Quantum Zenon effects are discussed in terms of a specific class of quantum trajectories, which are conditioned by continuous, mutually exclusive measurement signals. Such a conditioning is not restricted to simple systems but can be generalized to composite networks. In any case, the characteristic features of these trajectories tend to be washed out in the ensemble limit and thus require single system analysis. Only on a sufficiently small time-scale and for a coherent initial state, also the ensemble exhibits some Zenon effect. In this case, ironically, actual measurements are not required: a closed single composite system can emulate this behavior. Such a kind of quantum parallelism underlies also recent proposals for quantum computation.
Quantum Zenon effects are discussed in terms of a specific class of quantum trajectories, which are conditioned by continuous, mutually exclusive measurement signals. Such a conditioning is not restricted to simple systems but can be generalized to composite networks. In any case, the characteristic features of these trajectories tend to be washed out in the ensemble limit and thus require single system analysis. Only on a sufficiently small time-scale and for a coherent initial state, also the ensemble exhibits some Zenon effect. In this case, ironically, actual measurements are not required: a closed single composite system can emulate this behavior. Such a kind of quantum parallelism underlies also recent proposals for quantum computation.
We address the peculiarities of the quantum measurement process in the course of a continuous weak linear measurement (CWLM). As a tool, we implement an efficient numerical simulation scheme that allows us to generate single quantum trajectories of the measured system state as well as the recorded detector signal, and study statistics of these trajectories with and without post-selection. In this scheme, a linear detector is modelled with a qubit that is weakly coupled to the quantum system measured and is subject to projective measurement and re-initialization after a time interval at each simulation step. We explain the conditions under which the scheme provides an accurate description of CWLM. We restrict ourselves to a qubit non-demolition measurement. The qubit is initially in an equal-weight superposition of two quantum states. In the course of time, the detector signal is accumulated and the superposition is destroyed. The times required to resolve the quantum states and to destroy the superposition are of the same order. We prove numerically a rather counter intuitive fact: the average detector output conditioned on the final state does not depend on time. It looks like from the very beginning, the qubit knows in which state it is. We study statistics of decision times where the decision time is defined as time required for the density matrix along a certain trajectory to reach a threshold where it is close to one of the resulting states. This statistics is useful to estimate how fast a decisive CWLM can be. Basing on this, we devise and study a simple feedback scheme that attempts to keep the qubit in the equal-weight superposition. The detector readings are used to decide in which state the qubit is and which correction rotation to apply to bring it back to the superposition. We show how to optimize the feedback parameters and move towards more efficient feedback schemes.
The quantum Zeno effect of a single system is the effect of a series of measurements on the unitary time evolution of the system and on the ability to monitor this time evolution using the measurement results. This effect is shown to be equivalent to that of the indetermination of the quantum state of a single system [Phys. Rev. Lett. 74, 4106 (1995)], in which one considers the statistics of the results of a series of measurements performed on a single system, with no time evolution in between successive measurements.
A theoretical-computational procedure, recently proposed for modelling Vibrational Energy Relaxation (VER) processes of a molecule (Quantum Center, QC) embedded in a complex atomic-molecular system, is extended and applied for analyzing in detail the features of the QC density matrix (DM) temporal evolution. The results, obtained using aqueous azide ion as a case study, show the total lack of coherence in the DM, when the system is prepared to be initially in a pure vibrational eigenstate. This finding is fully in line with the statistical interpretation of the process typically adopted also in the experimental studies where the relaxation processes are all described within the typical schemes of chemical kinetics. Consistently, when the initial vibrational state corresponds to an eigenstate mixture, although initially coherent, the DM relaxes to a fully incoherent condition with a mean lifetime related to the one of the diagonal elements relaxation. These specific DM features turn out to be essentially governed by the thermal equilibrium condition of the atomic-molecular classical coordinates which drive the ensemble of the quantum-trajectories toward the observed statistical regime. Finally, from the analysis of a single long timescale quantum vibrational trajectory it also clearly emerges its ergodic behaviour.
Quantum trajectories are dynamical equations for quantum states conditioned on the results of a time-continuous measurement, such as a continuous-in-time current yt. Recently, there has been renewed interest in finite-interval dynamical maps for quantum trajectories, where the measurement record is available only over intervals of duration Δt. Guilmin \emph{et al.}\ (unpublished) derived such a map for the experimentally relevant case in which only a coarse-grained current (time-binned), denoted by It, is available. Remarkably, this coarse-grained record It still yields a conditioned state, termed the \emph{robinet} state, whose impurity scales as small as (Δt)3. We show, however, that the \emph{typical} trace distance between the robinet state and the fully conditioned state (obtained from the complete measurement record) scales as (Δt)3/2. We then introduce a higher-order finite-interval dynamical map, the ``nearly-exact'' map, which requires only one additional real statistic, φt, from each interval. We analytically show that the resulting conditioned state is much closer to the fully conditioned state, with a distance scaling as (Δt)2. We further numerically verify the distance scalings, measured relative to highly accurate approximations of the fully conditioned state, for the robinet map, our nearly-exact map, the lowest-order approximate Itô map, and two existing higher-order maps. Our results show that, for generic systems, whenever φt can be extracted experimentally alongside It, the nearly exact map provides the most accurate state estimate among the finite-interval maps considered.
We report universal statistical properties displayed by ensembles of pure states that naturally emerge in quantum many-body systems. Specifically, two classes of state ensembles are considered: those formed by (i) the temporal trajectory of a quantum state under unitary evolution or (ii) the quantum states of small subsystems obtained by partial, local projective measurements performed on their complements. These cases, respectively, exemplify the phenomena of “Hilbert-space ergodicity” and “deep thermalization.” In both cases, the resultant ensembles are defined by a simple principle: The distributions of pure states have maximum entropy, subject to constraints such as energy conservation, and effective constraints imposed by thermalization. We present and numerically verify quantifiable signatures of this principle by deriving explicit formulas for all statistical moments of the ensembles, proving the necessary and sufficient conditions for such universality under widely accepted assumptions, and describing their measurable consequences in experiments. We further discuss information-theoretic implications of the universality: Our ensembles have maximal information content while being maximally difficult to interrogate, establishing that generic quantum state ensembles that occur in nature hide (scramble) information as strongly as possible. Our results generalize the notions of Hilbert-space ergodicity to time-independent Hamiltonian dynamics and deep thermalization from infinite to finite effective temperature. Our work presents new perspectives to characterize and understand universal behaviors of quantum dynamics using statistical and information-theoretic tools. Published by the American Physical Society 2024