ICTS

Purification is a notable property of quantum trajectories. As time grows, mixed states tend to converge towards pure states. In 2005 Kümmerer and Maassen provided necessary and sufficient conditions for a full purification of quantum trajectories. In 2019, with some collaborators, we used purification to show that, under the assumptions of Kümmerer and Maassen, we can classify the full set of stationary distributions for quantum trajectories.

In this presentation, after reviewing these results, I will focus on what happens when the conditions of Kümmerer and Maassen are not fulfilled. Then, some dark subspaces appear. I will explain how we were able, with Anna Szczepanek and Clément Pellegrini, still, to classify all the stationary distributions of quantum trajectories. Since dark subspaces are also relevant to quantum error correction, I will try to put some bridges with our results. I will also mention ideas related to the same work but for imperfect measurements.

This presentation concerns the preprint arXiv:2409.18655.

This will be based on the textbook/lecture notes: https://textbooks.open.tudelft.nl/textbooks/catalog/book/85

If two parties have access to entangled parts of a quantum state, the common lore suggests that when measurements are made by one of the parties and its outcomes are classically communicated to the other party, it leaves telltale signatures on the state of the part accessible to the other party. Here we show that this lore is not necessarily true -- in generic scrambling dynamics within a tripartite setting (with the $R$, $S$ and $E$ labelling the three parts), a new kind of dynamical phase emerges, wherein local measurements on $S$ are invisible to one of the remaining two parts, say $R$, despite there existing non-trivial quantum correlations and entanglement between $R$ and $S$. At the heart of this lies the fact that information scrambling transmutes local quantum information into a complex non-local web of spatiotemporal quantum correlations. This non-locality in the information then means that ignorance of the state of part $E$ can leave $R$ and $S$ with sufficient information for them to be quantum correlated or entangled but not enough for measurements on $S$ to have a non-trivial backaction on the state of $R$. This new dynamical phase is sandwiched between two conventionally expected phases where the $R$ and $S$ are either disentangled from each other or are entangled along with non-trivial measurement backaction. This provides a new characterisation of entanglement phases in terms of their response to measurements instead of the more ubiquitous measurement-induced entanglement transitions. Our results have implications for the kind of tasks that can be performed using measurement feedback within the framework of quantum interactive dynamics.

We have tried to describe how the interplay between the system environment coupling and external driving frequency shapes the dynamical properties and steady state behavior in a periodically driven transverse field Ising chain subject to measurement. We have analyzed the fate of the steady state entanglement scaling properties as a result of a measurement induced phase transition. We have explained how such steady state entanglement scaling can be computed analytically using asymptotic features of the determinant of associated correlation matrix which turned out to be of block Toeplitz form. We have pointed out the differences from the Hermitian systems in understanding the entanglement scaling behav-ior in regimes where the asymptotic analysis can be performed using Fisher-Hartwig con-jecture. Finally we have discussed how the tuning of the drive frequency controls the do- main of applicability of the Fisher-Hartwig conjecture and the emergence of the long range ordering of the effective Floquet Hamiltonian governing the properties of the system.

While a generic open quantum system decays to its steady state, continuous time crystals (CTCs) develop spontaneous oscillation and never converge to a stationary state. Just as crystals develop correlations in space, CTCs do so in time. Here, we introduce a Continuous Quasi Time Crystals (CQTC). Despite being characterized by the presence of non-decaying oscillations, this phase does not retain its long-range order, making it the time analogous of quasi-crystal structures. We investigate the emergence of this phase in a system made of two coupled collective spin sub-systems, each developing a CTC phase upon the action of a strong enough drive. The addition of a coupling enables the emergence of different synchronized phases, where both sub-systems oscillate at the same frequency. In the transition between different CTC orders, the system develops chaotic dynamics with aperiodic oscillations. These chaotic features differ from those of closed quantum systems, as the dynamics is not characterized by a unitary evolution. At the same time, the presence of non-decaying oscillations makes this phenomenon distinct from other form of chaos in open quantum system, where the system decays instead. We investigate the connection between chaos and this quasi-crystalline phase using mean-field techniques, and we confirm these results including quantum fluctuations at the lowest order.

Structure of dynamical models describing  open quantum systems including measurement back-action and decoherence: discrete-time models based on quantum channels and  left stochastic matrices;  continuous-time  models driven by Wiener processes (weak measurement) and Poisson processes (quantum jump and counting measurement).

