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Paired Quantum Trajectories: Common Random Numbers for Protocol Selection in Continuously Monitored Quantum Systems

Jean-Jacques Dubray

Published Aug 31, 2026• 0 citations
Publisher

Abstract

A quantum jump trajectory is a discrete-event process in the literal sense: coherent evolution between jumps, a waiting time drawn from the decay of the no-jump norm, a channel selection at the jump. We take this correspondence in both directions. First, we build a monitored quantum laboratory (a driven emitter, detectors with efficiency and dead time, a click-driven feedback controller) inside a verification-first discrete-event engine that machine-checks the event structure and the bookkeeping on every transition, validated against exact finite-horizon renewal expectations and against QuTiP. Second, we import a comparison discipline that is standard in discrete-event simulation and that we did not find in the quantum trajectory literature: common random numbers. Addressing the randomness by event ordinal keeps two protocol arms on identical draws until their event histories diverge, and a mechanical audit verifies the pairing rather than assuming it. Pairing pays in proportion to trajectory alignment: for close protocol variants, where the selection decision is hard, the variance of the estimated protocol gap falls by a factor of 40 or more (replicated across five seeds), and the paired design resolves a 0.7% rate difference at $n = 400$ trajectories that the unpaired analysis of the same data cannot. Against the Fisher information of the click record, that paired design recovers about a tenth of the achievable precision at the count endpoint and the unpaired design under two percent, with the remaining gap attributable to the endpoint rather than the pairing. Two physics results follow. Dead-time losses for antibunched light sit well below the textbook Poisson correction, an effect reaching 21% that we compute exactly for resonance fluorescence and reproduce within the measurement intervals. And the rate-optimal feedback pulse after a click is not a $\pi$ pulse, and not because of latency: the mean wait is exactly $a_0 + a_1\cos\theta + a_2\sin\theta$ for every drive, detuning and latency, which puts the zero-latency optimum at our drive at exactly $3\pi/4$ and the deficit of the $\pi$ pulse at exactly $\sqrt{2} - 1$, both confirmed by measurement at the shortest latency we can represent. Latency lowers the optimum further, and $\pi$ is recovered only as the drive weakens. The optimum does not maximize post-pulse excitation, because the waiting time depends on coherence with the drive, not population alone.

