# Quantum Markov Semigroups in Analysis, Physics and Probability (15w5086)

Arriving in Oaxaca, Mexico Sunday, August 23 and departing Friday August 28, 2015

## Organizers

Franco Fagnola (Politecnico di Milano)

Roberto Quezada (Universidad Autonoma Metropolitana, Iztapalapa Campus)

Stephen Sontz (Centro de Investigación en Matemáticas, Guanajuato.)

Aurel Stan (Ohio State University at Marion)

## Objectives

The meeting aims to identify common research topics and stimulating joint research between mathematicians studying semigroups of completely positive maps on operator algebras and their physical applications to models of quantum open systems. Novel ideas and techniques are expected to emerge from the combination of tools developed in the study of special problems, providing a better understanding of the underlying mathematical models. Applications to open quantum systems, non-equilibrium (entropy methods), decoherence, quantization and orthogonal polynomials, (interacting) Fock spaces, play a key role and they will serve as a focal point for the presented talks and discussions.

The workshop is also aimed at introducing young researchers to the subject in order to establish a broad and continuous cooperation with internationally recognized experts.

The main purpose of this workshop is to collect small groups of experts who recently worked on problems involving quantum Markov semigroups in particular on the following subjects.

Motivated by non-equilibrium physical problems, starting from the notion of dynamical detailed balance emerged from the stochastic limit of quantum systems (Accardi and Imafuku), the Italian and Mexican group started investigations on the structure of QMSs which do not satisfy a quantum detailed balance condition. The goal is finding the algebraic structure of a class of generators sufficiently wide to cover non trivial non-equilibrium models but simple enough to allow one to find (possibly) explicit formulae for solutions and representations of currents and “quantum” cycles. This would give a full (i.e. independent of special models) understanding of the dynamical detailed balance condition.Preliminary results have been obtained by Accardi, Fagnola and Quezada in [1].

Entropy production plays a key role in the characterization of non-equilibrium QMSs, as shown by Fagnola and Rebolledo, and Bolanos and Quezada in recent papers, see [2] and [3]. They have proposed two similar definitions of entropy production rate. Moreover, they showed that the entropy production rate vanishes if and only if the QMS satisfies a quantum detailed balance condition, namely it is in an equilibrium state. We expect that the study of the structure of the entropy production formulae allows us to single out useful representations of the generator and, eventually, identify currents and quantum cycles.

Entropy and entropy inequalities were also fruitfully applied in the study of the behavior of open quantum systems in the Wigner function approach (Arnold and Carlen [4]). We plan to discuss their role in these problems in order to understand the applicability of these analytic methods in the operator algebraic framework.

A quantum system undergoes decoherence, roughly speaking, if its state at time

Several results on the asymptotic behavior of QMS on operator algebras are now available. Although they are often quite general and their application in concrete models often gives a limited information, we think they are nevertheless useful as starting points for further and deeper investigations. F. Mukhammedov is an expert in this field.

Most studied QMSs from a mathematical point of view are perhaps those obtained by second quantization because of the availability of simple explicit formulae and their close relationship with classical Ornstein-Uhlenbeck semigroups (Carbone [9], Lindsay[10]). On the other hand, proofs of deep results like hypercontractivity depend on entropy inequalities. We plan to discuss also the latest results on this topic and their generalizations to QMSs obtained from second quantization in interacting Fock spaces by means of orthogonal polynomial techniques. Aurel Stan is an expert in the last topic, see [14]-[15].

The stochastic limit of a quantum system interacting with a reservoir yields a QMS but also a quantum stochastic differential equation, where the driving noises are determined by “stochastic resonances” of the system with the reservoir. On the other hand, given a QMS, it is possible to construct its dilation, namely to represent it as the projection of a unitary (reversible) evolution. The dynamical detailed balance condition seems easier to understand when considering also the reservoir and so, in a way, dilations of the QMS itself.

References.

[1] Accardi L., Fagnola F. and Quezada R., Dynamical detailed balance and local KMS condition for non-equilibrium stationary states, in Proceedings of the International Conference in Memoriam of Shuichi Tasaki, Bussei Kenkyu vol. 97 (3), 318--356, 2011 (ISSN 0525-2997) Japan.

