Truly continuous quantum trajectories, obtained from homodyne or heterodyne readouts, can only ever be reconstructed approximately. The continuous measurement signal, needed for exact reconstruction, is averaged over bins of finite time $Δt$ during any analog to digital conversion step. The best reconstruction possible, knowing only this discrete record, was introduced recently and dubbed the Robinet state. In this article, we show how the Robinet state can be computed with a numerical discretization scheme that is completely positive, accurate to arbitrarily high order in $Δt$, and that does not rely on any other external solver. Our derivation relies on a dilation of the stochastic master equation into a system + transmission line setup, constructed in such a way that measuring what we call the "zero mode" of the line yields the Robinet state. We test the method on a challenging example with random Hamiltonian and jump operator, and verify its accuracy up to order $10$. Apart from its numerical interest, our approach provides a wealth of physical insights, extending in particular recent results on purity obtained by Wonglakhon, Chantasri, and Wiseman, that would be difficult to obtain in any other way.
Truly continuous quantum trajectories, obtained from homodyne or heterodyne readouts, can only ever be reconstructed approximately. The continuous measurement signal, needed for exact reconstruction, is averaged over bins of finite time Δt during any analog to digital conversion step. The best reconstruction possible, knowing only this discrete record, was introduced recently and dubbed the Robinet state. In this article, we show how the Robinet state can be computed with a numerical discretization scheme that is completely positive, accurate to arbitrarily high order in Δt, and that does not rely on any other external solver. Our derivation relies on a dilation of the stochastic master equation into a system + transmission line setup, constructed in such a way that measuring what we call the "zero mode" of the line yields the Robinet state. We test the method on a challenging example with random Hamiltonian and jump operator, and verify its accuracy up to order 10. Apart from its numerical interest, our approach provides a wealth of physical insights, extending in particular recent results on purity obtained by Wonglakhon, Chantasri, and Wiseman, that would be difficult to obtain in any other way.
The theory of continuous quantum measurement allows to reconstruct the state ρt of a system from a continuous stochastic measurement record It. However, this truly continuous-time signal It is never available in practice. In experiments, one generally has access to its digitization, i.e., to a series of time averages Ik over finite intervals of duration Δt. In this letter, we take this digitization seriously and define ρˉn as the best Bayesian estimate of the quantum state given (only) a digitized record (I1,…,In). We show that ρˉn+1 can be computed recursively from In+1 and ρˉn using an exact formula. The latter can be evaluated numerically exactly, or used as the basis for a perturbative expansion into successive powers of Δt. This allows reconstructing quantum trajectories in regimes of coarse Δt where existing methods fail, estimating parameters at fixed Δt without bias, and directly sampling digitized quantum trajectories with schemes of arbitrarily high order.
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.
Theoretical tools used in processing continuous measurement records from real experiments to obtain quantum trajectories can easily lead to numerical errors due to a noninfinitesimal 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 increments $\mathrm{\ensuremath{\Delta}}t$, 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 $\mathrm{\ensuremath{\Delta}}t$, 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.
Since precisely controlling dissipation in realistic environments is challenging, digital simulation of the Lindblad master equation (LME) is of great significance for understanding nonequilibrium dynamics in open quantum systems. However, achieving long-time simulations for complex systems with multiple dissipation channels remains a major challenge, both theoretically and experimentally. Here, we propose a 1-dilation digital scheme for simulating the LME based on quantum trajectory averaging without postselection. By rigorously matching the stochasticity inherent in quantum trajectories with the probabilistic outcomes of quantum measurements, our method effectively translates the classically established quantum jump algorithm into executable quantum circuits. A key advantage of our method is that it overcomes the exponential suppression of success probability seen in some existing postselection-dependent schemes, especially for long-time evolution or systems with numerous jump operators. Moreover, the scheme can be extended to a 2-dilation framework for the nonlinear LME with postselection, bridging the full LME and non-Hermitian Hamiltonian dynamics. This extended scheme provides a digital approach for exploring the interplay between non-Hermitian Hamiltonians and dissipative terms within a monitored quantum dynamics framework.
This tutorial note develops a self-contained engineering-oriented model that connects thecontinuous dynamics of a two-level quantum system to a classical binary readout. The purpose ispedagogical rather than foundational: no new interpretation of quantum mechanics, measurementpostulate, or quantum analog-to-digital conversion protocol is proposed. Instead, establishedelements of open-system dynamics, generalized measurement, stochastic master equations, pointer-state selection, and threshold detection are organized into a single chain that can be simulatedwith conventional state-space tools.A qubit driven by a Hamiltonian H = ℏ(Ωσx + ∆σz )/2 is first described by a Lindbladequation and by the corresponding Bloch equations. Repeated nonselective partial measurementsare represented by a completely positive trace-preserving dephasing channel, from which the exactdiscrete-time coherence decay rate Γϕ = −fs ln (1 − ϵ) is obtained. The distinction between thisensemble-averaged channel and a single experimental realization is then made explicit througha diffusive stochastic master equation. In that conditional description, the detector producesa continuous noisy record; integration over a finite acquisition time and comparison with athreshold yields the final Boolean symbol. The resulting engineering chain is therefore not “wavefunction to bit,” but rather qubit state → detector statistics → analog record → integration →binary decision.Three time scales are emphasized: coherent dynamics, measurement-induced dephasing, andreadout integration. A Nyquist-like sampling inequality is retained only as a temporal-resolutionguideline for ensemble-reconstructed expectation values, not as a sufficient theorem for single-shotquantum-state reconstruction. Finally, the unconditional Bloch equations are mapped ontoan analog state-space macromodel implementable in LTspice. The note includes a correctedreference netlist and a reproducible validation protocol comparing the discrete channel, thecontinuous master equation, averaged stochastic trajectories, and the SPICE realization.
We introduce jump-time unraveling as a distinct description of open quantum systems. As our starting point, we consider quantum jump trajectories, which emerge, physically, from continuous quantum measurements, or, formally, from the unraveling of Markovian quantum master equations. If the stochastically evolving quantum trajectories are ensemble-averaged at specific times, the resulting quantum states are solutions to the associated quantum master equation. We demonstrate that quantum trajectories can also be ensemble-averaged at specific jump counts. The resulting jump-time-averaged quantum states are then solutions to a discrete, deterministic evolution equation, with time replaced by the jump count. This jump-time evolution represents a trace-preserving quantum dynamical map if and only if the associated open system does not exhibit dark states. In the presence of dark states, on the other hand, jump-time-averaged states may decay into the dark states and the jump-time evolution may eventually terminate. Jump-time-averaged quantum states and the associated jump-time evolution are operationally accessible in continuous measurement schemes, when quantum jumps are detected and used to trigger the readout measurements. We illustrate the jump-time evolution with the examples of a two-level system undergoing relaxation or dephasing, a damped harmonic oscillator, and a free particle exposed to collisional decoherence.