Any nonzero detection loss qualitatively changes feedback-controlled purification under diffusive monitoring. In every finite dimension, we prove a sharp, dimension-independent ceiling on the decay of trajectory-averaged impurity moments, uniformly over admissible predictable feedback protocols.Below unit efficiency, this ceiling becomes independent of moment order above a critical value and is attained on extremal rank-two quantum-nondemolition (QND) faces. For generic observable spectra, a determinant-root law precludes every full-rank state from attaining this boundary rate over an explicit moment-order interval. For qubits at $0<η<1$, the frozen rate is the exact optimum, set by rare, persistently mixed trajectories. Parameter-free finite-action scaling functions resolve both the rounded QND moment-order transition and the near-unit QND--always-unbiased crossover.
Any nonzero detection loss qualitatively changes feedback-controlled purification under diffusive monitoring. In every finite dimension, we prove a sharp, dimension-independent ceiling on the decay of trajectory-averaged impurity moments, uniformly over admissible predictable feedback protocols.Below unit efficiency, this ceiling becomes independent of moment order above a critical value and is attained on extremal rank-two quantum-nondemolition (QND) faces. For generic observable spectra, a determinant-root law precludes every full-rank state from attaining this boundary rate over an explicit moment-order interval. For qubits at 0<η<1, the frozen rate is the exact optimum, set by rare, persistently mixed trajectories. Parameter-free finite-action scaling functions resolve both the rounded QND moment-order transition and the near-unit QND--always-unbiased crossover.
Continuous quantum feedback uses time-resolved measurement records to steer monitored systems toward pure states. Yet how the information available to a controller determines the ultimate purification speed remains unresolved. We establish this relation for a qubit under fixed-spectrum Hermitian monitoring with detector loss, obtaining the exact long-time impurity-moment spectrum optimized over causal basis controls at each horizon. Rare records with nearly canceled evidence then make all moments from half order upward decay at the Bhattacharyya information rate between two quantum nondemolition record laws. Aligned quantum nondemolition monitoring preserves that binary distinguishability and attains the limit, while complete detection restores an order-dependent branch. The mechanism extends to higher dimensions, where an attainable rank-two ceiling lies above the full-rank qutrit upper bound over a finite moment interval, establishing retained record distinguishability as a purification resource.
The usual way to quantify measurement incompatibility is noise robustness: how much noise must be mixed in before some joint measurement becomes possible. That is an extremal, worst-case quantity, computed by optimization, and it says little about what a real experiment would actually record if it just kept running. We ask a plainer question. A qubit is measured, over and over, in two projective bases separated by a Bloch angle θ. Once the resulting chain settles into steady state, how far does the state typically move between one measurement and the next? For sharp measurements this rate, r(θ), turns out to have the closed form 21sinθ1+sinθ. It agrees with the standard robustness threshold ν(θ)=1−1/1+sinθ to first order near θ=0, then departs from it everywhere else. For unsharp (POVM) measurements in the weak-measurement limit λ→0, full-rank Kraus operators preserve purity exactly, and this collapses the problem onto a one-dimensional diffusion on the Bloch circle. The resulting rate is c(θ)=sin(θ/2)/[2E(m)] with m=1−tan2(θ/2) and E the complete elliptic integral of the second kind, giving the clean value c(π/2)=1/π at the orthogonal point. A blind spot shows up along the way: at θ=π/2, r(θ) stops responding to noise entirely, for any Pauli-type or amplitude-damping channel, simply because sin45∘=cos45∘. That is unfortunate news for anyone hoping to use r(θ) as a calibration check at the angle experimentalists tend to prefer. Purification under repeated or simultaneous incompatible measurement has by now a sizable literature, but in every case we examined the angle θ itself is held fixed and something else, measurement strength or the number of axes, is varied instead. Treating θ as the free parameter and solving for it in closed form is the gap this paper tries to close. Past the qubit, the same construction becomes a random walk on CPd−1 rather than on a circle, and its closed form we leave open.