[2] F. Fagnola and R. Rebolledo, From classical to quantum entropy production, in Proceedings of the 29th Conference on Quantum Probability and Related Topics, QP-PQ Quantum Probability and White Noise Analysis, Vol. 25, (2010) 245-261.

[3] J. R. Bolaños-Servin and R. Quezada, A cycle representation and entropy production for circulant quantum Markov semigroups, Inf. Dim. Anal. Quant. Prob. Relat. Top. 16, (2013) 1350016 (23 pages)

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

[5] P. Blanchard and R. Olkiewicz, Decoherence as irreversible dynamical process in open quantum systems. in Open quantum systems III, Lect Notes in ath 1882, (2006), 117--159.

[6] R. Rebolledo, Decoherence of quantum Markov semigroups, Ann. Inst. H. Poincare Prob. Statist. 41 (2005), 349--373.

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

[8] R. Carbone, E. Sasso and V. Umanita, Decoherence for quantum markov semigroups on matrix algebras, Ann. Henri Poincare 14, (2013), 681--697.

[9] R. Carbone and E. Sasso, Hypercontractivity for a quantum Ornstein-Uhlenbeck semigroup, Probab. Theory Related Fields, 140 (2008), 505-522.

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

[11] D. Goswami, J.M. Lindsay, K.B. Sinha and S.J. Wills, Dilation of Markovian cocycles on a von Neumann algebra, Pacific J. Math. 211 (2003), 211-247.

[12] B.V.R. Bhat, Minimal isometric dilations of operator cocycles, Integral Equations Operator Theory 42 (2002), 125--141.

[13] B.V.R. Bhat, Dilations, cocycles and product systems, in Quantum independent increment processes I, Lecture Notes in Math. 1865, Springer (2005), 273--291.

[14] P. Da Pelo, A. Lanconelli and A. Stan, A Holder-Young-Lieb inequality for norms of Gaussian Wick products, Infin. Dimens. Anal. Quantum Probab, Relat. Top. 14 (2011), 375--407.

[15] L. Accardi, H.H. Kuo and A. Stan, An interacting Fock space charcterization of probability measures. Commun. Stoch. Anal. 3 (2009), 85--99.

The workshop is also aimed at introducing young researchers to the subject in order to establish a broad and continuous cooperation with internationally recognized experts.

**Expected topics for collaboration**The main purpose of this workshop is to collect small groups of experts who recently worked on problems involving quantum Markov semigroups in particular on the following subjects.

*QMSs and non-equilibrium phenomena*Motivated by non-equilibrium physical problems, starting from the notion of dynamical detailed balance emerged from the stochastic limit of quantum systems (Accardi and Imafuku), the Italian and Mexican group started investigations on the structure of QMSs which do not satisfy a quantum detailed balance condition. The goal is finding the algebraic structure of a class of generators sufficiently wide to cover non trivial non-equilibrium models but simple enough to allow one to find (possibly) explicit formulae for solutions and representations of currents and “quantum” cycles. This would give a full (i.e. independent of special models) understanding of the dynamical detailed balance condition.Preliminary results have been obtained by Accardi, Fagnola and Quezada in [1].

*Entropy methods*Entropy production plays a key role in the characterization of non-equilibrium QMSs, as shown by Fagnola and Rebolledo, and Bolanos and Quezada in recent papers, see [2] and [3]. They have proposed two similar definitions of entropy production rate. Moreover, they showed that the entropy production rate vanishes if and only if the QMS satisfies a quantum detailed balance condition, namely it is in an equilibrium state. We expect that the study of the structure of the entropy production formulae allows us to single out useful representations of the generator and, eventually, identify currents and quantum cycles.

Entropy and entropy inequalities were also fruitfully applied in the study of the behavior of open quantum systems in the Wigner function approach (Arnold and Carlen [4]). We plan to discuss their role in these problems in order to understand the applicability of these analytic methods in the operator algebraic framework.