DOI
10.20944/preprints202608.2297.v1

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A quantum jump trajectory is a discrete-event process in the literal sense: coherent evolution between jumps, a waiting time drawn from the decay of the no-jump norm, a channel selection at the jump. We take this correspondence in both directions. First, we build a monitored quantum laboratory (a driven emitter, detectors with efficiency and dead time, a click-driven feedback controller) inside a verification-first discrete-event engine that machine-checks the event structure and the bookkeeping on every transition, validated against exact finite-horizon renewal expectations and against QuTiP. Second, we import a comparison discipline that is standard in discrete-event simulation and that we did not find in the quantum trajectory literature: common random numbers. Addressing the randomness by event ordinal keeps two protocol arms on identical draws until their event histories diverge, and a mechanical audit verifies the pairing rather than assuming it. Pairing pays in proportion to trajectory alignment: for close protocol variants, where the selection decision is hard, the variance of the estimated protocol gap falls by a factor of 40 or more (replicated across five seeds), and the paired design resolves a 0.7% rate difference at n=400n = 400n=400 trajectories that the unpaired analysis of the same data cannot. Against the Fisher information of the click record, that paired design recovers about a tenth of the achievable precision at the count endpoint and the unpaired design under two percent, with the remaining gap attributable to the endpoint rather than the pairing. Two physics results follow. Dead-time losses for antibunched light sit well below the textbook Poisson correction, an effect reaching 21% that we compute exactly for resonance fluorescence and reproduce within the measurement intervals. And the rate-optimal feedback pulse after a click is not a π\piπ pulse, and not because of latency: the mean wait is exactly a0+a1cos⁡θ+a2sin⁡θa_0 + a_1\cos\theta + a_2\sin\thetaa0​+a1​cosθ+a2​sinθ for every drive, detuning and latency, which puts the zero-latency optimum at our drive at exactly 3π/43\pi/43π/4 and the deficit of the π\piπ pulse at exactly 2−1\sqrt{2} - 12​−1, both confirmed by measurement at the shortest latency we can represent. Latency lowers the optimum further, and π\piπ is recovered only as the drive weakens. The optimum does not maximize post-pulse excitation, because the waiting time depends on coherence with the drive, not population alone.
When the main source of qubit relaxation comes from its coupling to a photonic channel, each relaxation event is associated with the release of a photon. The corresponding discrete quantum jumps of the qubit state can be observed by counting the number of photons emitted by fluorescence. This discreteness of the quantum jumps is in fact related to the nature of the light detector. What does the evolution of the qubit state become if fluorescence is measured using a heterodyne detector instead? In this talk, an experiment will be discussed, in which the records of heterodyne measurements of fluorescence is used to reconstruct the quantum trajectories of a qubit during relaxation. Using a large number of experiments, it is shown that these trajectories can be used to better predict the probability to find given measurement outcomes during the evolution. This heterodyne measurement of the fluorescence is a quantum demolition continuous measurement, which is very different from the more common dispersive measurement sensitive to qubit occupation. These trajectories are thus expected to exhibit some exotic properties, particularly when using past and future knowledge. Besides, using measurement based feedback based on the fluorescence signal alone, it is possible to stabilize any chosen qubit state. This work thus demonstrates that relaxation into an efficiently monitored channel is not a limit for quantum information protocols but can instead be a resource.
Quantum physics was invented to account for two fundamental features of measurement results their discreetness and randomness. Emblematic of these features is the Bohr idea of quantum jumps between two discrete energy levels of an atom. Experimentally, quantum jumps were first observed in an atomic ion driven by a weak deterministic force while under strong continuous energy measurement. The times at which the discontinuous jump transitions occur are reputed to be fundamentally unpredictable. Can there be, despite the indeterminism of quantum physics, a possibility to know if a quantum jump is about to occur or not? Here, we answer this question affirmatively by experimentally demonstrating that the jump from the ground to an excited state of a superconducting artificial three-level atom can be tracked as it follows a predictable flight, by monitoring the population of an auxiliary energy level coupled to the ground state. The experimental results demonstrate that the jump evolution when completed is continuous, coherent, and deterministic. Furthermore, exploiting these features and using real-time monitoring and feedback, we catch and reverse a quantum jump mid-flight, thus deterministically preventing its completion. Our results, which agree with theoretical predictions essentially without adjustable parameters, support the modern quantum trajectory theory and provide new ground for the exploration of real-time intervention techniques in the control of quantum systems, such as early detection of error syndromes.
We consider the estimation of parameters encoded in the measurement record of a continuously monitored quantum system in the jump unraveling, corresponding to a single-shot scenario, where information is continuously gathered. Here, it is generally difficult to assess the precision of the estimation procedure via the Fisher Information due to intricate temporal correlations and memory effects. In this paper we provide a full set of solutions to this problem. First, for multi-channel renewal processes we relate the Fisher Information to an underlying Markov chain and derive a easily computable expression for it. For non-renewal processes, we introduce a new algorithm that combines two methods: the monitoring operator method for metrology and the Gillespie algorithm which allows for efficient sampling of a stochastic form of the Fisher Information along individual quantum trajectories. We show that this stochastic Fisher Information satisfies useful properties related to estimation on a single run. Finally, we consider the case where some information is lost in data compression/post-selection and provide tools for computing the Fisher Information in this case. All scenarios are illustrated with instructive examples from quantum optics and condensed matter.
A continuously measured quantum system with multiple jump channels gives rise to a stochastic process described by random jump times and random emitted symbols, representing each jump channel. While much is known about the waiting time distributions, very little is known about the statistics of the emitted symbols. In this letter we fill in this gap. First, we provide a full characterization of the resulting stochastic process, including efficient ways of simulating it, as well as determining the underlying memory structure. Second, we show how to unveil patterns in the stochastic evolution: Some systems support closed patterns, wherein the evolution runs over a finite set of states, or at least recurring states. But even if neither is possible, we show that one may still cluster the states approximately, based on their ability to predict future outcomes. We illustrate these ideas by studying transport through a boundary-driven one-dimensional XY spin chain.
How can deterministic microscopic dynamics reproduce quantum jump histories, including their timing, while creating records faithful to actual earlier events? This paper constructs a conditional realization of the minimal Bell jump process using a deterministic hybrid model on a fixed finite configuration graph. The model combines a canonical complex field, conservative packet production, carrier–packet reactions, finite-rate recombination, and a finite spatial gas. Under explicit initial calibration and gas-independence assumptions, the ordinary-configuration path law converges in total variation to the Bell jump process without rescaling physical time. Quantitative estimates control the gas contact history, recombination error, and population tracking, including wavefunction nodes, current reversals, and zero-current intervals. An autonomous quantum clock carries a finite measurement circuit that retains fuel, loss channels, copied archives, reset receivers, feedback, and an inaccessible reference. At specified copying gates, the Bell path crosses the gate exactly once during the first pass. The archive therefore records its own actual earlier configuration and preserves that record under subsequent label-preserving operations. An explicit resource hierarchy provides an input-uniform polynomial convergence bound for a fixed engineered circuit. The theorem depends on the stated interaction catalogue and statistical preparation premises. It does not derive those premises, establish laboratory realization, or embed the complete hybrid dynamics in a single smooth material Hamiltonian. A supplementary smooth construction covers isolated contact modules.
Introduction A very general concept of a quantum trajectory would be the path taken by the state of a quantum system over time. This state could be conditioned upon measurement results, as we considered in Chapter 1. This is the sort of quantum trajectory we are most interested in, and it is generally stochastic in nature. In ordinary use, the word trajectory usually implies a path that is also continuous in time. This idea is not always applicable to quantum systems, but we can maintain its essence by defining a quantum trajectory as the path taken by the conditional state of a quantum system for which the unconditioned system state evolves continuously. As explained in Chapter 1, the unconditioned state is that obtained by averaging over the random measurement results which condition the system. With this motivation, we begin in Section 4.2 by deriving the simplest sort of quantum trajectory, which involves jumps (that is, discontinuous conditioned evolution). In the process we will reproduce Lindblad's general form for continuous Markovian quantum evolution as presented in Section 3.6. In Section 4.3 we relate these quantum jumps to photon-counting measurements on the bath for the model introduced in Section 3.11, and also derive correlation functions for these measurement records. In Section 4.4 we consider the addition of a coherent field (the ‘local oscillator’) to the output before detection. In the limit of a strong local oscillator this is called homodyne detection, and is described by a continuous (diffusive) quantum trajectory.