We consider the problem of quantum error correction (QEC) for non-Markovian noise. We show that conditions for approximate QEC can be easily generalized for the case of non-Markovian noise, in the strong coupling regime where the noise map becomes non-completely-positive at intermediate times. While certain adaptive recovery schemes are ineffective against quantum non-Markovian noise, in the sense that the fidelity vanishes in finite time, a specific strategy based on the Petz map uniquely safeguards the code space even at the maximum noise limit. Focusing on the case of non-Markovian amplitude damping noise, we observe that the non-Markovian Petz map also outperforms the standard, stabilizer-based QEC code. Since implementing such a non-Markovian map poses practical challenges, we also construct a Markovian Petz map that achieves similar performance, with only a slight compromise on the fidelity.
[Based on arXiv:2411.09637]

Control and manipulation of quantum states by measurements and bath engineering in open quantum systems, and associated phenomena, such as measurement-induced phase transitions, have emerged as new paradigms in many-body physics. Here, taking a prototypical example of Josephson junction arrays (JJAs), I will discuss how repetitive monitoring can transform an insulating state in these systems to a superconductor and vice versa. To this end, we study the effects of continuous weak measurements and feedback control on isolated JJAs in the absence of any external thermal bath. The monitoring due to combined effect of measurements and feedback, inducing non-unitary evolution and dissipation, leads to a long-time steady state characterized by an effective temperature in a suitably defined semiclassical limit. However, we show that the quantum dissipation due to monitoring has fundamental differences with equilibrium quantum and/or thermal dissipation in the well-studied case of JJAs in contact with an Ohmic bath. In particular, using a variational approximation, and by considering the semiclassical, strong measurement/feedback and weak-coupling limits, we demonstrate that this difference can give rise to re-entrant steady-state phase transitions, resulting in transition from an effective low-temperature insulating normal state to superconducting state at intermediate temperature. Our work emphasizes the role of quantum feedback, that acts as an additional knob to control the effective temperature of non-equilibrium steady state leading to a phase diagram, not explored in earlier works on monitored and open quantum systems.

Quantum theory contravenes classical macrorealism by allowing a system to be in a superposition of two or more physically distinct states, producing physical consequences radically different from that of classical physics. Motivated by this, we construct superpositions between time evolution unitaries and study the dynamics of a qubit under such superposed unitary operators. We find that the superposition of unitaries significantly affects the trajectory of the qubit in the Bloch sphere by shifting the path of evolution and making the speed of evolution non-linear in time. The qubit spends more time near the poles of the Bloch sphere and passes through the equator rather quickly. This  remarkably enhances the endurance against dephasing noise, making the superposed unitaries suitable for robust quantum control tasks. Moreover, we observe an extreme violation of Leggett-Garg inequalities beyond the temporal Tsirelson's bound, which increases with increasing superposition between the unitaries. This shows stronger temporal correlations achieved by the superposed unitares. Using an NMR quantum register, we experimentally demonstrate the superposition of unitaries with the help of an ancillary qubit and verify our theoretical predictions.

This will be based on the textbook/lecture notes: https://textbooks.open.tudelft.nl/textbooks/catalog/book/85

Exploring continuous time crystals (CTCs) within the symmetric subspace of spin systems has been a subject of intensive research in recent times. Thus far, the stability of the time-crystal phase outside the symmetric subspace in such spin systems has gone largely unexplored. Here, I present results relating the effect of including the asymmetric subspaces on the dynamics of CTCs in a driven dissipative spin model. This results in multistability, and the dynamics becomes dependent on the initial state. Remarkably, this multistability leads to exotic synchronization regimes such as chimera states and cluster synchronization in an ensemble of coupled identical CTCs. Interestingly, it leads to other nonlinear phenomena such as oscillation death and signature of chaos.

(based on work with coauthors reported in Phys. Rev. Lett. 133, 260403, 2024)

Spin ensembles are promising quantum technological platforms, but their utility relies on the ability to perform quantum error correction (QEC) for decoherences in these systems. Typical QEC for ensembles requires addressing individually resolved qubits, but this is practically challenging in most realistic architectures. Here, we propose QEC schemes for unresolvable spin ensembles. By using degenerate superpositions of excited states, which are fundamentally mixed, we find codes that can protect against both individual and collective errors, including dephasing, decay, and pumping. We show how information recovery can be achieved with only collective measurement and control, and illustrate its applications in extending memory lifetime and loss-tolerant sensing.

I will explain a simple method for estimating the parameters of a continuously measured quantum system. The idea is to fit the correlation functions of the measured signal, using an exact formula derived from the theory of stochastic master equations (SME). This approach is applicable to any system whose evolution is described by a jump or diffusive SME. It allows the simultaneous estimation of many parameters for systems with large Hilbert space dimensions, and takes into account experimental constraints such as detector imperfections and signal filtering and digitisation.

