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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)

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

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.

Measurement-induced Phase Transitions (MiPTs) emerge from the interplay between competing local quantum measurements and unitary scrambling dynamics. While monitored quantum trajectories are inherently stochastic, post-selecting specific detector readouts leads to dynamics governed by non-Hermitian Hamiltonians, revealing distinct universal characteristics of MiPTs.

Here, we contrast the quantum dynamics of individual post-selected trajectories with their collective statistics behavior. We introduce a novel partially post-selected stochastic Schrödinger equation that enables the study of controlled subsets of quantum trajectories. Applying this formalism to a Gaussian Majorana fermions model, we employ a two-replica approach combined with renormalization group (RG) techniques to demonstrate that non-Hermitian MiPT universality persists even under limited stochasticity. Notably, we discover that the transition to MiPT occurs at a finite partial post-selection threshold. Our findings establish a framework for leveraging non-Hermitian dynamics to investigate monitored quantum systems while addressing fundamental challenges in post-selection procedures.

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.

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

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

With the advent of digital and analog quantum simulation experiments, it is now possible to experimentally simulate the dynamics of quantum many-body lattice systems and make site-resolved measurements. These experiments make it pertinent to consider the probability of getting any specific measurement outcome, which we call the signal, on placing multiple detectors at various sites while simulating the dynamics of a quantum many-body lattice system. In this work we formulate and investigate this problem, introducing the concept of quantum many-body detection probability (QMBDP), which refers to the probability of detecting a chosen signal at least once in a given time. We show that, on tuning some Hamiltonian parameters, there can be sharp transition from a regime where the QMBDP is approximately equal to one to a regime where the QMBDP is approximately equal to zero. Most notably, the effects of such a transition can be observed at a single trajectory level. This is not a measurement-induced transition, but rather a nonequilibrium transition reflecting opening of a specific type of gap in the many-body spectrum. We demonstrate this in a single-impurity nonintegrable model, where changing the many-body interaction strength brings about such a transition. Our findings suggest that instead of measuring expectation values, single-shot stroboscopic measurements could be used to observe nonequilibrium transitions.

 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.

In the original Matrix movie, the bulk of the human population lives not in the real world but inside a computer simulation called the Matrix. They are unable to detect this situation, except for the fact that certain agents can transcend the normal rules of physics. In this talk, I will explain how this is eerily similar to the world we live in. Certain people (quantum physicists) can transcend the normal rules by using entangled particles to do things that "should be" impossible. This makes the world a very puzzling place, even for quantum physicists. These “super-powers” are also central to the emerging field of quantum information technology. Finally, I will explain very recent work by myself and co-workers [1] that ties all of this together in order to show that the world is even more puzzling than we had thought. Much like the latest Matrix movie.

[1] K.-W. Bong, A. Utreras-Alarcón, et al., Nature Physics 16, 1199 (2020); E. G. Cavalcanti and H.M. Wiseman, Entropy 23, 925 (2021); H. M. Wiseman, E. G. Cavalcanti, and E.G Rieffel, Quantum 7, 1112 (2023).

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.