Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2509.13377 · Stochastic Processes and their Applications

Purification of quantum trajectories in infinite dimensions

Federico Girotti, A. Vitale

Published Jan 17, 2026• 0 citations
arXivHTMLPDF

Abstract

In this work we exhibit a class of examples that show that the characterization of purification of quantum trajectories in terms of ‘dark’ subspaces that was proved for finite dimensional systems ( Infin. Dimens. Anal. Quantum Probab. Relat. Top. , 06(02), 223-243, 2003 and IMS Lectures Notes-Monograph Series , 48, 252-261, 2006) fails to hold in infinite dimensional ones. Moreover, we prove that the new phenomenon emerging in our class of models and preventing purification to happen is the only new possibility that appears in infinite dimensional systems. Our proof strategy points out that the presence of new phenomena in infinite dimensional systems is due to the fact that the set of orthogonal projections is not sequentially compact. Having in mind this insight, we are able to prove that the finite dimensional result extends to a class of infinite dimensional models.

DOI
10.1016/j.spa.2026.104889

Related papers

The rate of purification of quantum trajectories

Maël Bompais, Nina H. Amini, Juan P. Garrahan +1 more

We investigate the behavior of quantum trajectories conditioned on measurement outcomes. Under a condition related to the absence of so-called dark subspaces, Kümmerer and Maassen had shown that such trajectories almost surely purify in the long run. In this article, we first present a simple alternative proof of this result using Lyapunov methods. We then strengthen the conclusion by proving that purification actually occurs at an exponential rate in expectation, again using a Lyapunov approach. Furthermore, we address the quantum state estimation problem by propagating two trajectories under the same measurement record--one from the true initial state and the other from an arbitrary initial guess--and show that the estimated trajectory converges exponentially fast to the true one, thus quantifying the rate at which information is progressively revealed through the measurement process.

quant-pharXiv:2601.14023•Jan 21, 2026•0 citations

Finite Controllability of Infinite-Dimensional Quantum Systems

Anthony M. Bloch, Roger W. Brockett, Chitra Rangan

Quantum phenomena of interest in connection with applications to computation and communication often involve generating specific transfers between eigenstates, and their linear superpositions. For some quantum systems, such as spin systems, the quantum evolution equation (the Schrödinger equation) is finite-dimensional and old results on controllability of systems defined on on Lie groups and quotient spaces provide most of what is needed insofar as controllability of non-dissipative systems is concerned. However, in an infinite-dimensional setting, controlling the evolution of quantum systems often presents difficulties, both conceptual and technical. In this paper we present a systematic approach to a class of such problems for which it is possible to avoid some of the technical issues. In particular, we analyze controllability for infinite-dimensional bilinear systems under assumptions that make controllability possible using trajectories lying in a nested family of pre-defined subspaces. This result, which we call the Finite Controllability Theorem, provides a set of sufficient conditions for controllability in an infinite-dimensional setting. We consider specific physical systems that are of interest for quantum computing, and provide insights into the types of quantum operations (gates) that may be developed.

arXiv:quant-ph/0608075•IEEE Transactions on Automatic Control•Mar 2, 2010•70 citations

The rate of purification of quantum trajectories

Maël Bompais, Nina H. Amini, Juan P. Garrahan +1 more

Abstract We investigate the behavior of quantum trajectories conditioned on measurement outcomes. Under a condition related to the absence of so-called dark subspaces, Kümmerer and Maassen showed that such trajectories almost surely purify in the long run. An alternative, simple proof of this result is first provided using Lyapunov methods. The result is then strengthened by showing that purification occurs at an exponential rate in expectation, again via a Lyapunov approach. The quantum filter stability problem is then considered: two trajectories are propagated under the same measurement record—one from the true initial state, the other from an arbitrary initial guess—and it is shown that the estimated trajectory converges exponentially fast to the true one. This quantifies the rate at which information about the current state is progressively revealed through the measurement process.