*Decoherence phenomena*A quantum system undergoes decoherence, roughly speaking, if its state at time

*t*, represented by a trace-one positive matrix (state), loses coherences, namely its off-diagonal matrix elements vanish as time goes to infinity. Several mathematical approaches (Blanchard and Olkiewicz [5], Rebolledo [6]) have been proposed to study this phenomenon, but all of them eventually entail the structure of the QMS and its generator. We plan to discuss the latest results (Carbone and Umanità [7]-[8]), also connected in a natural way to ergodicity, their relationship with non-equilibrium QMSs and their structure.*Ergodicity and asymptotic behavior*Several results on the asymptotic behavior of QMS on operator algebras are now available. Although they are often quite general and their application in concrete models often gives a limited information, we think they are nevertheless useful as starting points for further and deeper investigations. F. Mukhammedov is an expert in this field.

*Second quantization and orthogonal polynomials (deformations of commutation relations)*Most studied QMSs from a mathematical point of view are perhaps those obtained by second quantization because of the availability of simple explicit formulae and their close relationship with classical Ornstein-Uhlenbeck semigroups (Carbone [9], Lindsay[10]). On the other hand, proofs of deep results like hypercontractivity depend on entropy inequalities. We plan to discuss also the latest results on this topic and their generalizations to QMSs obtained from second quantization in interacting Fock spaces by means of orthogonal polynomial techniques. Aurel Stan is an expert in the last topic, see [14]-[15].

*Dilations and flows*The stochastic limit of a quantum system interacting with a reservoir yields a QMS but also a quantum stochastic differential equation, where the driving noises are determined by “stochastic resonances” of the system with the reservoir. On the other hand, given a QMS, it is possible to construct its dilation, namely to represent it as the projection of a unitary (reversible) evolution. The dynamical detailed balance condition seems easier to understand when considering also the reservoir and so, in a way, dilations of the QMS itself.

References.

[1] Accardi L., Fagnola F. and Quezada R., Dynamical detailed balance and local KMS condition for non-equilibrium stationary states, in Proceedings of the International Conference in Memoriam of Shuichi Tasaki, Bussei Kenkyu vol. 97 (3), 318--356, 2011 (ISSN 0525-2997) Japan.

[2] F. Fagnola and R. Rebolledo, From classical to quantum entropy production, in Proceedings of the 29th Conference on Quantum Probability and Related Topics, QP-PQ Quantum Probability and White Noise Analysis, Vol. 25, (2010) 245-261.

[3] J. R. Bolaños-Servin and R. Quezada, A cycle representation and entropy production for circulant quantum Markov semigroups, Inf. Dim. Anal. Quant. Prob. Relat. Top. 16, (2013) 1350016 (23 pages)

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

[5] P. Blanchard and R. Olkiewicz, Decoherence as irreversible dynamical process in open quantum systems. in Open quantum systems III, Lect Notes in ath 1882, (2006), 117--159.

[6] R. Rebolledo, Decoherence of quantum Markov semigroups, Ann. Inst. H. Poincare Prob. Statist. 41 (2005), 349--373.

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

[8] R. Carbone, E. Sasso and V. Umanita, Decoherence for quantum markov semigroups on matrix algebras, Ann. Henri Poincare 14, (2013), 681--697.

[9] R. Carbone and E. Sasso, Hypercontractivity for a quantum Ornstein-Uhlenbeck semigroup, Probab. Theory Related Fields, 140 (2008), 505-522.

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

[11] D. Goswami, J.M. Lindsay, K.B. Sinha and S.J. Wills, Dilation of Markovian cocycles on a von Neumann algebra, Pacific J. Math. 211 (2003), 211-247.

[12] B.V.R. Bhat, Minimal isometric dilations of operator cocycles, Integral Equations Operator Theory 42 (2002), 125--141.

[13] B.V.R. Bhat, Dilations, cocycles and product systems, in Quantum independent increment processes I, Lecture Notes in Math. 1865, Springer (2005), 273--291.

[14] P. Da Pelo, A. Lanconelli and A. Stan, A Holder-Young-Lieb inequality for norms of Gaussian Wick products, Infin. Dimens. Anal. Quantum Probab, Relat. Top. 14 (2011), 375--407.

[15] L. Accardi, H.H. Kuo and A. Stan, An interacting Fock space charcterization of probability measures. Commun. Stoch. Anal. 3 (2009), 85--99.