Gaussian Young inequalities, Infin. Dimens. Anal. Quantum Probab, Relat. Top. 19 (2016), 165001 37 pp.\\

[16] Da Pelo P., Lanconelli A. and Stan A., An extensi\'on of Beckner's type Poincar\'e inequality to convolution measures in abstract Wiener spaces, Stoch. Anal. Appl. 34 (2016), 47 - 64.\\

[17] Villani C., Hypocoercivity, arXiv:math/0609050v1 [math.AP] 1 Sep. 2006. \\

[18] Temme K., Non-commutative Nash inequalities, J. Math. Phys. 57 (2016), 015217, 11 pp.\\

[19] Cubbit T., Kastoryano M., Montanaro A. and Temme K., Quantum reverse hypercontractivity, J. Math. Phys, 56 (2015), 102204, 11 pp.