Nonlinear dynamics generate non-Gaussian states and operations, but they also make continuously monitored quantum systems difficult to track. A quantum filter performs this task by compressing a noisy measurement record into a set of evolving variables that predict future observables. For Gaussian dynamics, this compression is exact and finite. In nonlinear systems, conventional moment equations generally expand without bound, but the failure of one representation does not prove that every finite description is impossible. Here we formulate quantum filtering as a realization problem for general quantum input--output maps. We introduce the linearized observable Hankel operator KO, which maps changes in the past measurement record to changes in a future conditional observable. Any finite-dimensional smooth filter forces KO to have finite rank, so infinite rank provides a coordinate-independent proof that no exact finite-dimensional filter exists. Applied to single-mode polynomial bosonic systems under continuous quadrature monitoring, this yields a sharp boundary: Gaussian dynamics and the measurement-aligned Conditional Momentum Moment class admit finite realizations, whereas dynamics outside these classes produce infinitely many response directions and admit no finite-dimensional smooth exact filter. Exact impossibility, however, need not imply large practical complexity. The singular values of KO quantify how many response directions matter at a chosen local accuracy, defining the effective memory of an observable. For the Kerr and Duffing oscillators, these singular values decay rapidly despite infinite exact rank, showing that infinite exact memory can coexist with low finite-accuracy complexity. The framework therefore separates exact realizability from effective complexity in nonlinear quantum filtering.
The usual way to quantify measurement incompatibility is noise robustness: how much noise must be mixed in before some joint measurement becomes possible. That is an extremal, worst-case quantity, computed by optimization, and it says little about what a real experiment would actually record if it just kept running. We ask a plainer question. A qubit is measured, over and over, in two projective bases separated by a Bloch angle θ. Once the resulting chain settles into steady state, how far does the state typically move between one measurement and the next? For sharp measurements this rate, r(θ), turns out to have the closed form 21sinθ1+sinθ. It agrees with the standard robustness threshold ν(θ)=1−1/1+sinθ to first order near θ=0, then departs from it everywhere else. For unsharp (POVM) measurements in the weak-measurement limit λ→0, full-rank Kraus operators preserve purity exactly, and this collapses the problem onto a one-dimensional diffusion on the Bloch circle. The resulting rate is c(θ)=sin(θ/2)/[2E(m)] with m=1−tan2(θ/2) and E the complete elliptic integral of the second kind, giving the clean value c(π/2)=1/π at the orthogonal point. A blind spot shows up along the way: at θ=π/2, r(θ) stops responding to noise entirely, for any Pauli-type or amplitude-damping channel, simply because sin45∘=cos45∘. That is unfortunate news for anyone hoping to use r(θ) as a calibration check at the angle experimentalists tend to prefer. Purification under repeated or simultaneous incompatible measurement has by now a sizable literature, but in every case we examined the angle θ itself is held fixed and something else, measurement strength or the number of axes, is varied instead. Treating θ as the free parameter and solving for it in closed form is the gap this paper tries to close. Past the qubit, the same construction becomes a random walk on CPd−1 rather than on a circle, and its closed form we leave open.
Classical reverse diffusion is generated by changing the drift at fixed noise. We show that the quantum version of this principle obeys an exact law with a sharp phase boundary. For Gaussian pure-loss dynamics, the canonical model of continuous-variable decoherence, we prove that the unrestricted instantaneous reverse optimum exhibits a noiseless-to-noisy transition: below a critical squeezing-to-thermal ratio, reversal can be noiseless; above it, complete positivity forces irreducible reverse noise whose minimum cost we determine in closed form. The optimal reverse diffusion is uniquely covariance-aligned and simultaneously minimizes the geometric, metrological, and thermodynamic price of reversal. For multimode trajectories, the exact cost is additive in a canonical set of mode-resolved data, and a globally continuous protocol attains this optimum on every mixed-state interval. If a pure nonclassical endpoint is included, the same pointwise law holds for every t>0, but the optimum diverges as 2/t: exact Gaussian reversal of a pure quantum state is dynamically unattainable. These results establish the exact Gaussian benchmark against which any broader theory of quantum reverse diffusion must be measured.
We investigate the behavior of quantum trajectories conditioned on measurement outcomes. Under a condition related to the absence of so-called dark subspaces, Kümmerer and Maassen had shown that such trajectories almost surely purify in the long run. In this article, we first present a simple alternative proof of this result using Lyapunov methods. We then strengthen the conclusion by proving that purification actually occurs at an exponential rate in expectation, again using a Lyapunov approach. Furthermore, we address the quantum state estimation problem by propagating two trajectories under the same measurement record--one from the true initial state and the other from an arbitrary initial guess--and show that the estimated trajectory converges exponentially fast to the true one, thus quantifying the rate at which information is progressively revealed through the measurement process.