Journal of Physics A Mathematical and Theoretical•Sep 8, 2026•0 citations

Quantum trajectory of the one-atom maser

Tristan Benoist, Laurent Bruneau, Clément Pellegrini

The evolution of a quantum system undergoing repeated indirect measurements naturally leads to a Markov chain on the set of states which is called a quantum trajectory. In this paper we consider a specific model of such a quantum trajectory associated to the one-atom maser model. It describes the evolution of one mode of the quantized electromagnetic field in a cavity interacting with two-level atoms. When the system is non-resonant we prove that this Markov chain admits a unique invariant probability measure. We moreover prove convergence in the Wasserstein metric towards this invariant measure. These results rely on a purification theorem: almost surely the state of the system approaches the set of pure states. Compared to similar results in the literature, the system considered here is infinite dimensional. While existence of an invariant measure is a consequence of the compactness of the set of states in finite dimension, in infinite dimension existence of an invariant measure is not free. Furthermore usual purification criterions in finite dimension have no straightforward equivalent in infinite dimension.

arXiv:2403.20094•Probability and Mathematical Physics•Aug 11, 2025•2 citations

The controllability of infinite quantum systems

Roger W. Brockett, Chitra Rangan, Anthony M. Bloch

Quantum phenomena of interest in connection with computation and communication often deal with transfers between eigenstates. For systems having only a finite number of states, the quantum evolution equation is finite-dimensional and the results on controllability on Lie groups as worked out decades ago provide most of what is needed in so far as controllability of dissipationless systems is concerned. However, for infinite-dimensional evolution of quantum systems, many difficulties, both conceptual and technical, remain. In this paper we organize some recent results from the physics literature in control theoretic terms and emphasize the type of analysis needed to go beyond what basic differential geometry can provide. In particular, we analyze from a controllability point of view the important results of Law and Eberly, and discuss the problem of controllability of quantum systems subject to the constraint that the trajectories must lie in pre-defined subspaces. The study of this problem is motivated by recent work on the design of quantum computers that avoid loss of coherence by operating in decoherence free subspaces.

IEEE Conference on Decision and Control (CDC)•Jul 8, 2004•17 citations

Quantum Theory in Finite Dimension Cannot Explain Every General Process with Finite Memory

Marco Fanizza, Josep Lumbreras, Andreas Winter

Abstract Arguably, the largest class of stochastic processes generated by means of a finite memory consists of those that are sequences of observations produced by sequential measurements in a suitable generalized probabilistic theory (GPT). These are constructed from a finite-dimensional memory evolving under a set of possible linear maps, and with probabilities of outcomes determined by linear functions of the memory state. Examples of such models are given by classical hidden Markov processes, where the memory state is a probability distribution, and at each step it evolves according to a non-negative matrix, and hidden quantum Markov processes, where the memory is a finite-dimensional quantum system, and at each step it evolves according to a completely positive map. Here we show that the set of processes admitting a finite-dimensional explanation do not need to be explainable in terms of either classical probability or quantum mechanics. To wit, we exhibit families of processes that have a finite-dimensional explanation, defined manifestly by the dynamics of an explicitly given GPT, but that do not admit a quantum, and therefore not even classical, explanation in finite dimension. Furthermore, we present a family of quantum processes on qubits and qutrits that do not admit a classical finite-dimensional realization, which includes examples introduced earlier by Fox, Rubin, Dharmadikari and Nadkarni as functions of infinite-dimensional Markov chains, and lower bound the size of the memory of a classical model realizing a noisy version of the qubit processes.

