We derive three matched speed limits on the purification of a monitored system, valid in every finite dimension and for every strategy and efficiency. All three are set by one functional, the square root of the state's impurity, whose maximal decay rate is a state-independent constant. For a qubit this reduces to the square root of the determinant, whose protocol-independent decay at unit efficiency is a sharp null test. The three rate constants stand in ratio two to four to eight, and attainment freezes at that square root, the half-moment. Numerics confirm the bounds; for the qubit, a conservation law settles the local-versus-global optimality puzzle, and doubling feedback spends exactly twice the minimum.
We derive three matched speed limits on the purification of a monitored system, valid in every finite dimension and for every strategy and efficiency. All three are set by one functional, the square root of the state's impurity, whose maximal decay rate is a state-independent constant. For a qubit this reduces to the square root of the determinant, whose protocol-independent decay at unit efficiency is a sharp null test. The three rate constants stand in ratio two to four to eight, and attainment freezes at that square root, the half-moment. Numerics confirm the bounds; for the qubit, a conservation law settles the local-versus-global optimality puzzle, and doubling feedback spends exactly twice the minimum.
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.
We develop a calibration-based framework for effective irreversibility in continuously monitored open quantum systems. The central quantity is Irec(t)=DKL(P1(t)‖P0(t)), the Kullback–Leibler divergence between calibrated measurement-record laws under two controlled branches. Within the equal-covariance Gaussian class, Irec(t) is deterministic for fixed record laws, is computable from Kalman-filter innovations, and obeys an exact master curve for minimum branch-discrimination error. The trajectory-level log-likelihood ratio Λ(t) is stochastic: a candidate onset location t× begins a sustained evidence interval, whereas a causal declaration is available only at tdec, after the interval has been verified. Thus, Idec = Irec(tdec) is distributed across trajectories, although Irec(t) is deterministic at a fixed time. A calibrated reference-register construction assigns a one-nat excess-free-energy reference to the stored record. The primary new protocol result is that, in information time u = Irec(t), a hold interval specified in accumulated-information units yields a rate-invariant declaration distribution within the Gaussian benchmark, whereas a fixed physical-time hold interval generally does not. As a model-specific physical corollary, for pure coherent output pointers and a complete matched homodyne record, residual branch coherence obeys Cdet(t) = exp[−Irec(t)/4]; selecting Cdet = 1/2 gives the illustrative marker Irec = 4 ln 2 nats. For incomplete detected records, the mapping acquires an effective-efficiency correction. In practical terms, the framework tells an experimenter when a continuously recorded signal has acquired stable evidence and how that declaration changes as measurement quality varies. We also derive the deterministic reference-time correction for time-dependent information rates and state experimentally testable failure criteria.
Microscopic nonequilibrium can be conserved while restricted measurements become insensitive to it. We formulate observable quantum equilibrium for complete experimental records, including retained apparatus readings, clocks, and adaptive controls. The likelihood of a record is a conditional expectation of the microscopic relative density, so evidence already retained cannot decrease as a transcript grows. Under a quantified mixing hypothesis, we bound observable distinguishability by the product of dynamical decay and the sensitivity of the permitted record family. A Gaussian resolution estimate removes an unnecessary dimension-dependent power, and a Fisher-information bound expresses record sensitivity through calibrated device quantities. An explicit system–pointer Hamiltonian identifies the full Gaussian record with a single readout whose precision depends on pulse areas and the apparatus covariance. A driven Bohmian oscillator then gives an exact competition between unitary squeezing, retained pointer count, elapsed time, and reference drive energy. A separate two-dimensional square-well calculation follows 131,072 trajectories and shows finite-time relaxation across an eight-direction convex preparation family, with reproducible sampling and numerical checks. Sequential sufficiency distinguishes stable conditional predic- tions from accuracy of an indefinitely retained transcript; contraction yields a square-root horizon bound under a finite-alphabet noise floor. Together these results specify when Born-looking marginals support observable equilibrium and when additional controls or records expose the conserved deviation.
Abstract We study the one-shot reconstruction of a quantum state when a single copy of the source is measured, the outcome is stored in a classical record of fixed finite dimension, and a state is prepared at the destination from the record. This acquisition-limited problem is distinguished from a visible benchmark in which the source label is available to the encoder. For a finite ensemble we derive finite-record bounds on the visible distortion in terms of the cumulative prior mass covered by the record and the optimal discrimination probability, identify the regimes in which they are exact, and solve the two-state visible problem at every record dimension. For pure-state ensembles, exact acquisition-limited reconstruction is possible if and only if the states are perfectly distinguishable, so nonorthogonal targets retain a positive error at every record dimension. For two equiprobable nonorthogonal pure states we derive the optimal trace-distance distortion within the reflection-symmetric two-outcome class in closed variational form and find that unrestricted numerical optimization with up to six outcomes attains the same value; the resulting symmetric value lies strictly below the guess-and-prepare, unambiguous-discrimination, fidelity-optimal, and Haar-random strategies and is realized by one classical bit and one feed-forward rotation. Every classical-record reconstruction is shown to be entanglement breaking, with a quantitative bound on the loss of entanglement with an inaccessible reference, and the record dimension is related to the spectral rank of an energy-constrained memory. These results quantify the separation between a finite classical description and the information obtainable from a single quantum system. MSC Classification: 81P15 , 81P17 , 81P45 , 81P47
The uncontrolled endpoint channel can conceal how well a quantum process can protect a logical task under time-resolved control. We formulate a fixed-task resource theory of continuous process tensors with physical access interfaces, entrance-erasing free objects, and budgeted free conversions. Complete-record total-variation error measures performance; the calibrated capability is monotone under conversions that preserve a common free baseline. For unknown-state transmission the baseline is exactly 1−d−2. Two qubit processes with the same endpoint channel have optimal errors 0 and p=sin2(2gh) at the same finite pulse budget in an ideal-transparent system-only window architecture, forcing any endpoint-only predictor to err by at least p/2 on one of them. An exactly solvable family shows how the allocation of a fixed interaction phase around a control window sets the attainable error, and a nontrivial perfect echo requires and attains total control action π in that architecture. The separation extends directly to every qubit unitary target at a budget that includes its boundary gate, while unitary tasks in any dimension have baseline 1−d−2 whenever a budget-feasible free realization gives the depolarizing output. Complete-record implementation bounds quantify coherent and dissipative control imperfections. Continuous-process randomized compilation supplies physical instrument pullbacks; compatible completion conditions extend their action, and certified dynamical bounds yield achievable task-error and capability guarantees. The resulting theory identifies temporal structure and control investment that endpoint characterization alone cannot resolve.
Metastability in open quantum systems is usually inferred from spectral separation in the Liouvillian, which governs unconditional, ensemble-averaged dynamics. We show that this diagnosis is incomplete at the level of individual quantum trajectories: conditioned realizations of the same unconditional dynamics can bypass, transiently access, or operationally preserve a metastable memory, depending on the monitored channel and the observed record. We demonstrate these mechanisms in a driven-dissipative nonlinear oscillator realizing a quantum associative memory, by comparing spectra, target fidelities, and phase-space distributions. The non-Hermitian dynamics obtained by post-selecting on the absence of detected events supports metastable retrieval, but with a distinct long-time fate: normalization selects the least-decaying mode of the non-Hermitian spectrum rather than the addressed memory branch. In contrast, stochastic jump trajectories can preserve retrieval over extended times when the post-measurement update remains compatible with the coherent memory structure. Thus trajectory-level metastability is not determined by the averaged generator alone; it requires compatibility between the monitored channel, the measurement record, and the metastable manifold.