I will illustrate this approach in the context of superconducting circuits, beginning with an explanation of the typical workflow for characterising these systems today. I will then describe the challenges that lie ahead as we move towards larger and more complex systems, and why novel methods are needed. I will demonstrate the proposed approach in simulation on three examples: a driven anharmonic oscillator measured by heterodyne detection, a driven two-level system under photodetection, and a recent superconducting circuit experiment continuously monitoring a two-photon dissipative oscillator.

Feedback issues relying on classical controllers (optimizing QND measurement via Markovian feedback, quantum state stabilization via Bayesian feedback) and on quantum controllers (stabilization of Schrödinger cats via autonomous feedback).

I will briefly review the operational and algorithmic approach for digital quantum simulation using different forms of quantum walk and present the example for simulating Dirac equations [1], many-body systems dynamics, complex quantum networks and open quantum systems [2]. I will also present the progress made in experimentally realizing and controlling quantum walks which with a promise for performing universal quantum computation[3].

[1] Nature Communications 11, 3720 (2020)
[2] New J. Phys. 22, 123027 (2020) ; New Journal of Physics 23, 113013 (2021)
[3] EPJ Quantum Technology 10, 43 (2023); Physical Review A 110 (3), 032615 (2024)

We devise an autonomous quantum thermal machine consisting of three pairwise-interacting qubits, two of which are locally coupled to thermal reservoirs. The machine operates autonomously, as it requires no time-coherent control, external driving or quantum bath engineering, and is instead propelled by a chemical potential bias. Under ideal conditions, we show that this out-of-equilibrium system can deterministically generate a maximally entangled steady-state between two of the qubits, or any desired pure two-qubit entangled state, emerging as a dark state of the system. We study the robustness of entanglement production with respect to several relevant parameters, obtaining nearly-maximally-entangled states well-away from the ideal regime of operation. Furthermore, we show that our machine architecture can be generalised to a configuration with 2n−1 qubits, in which only a potential bias and two-body interactions are sufficient to generate genuine multipartite maximally entangled steady states in the form of a W state of n qubits.

In this work, we develop a panoramic schematic of a quantum thermoelectric circuit theory in the steady state regime. We establish the foundations of the said premise by defining the analogs of Kirchhoff's laws for heat currents and temperature gradients. We further show that our approach encompasses various circuits like thermal diode, transistor, and Wheatstone bridge. Additionally, we have been able to develop a model of a quantum thermal step transformer. We also construct a novel model of a thermal adder circuit, paving the way to develop thermal integrated circuits. This sheds new light on the present architecture of quantum device engineering.

 In the early 1960's Roy Glauber presented a theory to characterize the temporal fluctuations in photo-detection signals.  Such fluctuations can be signatures of non-classical properties, and the theory of photo-detection gave rise to the field of quantum optics with visions to control atomic light emitters to prepare and apply a variety of quantum states of light in experiments. In the past decades, "bits and pieces" of solid-state materials were manufactured with high purity and precision, enabling observation of similar phenomena with solid state spin systems and superconducting circuits, microwaves and acoustic waves as had been studied with single atoms and photons in quantum optics.
The talk will review more recent methods that refine and elaborate on Glauber's theories to describe the dynamics of open quantum systems, i.e., systems subject to interactions with their environment. These methods reintroduce, but with a plot twist, Niels Bohr's quantum jumps in modern quantum physics, and while being employed for quantum technology applications they imply delightful encounters with the famous discussions between Niels Bohr and Albert Einstein on the interpretation of quantum theory.

The Fisher information quantifies to what precision an unknown parameter can be learned from stochastic data. In the case of independent and identically-distributed random variables the precision scales linearly with their number. The i.i.d. assumption, however, is not always justified especially for temporal data where correlations are to be expected, such as in the outcomes of continuous measurement of a quantum system. In this talk, I will show how estimation precision behaves in the presence of temporal correlations and show that the scaling remains linear for processes with finite Markov order and with what rate. The second part of the talk will focus on parameter estimation in the quantum jump unravelling of a quantum master equation.
This talk is based on:
Radaelli, M., Landi, G. T., Modi, K., & Binder, F. C. Fisher information of correlated stochastic processes. New Journal of Physics 25, 053037 (2023). https://doi.org/10.1088/1367-2630/acc01d
Radaelli, M., Smiga, J. A., Landi, G. T., & Binder, F. C. Parameter estimation for quantum jump unraveling. ArXiv:2402.06556 (2024). https://doi.org/10.48550/arXiv.2402.06556
Radaelli, M., Landi, G. T., & Binder, F. C. Gillespie algorithm for quantum jump trajectories. Physical Review A 110, 062212 (2024). https://doi.org/10.1103/PhysRevA.110.062212

Hybrid quantum systems based on collective states of a spin ensemble have served as exciting platforms for quantum technology ranging from quantum communication protocols to processing and storage of quantum information. In this talk, we present a theoretical approach to study the open dynamics of states of the spin ensemble-cavity system based on tensor-network methods. We show how these methods allow us to design and demonstrate high-fidelity transfer and protection of information in collective states of a hybrid quantum system.

