In this review we discuss some results on the asymptotic dynamics of finite-dimensional open quantum systems in the Heisenberg picture. Both the spectral and algebraic approaches to this topic are addressed, with particular emphasis on their relationship. The analysis is conducted in both the discrete-time and the continuous-time Markovian settings. In the final part of the work, some issues emerging in the infinite-dimensional case are also discussed. In particular, we provide an example of a Markovian evolution whose decoherence-free algebra is a type III von Neumann algebra.In this review we discuss some results on the asymptotic dynamics of finite-dimensional open quantum systems in the Heisenberg picture. Both the spectral and algebraic approaches to this topic are addressed, with particular emphasis on their relationship. The analysis is conducted in both the discrete-time and the continuous-time Markovian settings. In the final part of the work, some issues emerging in the infinite-dimensional case are also discussed. In particular, we provide an example of a Markovian evolution whose decoherence-free algebra is a type III von Neumann algebra.

quant-pharXiv:2511.18021Springer INdAM seriesNov 25, 20252 citations

An Approach by Representation of Algebras for Decoherence-Free Subspaces

Marco A. S. Trindade, Eric Pinto, J. D. M. Vianna

arXiv:1405.0611Advances in Applied Clifford AlgebrasNov 12, 20151 citation

Quantum dynamics and algebras of unbounded operators

Werner Timmermann

De Gruyter eBooksDec 31, 19900 citations

Constructing Large and Structured Decoherence-Free Subspaces in Hybrid Quantum Systems

Catalin-Mihai Halati, Tony Jin, Roderich Moessner

We develop a framework to construct large decoherence-free subspaces with a nontrivial structure. The construction is based on hybrid quantum systems in which quantum matter is coupled to a dissipative bosonic mode. Dissipation imposes a global constraint on the matter by selecting the matter null states in the long-time limit. The decoherence-free subspaces spanned by matter null states can exhibit exotic quantum properties and emergent symmetries that do not characterize the full Liouvillian dynamics. We show how to construct many-body operators with physically interesting null states by employing representation theory of Lie algebras. This approach allows us to choose models based on their underlying algebraic structure. We consider explicitly the example of coupling a lossy optical cavity to the directed tunneling operator in one-dimension. In this case, the symmetries of the decoherence-free subspace originate in an $\mathfrak{sl}(2,\mathbb{C})$ algebra, leading to an exponentially large number of steady states. We characterize the null states for both fermionic and bosonic particles, employing analytical and numerical methods. The states exhibit several properties that underline their complex quantum nature: long-range kinetic correlations and volume-law entanglement. The nonreciprocal nature of the directed tunneling dynamics leads to the emergence in the many-body regime of a symmetry-constrained Liouvillian skin effect. The open light-matter system exhibits a strong symmetry stemming from spinless $η$ -pairing algebras. We show how the nontrivial properties emerge in the dissipative dynamics from generic initial states, by performing the quantum evolution of the hybrid open system with time-dependent matrix product state. Our framework opens avenues for dissipatively engineering complex quantum correlations and harnessing nonreciprocity in a controlled manner.We develop a framework to construct large decoherence-free subspaces with a nontrivial structure. The construction is based on hybrid quantum systems in which quantum matter is coupled to a dissipative bosonic mode. Dissipation imposes a global constraint on the matter by selecting the matter null states in the long-time limit. The decoherence-free subspaces spanned by matter null states can exhibit exotic quantum properties and emergent symmetries that do not characterize the full Liouvillian dynamics. We show how to construct many-body operators with physically interesting null states by employing representation theory of Lie algebras. This approach allows us to choose models based on their underlying algebraic structure. We consider explicitly the example of coupling a lossy optical cavity to the directed tunneling operator in one-dimension. In this case, the symmetries of the decoherence-free subspace originate in an $\mathfrak{sl}(2,\mathbb{C})$ algebra, leading to an exponentially large number of steady states. We characterize the null states for both fermionic and bosonic particles, employing analytical and numerical methods. The states exhibit several properties that underline their complex quantum nature: long-range kinetic correlations and volume-law entanglement. The nonreciprocal nature of the directed tunneling dynamics leads to the emergence in the many-body regime of a symmetry-constrained Liouvillian skin effect. The open light-matter system exhibits a strong symmetry stemming from spinless $η$-pairing algebras. We show how the nontrivial properties emerge in the dissipative dynamics from generic initial states, by performing the quantum evolution of the hybrid open system with time-dependent matrix product state. Our framework opens avenues for dissipatively engineering complex quantum correlations and harnessing nonreciprocity in a controlled manner.

cond-mat.quant-gasCross-listedarXiv:2610.12348Oct 9, 20260 citations

Entanglement production and decoherence-free subspace of two single-mode cavities embedded in a common environment

Jun‐Hong An, Shun-Jin Wang, Hong‐Gang Luo

A system consisting of two identical single-mode cavities coupled to a common environment is investigated within the framework of algebraic dynamics. Based on the left and right representations of the Heisenberg–Weyl algebra, the algebraic structure of the master equation is explored and exact analytical solutions of this system are obtained. It is shown that for such a system, the environment can produce entanglement in contrast to its commonly believed role of destroying entanglement. In addition, the collective zero-mode eigensolutions of the system are found to be free of decoherence against the dissipation of the environment. These decoherence-free states may be useful in quantum information and quantum computation.A system consisting of two identical single-mode cavities coupled to a common environment is investigated within the framework of algebraic dynamics. Based on the left and right representations of the Heisenberg–Weyl algebra, the algebraic structure of the master equation is explored and exact analytical solutions of this system are obtained. It is shown that for such a system, the environment can produce entanglement in contrast to its commonly believed role of destroying entanglement. In addition, the collective zero-mode eigensolutions of the system are found to be free of decoherence against the dissipation of the environment. These decoherence-free states may be useful in quantum information and quantum computation.

arXiv:quant-ph/0505087Journal of Physics A Mathematical and GeneralApr 6, 200526 citations

\mathfrak{sl}(2,\mathbb{C})

algebra, leading to an exponentially large number of steady states. We characterize the null states for both fermionic and bosonic particles, employing analytical and numerical methods. The states exhibit several properties that underline their complex quantum nature: long-range kinetic correlations and volume-law entanglement. The nonreciprocal nature of the directed tunneling dynamics leads to the emergence in the many-body regime of a symmetry-constrained Liouvillian skin effect. The open light-matter system exhibits a strong symmetry stemming from spinless

η

-pairing algebras. We show how the nontrivial properties emerge in the dissipative dynamics from generic initial states, by performing the quantum evolution of the hybrid open system with time-dependent matrix product state. Our framework opens avenues for dissipatively engineering complex quantum correlations and harnessing nonreciprocity in a controlled manner.