arXiv:2209.11225•Communications in Mathematical Physics•Feb 1, 2024•4 citations
In this work we exhibit a class of examples that show that the characterization of purification of quantum trajectories in terms of ‘dark’ subspaces that was proved for finite dimensional systems ( Infin. Dimens. Anal. Quantum Probab. Relat. Top. , 06(02), 223-243, 2003 and IMS Lectures Notes-Monograph Series , 48, 252-261, 2006) fails to hold in infinite dimensional ones. Moreover, we prove that the new phenomenon emerging in our class of models and preventing purification to happen is the only new possibility that appears in infinite dimensional systems. Our proof strategy points out that the presence of new phenomena in infinite dimensional systems is due to the fact that the set of orthogonal projections is not sequentially compact. Having in mind this insight, we are able to prove that the finite dimensional result extends to a class of infinite dimensional models.
We investigate the behavior of quantum trajectories conditioned on measurement outcomes. Under a condition related to the absence of so-called dark subspaces, Kümmerer and Maassen had shown that such trajectories almost surely purify in the long run. In this article, we first present a simple alternative proof of this result using Lyapunov methods. We then strengthen the conclusion by proving that purification actually occurs at an exponential rate in expectation, again using a Lyapunov approach. Furthermore, we address the quantum state estimation problem by propagating two trajectories under the same measurement record--one from the true initial state and the other from an arbitrary initial guess--and show that the estimated trajectory converges exponentially fast to the true one, thus quantifying the rate at which information is progressively revealed through the measurement process.
Quantum phenomena of interest in connection with applications to computation and communication often involve generating specific transfers between eigenstates, and their linear superpositions. For some quantum systems, such as spin systems, the quantum evolution equation (the Schrödinger equation) is finite-dimensional and old results on controllability of systems defined on on Lie groups and quotient spaces provide most of what is needed insofar as controllability of non-dissipative systems is concerned. However, in an infinite-dimensional setting, controlling the evolution of quantum systems often presents difficulties, both conceptual and technical. In this paper we present a systematic approach to a class of such problems for which it is possible to avoid some of the technical issues. In particular, we analyze controllability for infinite-dimensional bilinear systems under assumptions that make controllability possible using trajectories lying in a nested family of pre-defined subspaces. This result, which we call the Finite Controllability Theorem, provides a set of sufficient conditions for controllability in an infinite-dimensional setting. We consider specific physical systems that are of interest for quantum computing, and provide insights into the types of quantum operations (gates) that may be developed.
Abstract We investigate the behavior of quantum trajectories conditioned on measurement outcomes. Under a condition related to the absence of so-called dark subspaces, Kümmerer and Maassen showed that such trajectories almost surely purify in the long run. An alternative, simple proof of this result is first provided using Lyapunov methods. The result is then strengthened by showing that purification occurs at an exponential rate in expectation, again via a Lyapunov approach. The quantum filter stability problem is then considered: two trajectories are propagated under the same measurement record—one from the true initial state, the other from an arbitrary initial guess—and it is shown that the estimated trajectory converges exponentially fast to the true one. This quantifies the rate at which information about the current state is progressively revealed through the measurement process.
The evolution of a quantum system undergoing repeated indirect measurements naturally leads to a Markov chain on the set of states which is called a quantum trajectory. In this paper we consider a specific model of such a quantum trajectory associated to the one-atom maser model. It describes the evolution of one mode of the quantized electromagnetic field in a cavity interacting with two-level atoms. When the system is non-resonant we prove that this Markov chain admits a unique invariant probability measure. We moreover prove convergence in the Wasserstein metric towards this invariant measure. These results rely on a purification theorem: almost surely the state of the system approaches the set of pure states. Compared to similar results in the literature, the system considered here is infinite dimensional. While existence of an invariant measure is a consequence of the compactness of the set of states in finite dimension, in infinite dimension existence of an invariant measure is not free. Furthermore usual purification criterions in finite dimension have no straightforward equivalent in infinite dimension.
Quantum phenomena of interest in connection with computation and communication often deal with transfers between eigenstates. For systems having only a finite number of states, the quantum evolution equation is finite-dimensional and the results on controllability on Lie groups as worked out decades ago provide most of what is needed in so far as controllability of dissipationless systems is concerned. However, for infinite-dimensional evolution of quantum systems, many difficulties, both conceptual and technical, remain. In this paper we organize some recent results from the physics literature in control theoretic terms and emphasize the type of analysis needed to go beyond what basic differential geometry can provide. In particular, we analyze from a controllability point of view the important results of Law and Eberly, and discuss the problem of controllability of quantum systems subject to the constraint that the trajectories must lie in pre-defined subspaces. The study of this problem is motivated by recent work on the design of quantum computers that avoid loss of coherence by operating in decoherence free subspaces.
Abstract Arguably, the largest class of stochastic processes generated by means of a finite memory consists of those that are sequences of observations produced by sequential measurements in a suitable generalized probabilistic theory (GPT). These are constructed from a finite-dimensional memory evolving under a set of possible linear maps, and with probabilities of outcomes determined by linear functions of the memory state. Examples of such models are given by classical hidden Markov processes, where the memory state is a probability distribution, and at each step it evolves according to a non-negative matrix, and hidden quantum Markov processes, where the memory is a finite-dimensional quantum system, and at each step it evolves according to a completely positive map. Here we show that the set of processes admitting a finite-dimensional explanation do not need to be explainable in terms of either classical probability or quantum mechanics. To wit, we exhibit families of processes that have a finite-dimensional explanation, defined manifestly by the dynamics of an explicitly given GPT, but that do not admit a quantum, and therefore not even classical, explanation in finite dimension. Furthermore, we present a family of quantum processes on qubits and qutrits that do not admit a classical finite-dimensional realization, which includes examples introduced earlier by Fox, Rubin, Dharmadikari and Nadkarni as functions of infinite-dimensional Markov chains, and lower bound the size of the memory of a classical model realizing a noisy version of the qubit processes.