In this lecture, we dive more into quantum trajectories, and discuss what kind of states are produced, and what kind of dynamics is observed when we monitor quantum systems.

Quantum state engineering plays a vital role in various applications in the field of quantum information. Different strategies, including drive-and-dissipation, adiabatic cooling, and measurement-based steering, have been proposed for state generation and manipulation, each with its upsides and downsides. Here, we address a class of measurement-based state engineering protocols where a sequence of generalized measurements is employed to steer a quantum system toward a desired (pure or mixed) target state. Previously studied measurement-based protocols relied on idealized procedures and avoided exploration of the effects of various errors stemming from imperfections of experimental realizations and external noise. We employ the quantum trajectory formalism to provide a detailed analysis of the robustness of these steering protocols against multiple classes of errors. We study a set of realistic errors that can be classified as dynamic or static, depending on whether they remain unchanged while running the protocol. More specifically, we investigate the impact of the erroneous choice of detector-system coupling, erroneous reinitialization of the detector state following a measurement step, fluctuating steering directions, and environmentally induced errors in the detector-system interaction. We show that the protocol remains fully robust against the erroneous choice of detector-system coupling parameters and presents reasonable robustness against other types of errors. Our analysis employs various quantifiers such as fidelity, trace distance, and linear entropy to characterize the protocol’s robustness and provide analytical results for these quantifiers against various errors. We introduce averaging hierarchies of stochastic equations describing individual quantum trajectories associated with detector readouts. Subsequently, we demonstrate the commutation between the classical expectation value and the time-ordering operator of the exponential of a Hamiltonian with multiplicative white noise, as well as the commutation of the expectation value and the partial trace with respect to detector outcomes. Our ideas are implemented and demonstrated for a specific class of steering platforms, addressing a single qubit.

It is shown that linearity of classical/quantum/hybrid ensemble dynamics follows from bases of statistics. Hybrid classical- -quantum trajectories and their hybrid master equations are discussed. We stress that the interaction between a classical and a quantum subsystem requires monitoring the quantum subsystem because its action on the classical subsystem can only be realized by the emerging classical signal.

* Historical overview by me of 5 independent streams leading to quantum trajectory theory. 
* My various small contributions as a PhD student when these all (more or less) came together in 1993.
* A unified description of jump and diffusion unravellings (presented by Mr Pierre Guilmin).
* My fascination with the different unravellings of simple quantum systems, and how it lead to the idea of quantum steering.
* How this would allow us to prove experimentally that there is no objective (measurement-independent) unraveling.

Superconducting qubits have provided a fertile landscape for pioneering work examining experimental quantum trajectories. Here, continuous monitoring of a quantum system's environment can be used to unravel individual quantum trajectories of the open system evolution. Many fascinating extensions of these trajectories have been explored, including quantum state smoothing, retrodiction, parameter estimation, connections to thermodynamics, topological transitions, quantum feedback, and much more. Resisting the temptation to discuss all of these topics at breakneck pace, this talk will instead focus on a simple case: a quantum system interacting with its environment via radiative decay. This is typically and ultimately characterized by quantum jumps of the system to a lower energy level. What happens before these quantum jumps occur? Here, in the absence of quantum jumps, the dissipative interaction results in coherent, yet non-unitary evolution described by an effective non-Hermitian Hamiltonian. I will survey our recent work that explores the rich landscape of these non-Hermitian dynamics highlighting connections to quantum measurement dynamics along the way.

We introduce a time-energy uncertainty relation within the context of restarts in monitored quantum dynamics [1] . Previous studies have established that the mean recurrence time, which represents the time taken to return to the initial state, is quantized as an integer multiple of the sampling time, displaying pointwise discontinuous transitions at resonances. Our findings demonstrate that the natural utilization of the restart mechanism in laboratory experiments [2], driven by finite data collection time spans, leads to a broadening effect on the transitions of the mean recurrence time. Our proposed uncertainty relation captures the underlying essence of these phenomena, by connecting the broadening of the mean hitting time near resonances, to the intrinsic energies of the quantum system and to the fluctuations of recurrence time. Our uncertainty relation has also been validated through remote experiments conducted on an International Business Machines Corporation (IBM) quantum computer. We then discuss fractional quantizatization of the recurrence time for interacting spin systems using sub-space measurements [3].
References
[1] R. Yin, Q. Wang, S. Tornow, and E. Barkai, Restart uncertainty relation for monitored quantum dynamics Proceedings of the National Academy of Sciences 122 (1) e2402912121, (2025).
[2] R. Yin, E. Barkai Restart expedites quantum walk hitting times Phys. Rev. Lett. 130, 050802 (2023).
[3] Q. Liu, S. Tornow, D. Kessler, and E. Barkai Properties of Fractionally Quantized Recurrence Times for Interacting Spin Models arXiv:2401.09810 [condmat.stat-mech] (submitted)

In this lecture, we will see how quantum trajectory theory is also the theory of sensing with quantum systems, i.e., the estimation of physical influences acting on the system. 

Measurements are able to fundamentally affect quantum dynamics. We show that a continuously measured quantum many-body system can undergo a spontaneous transition from asynchronous stochastic dynamics to noise-free stable synchronization at the level of single trajectories. We formulate general criteria for this quantum phenomenon to occur and demonstrate that the number of synchronized realizations can be controlled from none to all. We additionally find that ergodicity is typically broken, since time and ensemble averages may exhibit radically different synchronization behavior. We further introduce a quantum type of multiplexing that involves individual trajectories with distinct synchronization frequencies. Measurement-induced synchronization appears as a genuine nonclassical form of synchrony that exploits quantum superpositions.

We present a protocol for measurement-induced transition, where at each time-step we evolve a quantum Ising chain under transverse Ising Hamiltonian for time τ, and then make a global measurement with certainty. For system size ≤28, there is a transition in entanglement at some critical value τ = τc. We also calculate survival probability of the initial state and find that some quantity derived from this probability also shows a peak at τc. Using a recurrence relation, one can compute survival probability in our set-up for size upto 1000. It is found that the critical value of τ follows a scaling τc ∝1/√N. Hence, the transition occurs for finite size only. It will be interesting to investigate the size-dependence of critical point in other measurement-induced transitions.

1) Introduction to quantum superconducting circuits: resonators, qubits, readout methods
2) Measurement apparatus and their modeling: amplifiers, homodyne and heterodyne measurements, photon detectors, photon counters, quantum efficiency
3) Quantum trajectories of superconducting qubits and cavities: quantum jumps, diffusive trajectories using dispersive measurement and/or fluorescence, past quantum states approach
4) Measurement-based feedback:  stabilization of qubit states and trajectories, stabilization of cavity states, use of neural networks, pros and cons of feedback control compared to reservoir engineering techniques, applications

One-dimensional atoms (1D atoms) refer to quantum emitters interacting with light fields confined in a single dimension of space. Owing to the huge number of degrees of freedom of the field, the dynamics of such devices is usually solved in the quantum open system paradigm where the atom (the field) is the system under study (the bath). Recently, so-called Autonomous Collisional Models (ACM) have provided Hamiltonian solutions to the dynamics of 1D atoms, where the atom and the field are two parts of a closed and isolated system. In addition to the interest of providing exact light-atom states, such models are autonomous: the global energy of the system is conserved, allowing for accurate energy balances.

In this talk, I will present a new framework dubbed Bipartite Quantum Energetics (BQE), which allows us to analyse energy exchanges within closed, isolated bipartite systems, and apply it to 1D atoms. In BQE, b-work (b-heat) refer to energy flows induced by effective unitaries (correlations) between systems. I will show that b-work and b-heat are experimentally accessible through -dyne or photon-counting experiments. Focusing on Optical Bloch Equations, I will compare the usual thermodynamic analyses conducted in the open system paradigm to the BQE framework. The two analyses differ by a self-work which yields a tighter expression of the second law, a tightening which I will quantitatively relate to the increased knowledge of the field state. I will finally present experimental results, where energy exchanges between semiconducting quantum dots and light fields have been fully characterized and the self-work was measured. ”

In the presence of environmental decoherence, achieving unit fidelity in quantum state preparation is often unattainable. Monitoring the environment and performing feedback based on the results can enhance the maximum achievable fidelity, yet unit fidelity remains elusive in many scenarios. We derive a theoretical bound on the average fidelity in the ideal case of perfect environmental monitoring. The work focuses on the challenge of preparing Dicke states under collective noise, employing machine learning techniques to identify optimal control protocols. These protocols are then compared against the derived theoretical bound, offering insights into the limits of fidelity in continuously monitored quantum systems.

The investigation of ergodicity or lack thereof in isolated quantum many-body systems has conventionally focused on the description of the reduced density matrices of local subsystems in the contexts of thermalization, integrability, and localization. Recent experimental capabilities to measure the full distribution of quantum states in Hilbert space and the emergence of specific state ensembles have extended this to questions of deep thermalization, by introducing the notion of the projected ensemble – ensembles of pure states of a subsystem obtained by projective measurements on its complement. While previous work examined chaotic unitary circuits, Hamiltonian evolution, and systems with global conserved charges, we study the projected ensemble in systems where there are an extensive number of conserved charges all of which have (quasi)local support. We employ a strongly disordered quantum spin chain which shows many-body localized dynamics over long timescales as well as the ℓ-bit model, a phenomenological archetype of a many-body localized system, with the charges being 1-local in the latter. In particular, we discuss the dependence of the projected ensemble on the measurement basis. Starting with random direct product states, we find that the projected ensemble constructed from time-evolved states converges to a Scrooge ensemble at late times and in the large system limit except when the measurement operator is close to the conserved charges. This is in contrast to systems with global conserved charges where the ensemble varies continuously with the measurement basis. We relate these observations to the emergence of Porter-Thomas distribution in the probability distribution of bitstring measurement probabilities.

Tensor products of normed spaces. From matrix to tensor norms.

After introducing the basic notions about tensors, I will discuss different aspects of quantum entanglement in the framework of tensor norms. I will show how this point of view can bring new insights to this fundamental notion of quantum theory and how new entanglement criteria can be naturally obtained in this way.

Noise is an inherent feature in many natural systems, making it crucial to understand its effects. To achieve this, we introduce a new observable called the stochastic operator variance (SOV), which measures the spread of different stochastic trajectories within the space of operators. The SOV satisfies an uncertainty principle and allows us to identify the initial state that minimizes this spread.
Our analysis reveals that the dynamics of the SOV is closely tied to that of out-of-time-order correlators (OTOCs), which define a fundamental quantity known as the quantum Lyapunov exponent λ. We demonstrate our findings through both analytical and numerical calculations in the context of a stochastic Lipkin-Meshkov-Glick (sLMG) Hamiltonian undergoing energy dephasing.

The amount of work that can be extracted from a quantum system can be increased by exploiting the information obtained from a measurement performed on a correlated ancillary system. The concept of daemonic ergotropy has been introduced to properly describe and quantify this work extraction enhancement in the quantum regime. We explore the application of this idea in the context of continuously monitored open quantum systems, where information is gained by measuring the environment interacting with the energy-storing quantum device. We show that the corresponding daemonic ergotropy takes values between the ergotropy and the energy of the corresponding unconditional state. The upper bound is achieved by assuming an initial pure state and a perfectly efficient projective measurement on the environment, independently of the kind of measurement performed. On the other hand, if the measurement is inefficient or the initial state is mixed, the daemonic ergotropy is generally dependent on the measurement strategy. We will first theoretically investigate this scenario via a paradigmatic example of an open quantum battery: a two-level atom driven by a classical field and whose spontaneously emitted photons are continuously monitored via either homodyne, heterodyne, or photodetection. We will then present a work-of-principle experimental demonstration of daemonic work extraction by simulating a continuously monitored collision model on an IBM quantum computer.

Entanglement of pure and mixed quantum states.

After introducing the basic notions about tensors, I will discuss different aspects of quantum entanglement in the framework of tensor norms. I will show how this point of view can bring new insights to this fundamental notion of quantum theory and how new entanglement criteria can be naturally obtained in this way.

Theoretical tools used in processing continuous measurement records from real experiments to obtain quantum trajectories can easily lead to numerical errors due to a non-infinitesimal time resolution. In this work, we propose a systematic assessment of the accuracy of a map. We perform error analyses for diffusive quantum trajectories, based on single-time-step Kraus operators proposed in the literature, and find the orders in time increment to which such operators satisfy the conditions for valid average quantum evolution (completely positive, convex-linear, and trace-preserving), and the orders to which they match the Lindblad solutions. Given these error analyses, we propose a Kraus operator that satisfies the valid average quantum evolution conditions and agrees with the Lindblad master equation, to second order in the time increment, thus surpassing all other existing approaches. In order to test how well our proposed operator reproduces exact quantum trajectories, we analyze two examples of qubit measurement, where exact maps can be derived: a qubit subjected to a dispersive (z-basis) measurement and a fluorescence (dissipative) measurement. We show analytically that our proposed operator gives the smallest average trace distance to the exact quantum trajectories, compared to existing approaches.

Quantum experiments are performed in noisy platforms. In NISQ devices, realistic setups can be described by open systems or noisy Hamiltonians. Using this setup, we explore a number of dynamical schemes and control techniques. First, starting from a generic noisy Hamiltonian, I will show how noise can help simulate long-range and many-body interaction in a quantum platform [1]. Second, in the setup of shortcut to adiabaticity extended to open quantum systems, we adapt our noisy Hamiltonian to control the thermalization of a harmonic oscillator [2] and generate a squeezed thermal state [3] in arbitrary time. 

Third, adding non-Hermiticity in the picture [3], I will show how noise allows for a rich control of the dynamics, and induced a new phase in which the lossy state becomes stable. More generally, we characterize the quantum dynamics generated by a non-Hermitian Hamiltonian subject to stochastic perturbations in its anti-Hermitian part, describing fluctuating gains and losses. 

Finally, I will briefly show results where we do not look at the noise-averaged density matrix but at an observable introduced as the stochastic operator variance (SOV), which characterizes the deviations of any operator from the noise-averaged operator in a stochastic evolution governed by the Hamiltonian.  Surprisingly, we find that the evolution of the noise-averaged variance relates to an out-of-time-order correlator (OTOC), which connects fluctuations of the system with scrambling, and thus allows computing the Lyapunov exponent.

[1] A. Chenu, M. Beau, J. Cao, and A. del Campo. Phys. Rev. Lett. 118:140403 (2017)
[2] L. Dupays, I. L. Egusquiza, A. del Campo,  and A. Chenu. Superadiabatic thermalization of a quantum oscillator by engineered dephasing, Phys. Rev. Res. 2:033178 (2020)
[3] L. Dupays and A. Chenu. Dynamical engineering of squeezed thermal state, Quantum 5:449 (2021)
[4] P. Martinez-Azcona, A.Kundu, A. Saxena, A. del Campo, and A. Chenu, ArXiv 2407.07746 
[5] P. Martinez-Azcona, A.Kundu, A. del Campo, and A. Chenu, Phys. Rev. Lett. 131:16202 (2023). 

We consider weak unitary symmetries of Markovian open quantum systems at the level of the joint dynamics of the system and its environment described by a continuous matrix product state, as well as for stochastic quantum trajectories of the system, obtained by conditioning on counting measurements of the environment. We derive necessary and sufficient conditions under which the dynamics of these different descriptions exhibit a weak symmetry, in turn characterising the resulting symmetries of their generators. In particular, this depends on whether the counting measurement satisfies the conditions we derive. In doing so we also consider the possible gauge transformations for generators of quantum trajectories, i.e. when two representations of the master operator produce equivalent trajectory ensembles.

In this lecture, we provide an introduction to quantum trajectory theory. We present various mathematical problems that arise within this context. In particular, we introduce approaches for analyzing the asymptotic behavior, convergence speed, and stabilization of quantum trajectories toward different states or subspaces through feedback control strategies. Our study includes both quantum non-demolition (QND) measurements and generic (non-QND) measurements in discrete-time and continuous-time settings.

In this talk, we study how noise and quantum monitoring shape symmetry breaking and chaos. We discuss the simulation of complex open quantum systems using classical noise and how noise limits adiabatic strategies, giving rise to anti-Kibble-Zurek scaling. We further show how spontaneous symmetry breaking is modified in the presence of an observer whose action is described by continuous quantum measurements. In the second part of the talk, we show how the signatures of Hamiltonian quantum chaos in the spectral form factor are suppressed energy dephasing while they are enhanced when the dynamics is conditioned to the absence of con quantum jumps.

A variable-range interacting Ising model with spin-$s$ particles exhibits distinct behavior depending on the fall-off rates in the range of interactions, notably non-local (NL), quasi-local (QL), and local, which are based on the equilibrium properties.  It is unknown if such a transition is respected in the dynamical framework. We use an analytically solvable model in arbitrary spatial dimension ($D$), to establish a dynamical non-local (dNL) behavior, which does not agree with the known result of equilibrium NL behavior. We analyze the profiles of topological entanglement entropy (TEE), mutual information, Lieb-Robinson bound (LRB) and genuine multipartite entanglement (GME) of the weighted graph state (WGS), prepared when the multi-level maximally coherent state at each site evolves according to the long-range spin-$s$ Ising Hamiltonian. Specifically, we demonstrate that the connection between the LRB profile and the divergence in the first derivative of GME with respect to the fall-off rate in the WGS can indicate the transition point from dNL to a dynamical local/quasi-local (dQL) regimes.

In this lecture, we provide an introduction to quantum trajectory theory. We present various mathematical problems that arise within this context. In particular, we introduce approaches for analyzing the asymptotic behavior, convergence speed, and stabilization of quantum trajectories toward different states or subspaces through feedback control strategies. Our study includes both quantum non-demolition (QND) measurements and generic (non-QND) measurements in discrete-time and continuous-time settings.

Rydberg atoms are giant atoms with the outer electron in a highly excited state with large values of the principal quantum number n. Rydberg atoms are highly sensitive to external fields, imparting these atoms extraordinary characteristics for Quantum sensing of electromagnetic fields.

In this talk, I will describe our recent results on Doppler-enhanced Quantum magnetometry with Rydberg atoms. We demonstrate in this work that one can harness Doppler shifts in an unconventional configuration of laser beams for Rydberg excitation to produce an order-of-magnitude enhanced response to a magnetic field as compared to the commonly used conventional configuration. We explain and generalize our findings with theoretical modelling and simulations based on a Lindblad master equation.
I will also discuss our recent observations on the effect of interatomic interaction in Autler-Townes splitting in ultra-cold Rydberg atoms. Our measurements on highly excited Rydberg atoms are directed towards Quantum sensing, Quantum computing and Quantum simulation of many-body physics with individual Rydberg atoms in an array of optical tweezers.

We theoretically study the transport signatures of unpaired Floquet Majorana bound states in the Josephson current of weakly linked, periodically driven topological superconductors. We obtain the occupation of the Floquet Majorana modes in the presence of weak coupling to thermal leads analytically, and show that, similar to static superconductors, the Josephson current involving Floquet Majorana bound states is also 4π-periodic in the phase difference across the junction, and also depends linearly on the coupling between superconductors. Moreover, unlike the static case, the amplitude of the Josephson current can be tuned by setting the unbiased chemical potential of the driven superconductors at multiple harmonics of the drive frequency. As a result, we uncover a Josephson Floquet sum rule for driven superconductors. We confirm our analytical expressions for Josephson current, the occupation of Floquet bands, and a perturbative analysis of the quasienergies with numerically exact results.

Steady-state quantum thermal machines are typically characterized by a continuous flow of heat between different reservoirs. However, at the level of discrete stochastic realizations, heat flow is unraveled as a series of abrupt quantum jumps, each representing an exchange of finite quanta with the environment. In this work, we present a framework that resolves the dynamics of quantum thermal machines into cycles classified as engine-like, cooling-like, or idle. We analyze the statistics of individual cycle types and their durations, enabling us to determine both the fraction of cycles useful for thermodynamic tasks and the average waiting time between cycles of a given type. Central to our analysis is the notion of intermittency, which captures the operational consistency of the machine by assessing the frequency and distribution of idle cycles. Our framework offers a novel approach to characterizing thermal machines, with significant relevance to experiments involving mesoscopic transport through quantum dots.

Achieving high-fidelity and fast quantum state manipulation under realistic dissipation conditions remains a pivotal challenge in quantum computing and quantum information processing. Real-world quantum systems face dual sources of dissipation: environmental noise and drive-induced effects, which are often overlooked in existing control protocols. These limitations hinder the practical implementation of high-speed, accurate quantum operations.

In this work, we propose a method for designing pulse profiles that drive a quantum system from an initial state to a target state with both high fidelity and minimal time. Leveraging the GRAPE algorithm, our approach explicitly accounts for both environmental and drive-induced dissipation, ensuring robust performance across diverse quantum platforms.

Our findings highlight two critical insights: (1) the existence of an optimal evolution time that maximizes fidelity and (2) the counterintuitive enhancement of fidelity at lower drive strengths. These results pave the way for robust quantum control in open systems, addressing key obstacles to scaling quantum technologies. By improving the efficiency and accuracy of quantum operations, our method contributes to the realization of practical quantum computers and advanced quantum sensing technologies, even in the presence of realistic dissipation.

 I will present a modification of the Sachdev-Ye-Kitaev model which demonstrates ergodicity breaking phenomenon, while retaining its' solvable structure (which is one of the signatures of the model). I will present results from various probes that detect the ergodicity breaking, and demonstrate a roadmap that will lead to a solution.

The question of the long-time behavior of quantum trajectories has been recently solved in the finite-dimensional case. In infinite dimension, the problem is still open. In this talk, we consider the particular model of the "one-atom maser," an infinite-dimensional system with many applications in quantum mechanics. We completely describe its long-time behavior by comparing it with a classical birth-and-death process. This is a joint work with T. Benoist and L. Bruneau.